Sponsor Area
Under what condition is the relation s=ut is correct?
This condition is only true if body is moving in uniform velocity, as there will be no acceleration and distance covered per unit time will be constant.
State relation between impulse and momentum?
Momentum is mass in motion, and any moving object can have momentum. An object's change in momentum is equal to its impulse. Impulse is a quantity of force times the time interval.
Action and reaction forces do not balance each other why?
Action and reaction exist in pair and are always equal in magnitude and opposite in direction. But they act on different bodies, hence they cannot balance each other.
Two billiard balls each of mass 0.05kg moving in opposite directions with speed 6m/s collide and rebound with the same speed. What is the impulse imparted to each ball due to other
Impulse (I) = Change in linear momentum
Initial momentum of ball ( both) = mv =
0.05×6 = 0.30 Kg m/s
Final momentum of both balls = -(0.05×6)Kg-m/s .
here minus sign is because the direction of final momentum is just opposite to the direction of initial momentum (linear momentum is a vector )
Now impulse imparted to each ball ;
I = -0.30 - ( 0.30) = -0.60 Kg-m/s
How is centripetal force provided in case of the following?
A Carnot engine develops 100 H.P. and operates between 270C and 2270C. Find
1) thermal efficiency
2) heat supplied
3) heat rejected?
Here, energy = W = 100 H.
Sponsor Area
Force is directly proportional to the acceleration of the body.
Hence, acceleration produced on the bod is 1 m/s2.
Sponsor Area
kg, is the reading of the weighing machine. Name the law of motion which forms basis of law of conservation of momentum.
A bomb at rest explodes into two fragments. If 
be the directions of motion of two fragments, then what is the relation between 
?
Two velocity will be equal and opposite. 
time. Sponsor Area
The velocity acquired by the rocket in terms of exhaust velocity of burnt gases, initial mass of rocket at any instant is given by,
v = ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037327.png)
Acceleration of the rocket during the burning of fuel is given by,
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037329.png)
Rate of burning of fuel = m kg/s
Velocity at which the gas leaves the rocket = u m/s
So, force experienced by rocket due to exhaust of gases = mu
(i) Any external imbalance that changes or tends to change the state of rest or uniform motion in a straight line or the direction of motion is called as Force.
Force always produces acceleration in the body.
(ii) Inertia is the ability of the body to change the state of rest or uniform motion, unless acted upon by an external force.
Inertia is categorized into three types:
i) Inertia of rest,
ii) Inertia of motion, and
iii) Inertia of direction.
Inertia of rest: It is the property of body by virtue of which it cannot change the state of rest by itself.
Inertia of motion: It is the tendency of the body to remain in the state of uniform motion in a straight line.
Inertia of direction: It is the property of body by virtue of which it cannot change the direction of motion by itself.
(iii) Momentum is a measure of the quantity of motion contained in the body and is equal to the product of mass and velocity.
Momentum is a vector quantity and it's direction is same as that of velocity.
Mathematically, momentum is given by,
Newton's law of motion are as follows:
First law states that every body in the universe continues to be in the state of rest or uniform motion in a straight line until and unless an external force compels it.
Second law states that the rate of change of momentum of body is directly proportional to the external force applied.
Third law states that too every action there is an equal and opposite reaction.
Sponsor Area
Absolute unit of force:
In SI system, the unit of force is newton.
One Newton of force is the force which produces 1 m/sec2acceleration in 1 kg mass.
In cgs system, the unit of force is dyne.
1 dyne is the force which produces 1cm/sec2 acceleration in 1 gm of mass.
Gravitational unit of force:
In SI system, gravitational unit of force is kgf (kilogram force).
It is that force which accelerates 1 kg mass with acceleration 9.8 m/s2.![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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In cgs system, unit of force is g-f (gram force).
1 g-f is the force that accelerates 1 gm of mass with acceleration 980 cm/s2. 
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Second law is the real law of motion:
This can be proved by showing that first law and third law are contained in second law of motion.
Let's prove that first law is contained in the second alw of motion.
According to the second law of motion,
That is, if no force is acting on the body then its acceleration is zero. This means if a body is at rest, it remains at rest and if the body is moving in straight line with constant velocity, it continues to do so. This is what the first law states. Hence first law is contained in the second law.
(ii) Let's prove that Third law is contained in second law.
According to the second law of motion, 
So, in th absence of external force, total momentum of the system remains constant.
Let, us consider an isolated system of two bodies of masses m1 and m2. Let, 
be the force exerted by mass m1 on m2 which is the reaction.
Let, 
be the force exerted by mass m1 on m2, which is the reaction.
Let, due to 
,
Rate of change of momentum of m1 = 
Rate of change of momentum of m2 = 
Therefore, 
Here, 
is the total momentum of the system.
According to the second law of motion, in the absence of external force, momentum remains conserved.
Therefore, 
Hence, the third law of motion is proved.
Therefore, Newtons' second law of motion is the real law of motion.
According to Newton's third law, action and reaction are equal and opposite and acts on two different bodies.
Here, in this case fan and sail are on two different boats. Therefore, air thrown by fan on the sail will exert external force. So, reaction will be on the second boat on which the fan is placed. Thus action and reaction are on two different boats. Hence, the boats will be set into motion.
As the raindrop is falling with a constant speed, acceleration = 0. Hence, net force ma= 0.
As the cork is floating on water, it's weight is being balanced by the upthrust which is equal to the weight of the water displaced. Hence, net force acting on the cork is zero.
Given, the velocity with which the car is moving is constant. So, acceleration=0. Hence, net force on the car is zero.
Given that, no field is acting on the electron. Therefore, net force on the particle is equal to zero.
Since, 
reaction is due to weight 5 kg
and
reaction is due to two weights 5 kg and 10 kg.
∴ ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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Thus, 
1:3 is the required ratio of the reaction R1 and R2.
An electron is subjected to a force 
How long it will travel in 1.2 ms?
Given,
Force acting on the electron, F = 
Mass of electron,
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The acceleration of electron is given by, 
Now, using the second equation of motion,
We have,
s
Mass of bullet is, m = 30 gm = 0.03 kg
Velocity of bullet, v = 400 m/s
Frequency of firing the bullets is, n = 20 bullets/second
Change in momentum of one bullet is,
mv = 0.03 x 400 = 12 Ns
Change in momentum of 20 bullets is,
12 x 20 = 240 Ns
As 20 bullets are fired in one second, therefore 240 Ns is change in momentum of bullets in 1 second.
Since change in momentum of body in one second is equal to force acting on the body, therefore force exerted by gun on the bullets is 240N.
Now according to Newton’s third law, the bullets exert equal and opposite force on the gun.
Thus the force required to hold the gun is 240N.
Given,
Mass of first body, M1 = 12 kg
Velocity of first body, v1 = 5 m/s
Mass of second body, M2 = 5 kg
Velocity of second body, v2 = 12 m/s
Therefore,
The momentum of Mass 
is,
The momentum of mass 
is,
Since the momentum of both the bodies is same, thus both the bodies have same quantity of motion.
Ans.
(i) For ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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,
The force acting on the body is constant. Thus, the body has uniform accelerated motion.
(ii) For
,
Force on the body is zero, therefore body has uniform motion.
(iii) For 
,
The force is negative and increases linearly with time in magnitude, thus the motion is non-uniformly retarded motion.
(iv) For 
,
The force is positive and increases linearly with time, thus the motion is non-uniformly accelerated motion.
(v) For ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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,
The force acting on the body is constant, thus the body has uniform accelerated motion.
(vi) For 
,
The force acting on the body is constant and negative, thus the body has uniformly retarded motion.
According to Newton's first law of motion,
F = ma
i.e., m = F/a
Force acting on two bodies, F = 15 N
∴
Mass of first body, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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Mass of second body, 
Now the acceleration produced by 30N force when both masses are tied together is,
A man weighing M kg, stands on a weighing machine inside a lift. What will be the reading of the machine if it:
(a) descends with acceleration a,
(b) ascends with an acceleration a and
(c) moves with constant velocity in upward direction?
(a)
The true weight of body = Mg
Reaction of floor = R
Net force acting on the body is,
F = Mg - R
If a is the acceleration with which lift moves downward, then
F = M a
∴ Ma = Mg - R
i.e., R = M(g - a)
Thus the apparent weight of the body is,
(b)
If lift is ascending in upward direction with acceleration a then
Ma = R - Mg
or 
(c)
If lift moves with constant velocity then acceleration, a = 0.
∴ 
So, reading on the machine = Mg
Given,
Initial velocity, u = 2m/s
Final velocity, v = 0 m/s
Distance, s = 1.6m
The acceleration of elevator is,
The tension in the cable is,
T = m(g - a)
= 400[9.8 - (-1.25)]
= 4420 N
The tension in the cable is greater than the strength of string, therefore the cable will break.
Thus it will not be wise to stop the elevator within 1.6m.
Law of conservation of momentum states that total momentum of system remains conserved in the absence of external force.
Proof:
Consider a body of mass m1 moving with veocity 
striking against another body of mass m2 moving with velocity 
.
Let, the two bodies remain in contact with each other for a small interval 
.
Let, 
be the average force exerted by mass m1 on m2 and let 
be the force on m2 due to m1.
Let, v1 and v2 be the velcoities of two bodies after collision. 
Momentum of mass m1 before collision = m1u1
Momentum of mass m2 after collision = m2
Momentum of mass m1 after collision = m1v1
By using the defintion of impulse, change in momentum of mass m1 is, 
be the velocity of the bullet and gun respectively.
be the mass and velocity of light boy, and 
the mass and velocity of heavy boy, respectively. 
When a body is falling, it's momentum increases in the downward direction. But, this will not violate the law of conervation of momentum because there is external gravitational force acting on body, which increases the momentum of body.
But if we consider body and the earth as a system then total momentum of body and the earth remains constant because then gravitational force will be internal force.
Initial momentum of block = M x 0 = 0
Initial momentum of bullet = mu
Therefore,
Initial momentum of system (bullet + block)
before the bullet is embedded into block = mu
Let V be the velocity of system (bullet + block)
after bullet gets embedded into block.
∴ Final momentum of system (bullet + block)
after bullet gets embedded into block = (m + M)V
According to law of conservation of momentum,
(m+M)V = mu
This is the final velocity of the system.
Here,
Mass of shell = m = 0.020 kg
Mass of gun = M = 100 kg
Velocity of shell = u = 80 m/s
Let V be the velocity of gun.
According to the law of conservation of linear momentum,
mu + MV = 0
= -0.016 m/s
Negative sign means the gun recoils.
A bomb at rest explodes into three parts of same mass. The momentum of two parts is 
Find the magnitude of momentum of third point.
Let 
be the momentum of third point.
According to law of conservation of linear momentum,
i.e., 
, is the required magnitude of momentum of the third point.
After explosion,
Let 2 kg mass after explosion fly off along X-axis and 3 kg mass fly off along Y-axis perpendicular to direction of motion of 2 kg mass.
Suppose the velocity of third fragment after explosion is ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037469.png)
∴ Momentum of 2 kg mass is,
Momentum of 3 kg mass is,
Momentum of 5 kg mass is,
According to the law of conservation of linear momentum,
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037469-6.png)
∴ 
This is the required velocity of the third fragment.
Given,
Mass of the boy, m1 = 50 kg
Mass of the cart, m2 = 100 kg
Velocity of boy,
Velocity of cart, 
Let V be the velocity of boy - cart system.
Using law of conservation of linear momentum,
50 x 24 - 100 x 12 = (50 + 100) x V
V = 0
Therefore, the velocity of the boy-cart system becomes zero after the boy lands on the cart.
Given,
Initial velocity, u = 100 m/s
angle of projection of the particle, 
Therefore the velocity of body at highest point is,
At the highest point one of the fragment retraces its path, therefore after explosion the direction of the velocity of fragment reverses without change in the magnitude of velocity.
Therefore the velocity of fragment that retraces its path after explosion becomes 50 m/s.
Let V be the velocity of second fragment.
Now according to the law of conservation of linear momentum,
V = 150 m/s, is the velocity of the second fragment.
V = 100 m/s
The velocity of a rocket depends on:
(i) Initial mass of rocket.
(ii) Velocity of ejecting gases.
(iii) Rate of ejection of burnt gases.
The thrust on a rocket depends on:
(i) Velocity of ejecting gases w.r.t. rocket.
(ii) Rate of ejection of burnt gases.
Consider, a rocket of mass m0 take off with velocity v0 from ground. 
Let the fuel burn at the rate of dm/dt and burnt gases eject with velocity u w.r.t. rocket.
Let at any instant t, v be the velocity of rocket and m be the mass of rocket.
Therefore velocity of burnt gas ejected w.r.t. ground is (v – u).
Here, fuel is burning at the rate of dm/dt.
Therefore in time dt, dm mass of the fuel will burn and velocity of rocket increases by dv.
Here, we have
'm' is the mass of rocket, and
dm is the mass of gas ejected.
As the fuel burns the mass of rocket decreases.
Therefore, to calculate the velocity of rocket at any instant in terms of mass of rocket, m must be replaced by ‘dm’. 
Here,
Initial mass of rocket = 5000 kg
Velocity with the burnt gas is leaving the rocket, u = 1000m/s
Rate of consumption of fuel, 
The thrust on the rocket is,
8000 x 9.8 = u x 16
... (1)
Given,
Rate of ejection of gases, 
Initial speed of ejection of gases from rocket, 
∴ Accelerating force on the rocket is,
A slow change in momentum reduces force of action and reaction, causing less injury.
(i) Seat of cycles is provided with springs. The spring slows down the rate of change of momentum and reduces the hurt to the cyclist.
(ii) The paper cutting used in packing the glasswares decreases the rate of change of momentum and avoids their damages.
Given,
Mass of the ball, m = 180gm = 0.18kg
Initial velocity of ball is, u = 54 km/hr = 15 m/s
Final velocity of the ball, v = 0
∴ Impulse of the ball, I = m(u - v)
= 0.18 x 15
= 2.7 Ns
Time of catching the ball, 
Thus,
Force exerted by player on the ball, 
Mass of the ball, m = 0.125 kg
Velocity of the ball, v = 4 m/s
Initial momentum of ball is, 
After collision, the ball rebounds with the same speed. That is it is travelling in an opposite direction.
Momentum of ball after collision is,
Therefore impulse of ball is 
Area under Force–time graph is equal to the impulse or change in the momentum of the body.
Now, Area under F–t graph,
From, t = 0 to t = 10s = Area of trapezium 1+ Area of trapezium 2 + Area of rectangle 3
= 34 + 28.5 + 21 = 83.5
∴ Change in momentum of body = 83.5 Ns
i.e. m(v - 0) = 83.5 
7.59 x v = 83.5 
v = 11 m/s, is the velocity of the body.
Momentum of the ball, m = 12 Ns
Initial momentum of ball, 
along 
Final momentum of ball is, 
along ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037507-4.png)
∴ Impulse imparted to ball, 
From the given figure, the angle between vector 
and 
is 
.
So, impulse imparted to the ball is, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037507-9.png)
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= 9.18 Ns
The four basic interactions in the universe are:
(i) Gravitational interaction.
(ii) Electromagnetic interaction.
(iii) Strong interaction.
(iv) Weak interaction.
The similarities between gravitaional and electromagnetic forces are:
(i) Both the forces follow inverse square law.
(ii) Both the forces are central forces.
(iii) Both the forces are conservative forces.
Let the cart be connected to horse through a string, as shown in the figure.
Let 
be the tension in the string.
For the cart to move, the horse presses the ground and the horse gets equal and opposite reaction 
Let 
be the force of friction between cart and ground, and
Acceleration of system = 
Resolve 
into two rectangular components ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037516-6.png)
From the free body diagram of horse given above
V = mg
and H - T = ma ...(1)
From the free body diagram of the cart,
T - F = Ma ...(2)
Adding (1) and (2), we have,
H - F = (m+M)a
The system will move if and only if a>0.
i.e. H>F, for the cart to move.
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037517-2.png)
and, 
and
Let 'T' be the tension in the string and 'a' be the acceleration of mass m1 and m2.
Equations of motion of masses m1 and m2 are,
...(1)
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...(2)
Solving (1) and (2), we get
Acceleration of the masses is given by,
and
Tension in the string is,
The acceleartion acting on the system is irrespective of whether force is applied on mass M or m.
If force F is applied on mass m then contact force F1 is,
If force F is applied on Mass M then contact force F2 is,
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Given,
Mass of first body, 
Mass of second body, 
The acceleration of system of masses is, 
i.e. The mass 3 kg accelerates upward with acceleration 2.5 m/s2.
Let the motion of 3 kg mass start from A and accelerate for 4s and reach the point B.
At point B the string breaks and 3kg mass rises to point C.
Motion from A to B:
Initial velocity, 
Rate of acceleration, 
Time of travel, 
∴ Velocity attained during the motion is given by first equation of motion,
Motion from B to C:
Velocity, 
Acceleration, 
∴ Distance travelled by the mass, 
The mass would go 5m high.
...(1)
is, 
...(2)
, is the acceleration of the system.
...(1) 
...(2) 
, is the required angle of inclination of inclined plane. Inertial reference frames:
Inertial reference frames are the frames in which Newton’s laws of motion hold good. These are the frames which are at rest or move with constant velocity with respect to one another.
Non-inertial reference frames:
The frames in which Newton’s laws of motion do not hold good are called non-inertial reference frames. All accelerated frames are non-inertial reference frames.
...(1) 
is, 
...(2) 
is, 
...(3) 
...(4) 
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037594-8.png)
For ordinary terrestrial experiments which of the following observers are inertial and which are non-inertial:
(a) A child revolving in a giant wheel.
(b) A driver in a sports car moving with a constant high speed of 200 km/hr on straight road.
(c) The pilot of an aeroplane, which is taking off.
(d) A cyclist negotiating a sharp turn.
(e) The guard of a train, which is slowing down to stop at a station?
(a) Non-inertial
(b) Inertial
(c) Non-inertial
(d) Non-inertial
(e) Non-inertial.
Any body which is moving with a constant velocity, does not experience any unbalanced force. When the soundproof room is moving with constant velocity, then we cannot detect the motion of room.
If the room is uniformly accelerated then there is net unbalanced force and in order to keep the bodies stable inside the room one has to exert the force. Hence accelerated motion of room can be detected.
R
Angle of friction for the rough surface is,
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037615.png)
Given that angle of inclination is greater than angle of repose.
So,
Acceleration of the body is,
.
When a body is moving down the inclined plane, coefficient of friction = 
When the block is at rest on the plane, frictional force is mg sin 
When the block is moving down the plane, f = u mg cos 
Retardation of the body is given by, 
The force of friction between the body and the inclined plane is ,
f =
A body placed on the rough inclined plane has angle of inclination greater than angle of repose.
So,
Force of friction between the body and the inclined plane is given by,
f = 
Angle of inclination = 
Angle of repose = 
Here, angle of inclination is greater than angle of repose.
Minimum force required to prevent the body from sliding is,
F= 
Given angle of inclination 
is greater than angle of repose 
.
So, minimum force required to take the body upto inclined plane = 
Since the body is at rest, the force of friction acting between the body and the surface is zero.
No, when friction is not present, the machine will not perform.
The force of friction acting on a body of mass m placed on the rough horizontal surface is zero.
Normal force, N = 200 N
= 0.3 
When a body actually moves or tends to move over the surface of another body, frictional force comes into play as the retarding force.
The surface of a moving body is not perfectly smooth. even though it appears to be smooth, there are irregularities on the surface.
When a body is placed on the second body, the irregularities of the two surfaces get interlocked. To break the interlocking, we have to apply the force. Therefore, the cause of friction is interlocking of irregularities.
Limiting friction is the maximum static friction which comes into play when the body is just at the point of motion.
Dynamic friction is the friction that comes into play when the body is in motion.
When the body is at rest, the cavities of two bodies are fully interlocked and we require large force to overcome the cavities.
Once the sliding has begun, the two surfaces have not sufficient time to fully settle down and hence cavities will not fully interlock and we require lesser force to keep the body in motion.
Therefore, dynamic friction is less than static friction.
Laws of limiting friction:
(i) The force of limiting friction depends upon the nature of surfaces in contact and-acts tangentially to the interface between the two surfaces.
(ii) The force of limiting friction between two surfaces in contact is independent of the area of contact.
(iii) Limiting friction is directly proportional to the magnitude of normal reaction N between two surfaces in contact,
i.e.
where n is constant of proportionality and known as coefficient of friction.
(iv)The direction of force of friction is always opposite to the direction in which the body moves or tends to move.
(i) When a wheel or rolling object is placed on the surface, it depresses the surface and causes a slight elevation in the surrounding of depression as shown in the fig. above.
Therefore, body has to continuously climb up the hill and force is required to do so.
(ii) There are adhesion forces between rolling body and the surface. So, the rolling body has to detach itself from the surface on which it rolls and the force is required to detach the body.
Angle of friction is the angle which the resultant of force of friction and normal reaction make with the normal reaction.
Let a body of weight 'W' be placed on the horizontal surface and pulled by force F.
Weight of the body is acting in vertically downward direction and there is no component of force F in vertical direction, therefore, normal reaction will be equal to weight.
Let μ be coefficient of friction.
Therefore, force of friction f = μW, will act in horizontal direction opposite to the direction of F applied.
Let the resultant of 
make an angle 
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037656-3.png)
∴ 
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037656-5.png)
Hence, proved.
Angle of repose is the angle that an inclined plane makes with the horizontal when a body placed on it just starts sliding down.
Let a body of weight W be placed on rough inclined plane and the body just starts sliding when angle of inclination is ϕ. 
W can be resolved into two components:
(i) W sinϕ along the plane.
(ii) W cosϕ perpendicular to the plane.
When body is just at the point of sliding, then
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037658-2.png)
...(1)
We know the relation between angle of friction![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037658-3.png)
is,
...(2)
So, from (1) and (2),
Why are lubricants used in machines?
Friction is necessary because it is helpful to us in many ways. For example,
(i) Without friction we cannot write.
(ii) Speed of vehicles can be controlled only due to friction.
(iii) We cannot rotate the pulley without friction between belt and pulley.
(iv) It is impossible to climb up with rope without friction between hand and rope. Frictional force helps the person to move forward.
(v) Without friction we even cannot walk.
When we push or pull the lawn mower with force F at angle θ with horizontal, the force of push or pull can be resolved into two rectangular components:
(i) F cosθ, the horizontal component which helps it to move in forward direction.
(ii) F sinθ, the vertical component of force.
In case of pull, Fsinθ acts in vertically upward direction which decreases the normal reaction and hence force of friction decreases.
Since in case of pull, force of friction is less than that in the case of push. Hence it is easy to pull than to push.
Given, 
Mass of the block, m = 4kg
Coefficient of friction,
∴ Force of limiting friction is,
= 23.52 N
The minimum force required to set the body into motion is equal to limiting friction. Thus, force required to set the body into motion is 23.52N.
Given,
Initial velocity, u = 54km/h = 15 m/s
Final velocity, v = 0
The wagon comes to rest in, t = 12s
Mass of the wagon, m = 2000 kg
The retarding force on the wagon is,
= -2500N
Therefore coefficient of friction is,
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037676-1.png)
When a force of 29.4N applied on body of mass 
it just starts moving. Find the angle of friction.
Givem, 
Angle of friction, 
be the coefficient of friction between block and horizontal surface. 
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037681-3.png)
Given,
Mass,
Mass, 
Force acting on the block, F = 24N
Tension in the block, T = 8N
Acceleration, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037682-3.png)
Let 
be the force of frictions on masses 
respectively.
Therefore by using free body diagram,
and 
Substituting the given values,
and 
On solving we get,
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037682-11.png)
f1 and f2 are the required force of frcition.
Let a body of mass 'm' be placed on the inclined plane of inclination θ as shown in the figure. Since tan θ > μ.
Therefore,
Body will accelerate in downward direction and is in equilibrium perpendicular to inclined plane.
Now,
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#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037684-1.png)
Net force acting on the body in downward direction along the plane is, 
Acceleration of body along inclined plane is,
If tanθ < μ, then limiting friction μgcos θ will be greater than mgsinθ, the force that tends to slide the body down.
Thus, the body will not slide down. That is, the force has to be applied to slide the body down or up the inclined plane.
Case I:
Force should be applied parallel to the plane in downward direction, to slide the body down the inclined plane.
Here, force of friction is acting upwards, therefore the force required to just slide the body down the inclined plane is,
f = mg sinθ
Case II:
Force should be applied parallel to the plane in the upward direction in order to slide the body up the inclined plane. 
In this case, force of friction is acting downwards. Therefore, the force required to slide the body up the inclined plane is,
F' = f + mg sin θ
i.e., F' = 
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037690-1.png)
,
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037690-5.png)
as shown. Find the acceleration with which the body will accelerate.
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037692-4.png)
then acceleration will be zero, and![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037692-6.png)
then body will accelerate and is given by
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037694.png)
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037694-1.png)
, is the minimum rotational speed.
(ii) mgcosθ is the component perpendicular to plane.
Since body is in equilibrium perpendicular to the plane,
Since the body is displaced up the inclined plane, therefore force of friction will be down the inclined plane.
Force of friction, 
∴ Work done against friction is,
Given,
MAss of the block, m = 2kg
Angle of inclination between the block and horizontal line is, 
Coefficient of friction, 
The force that tends to slide the body on the inclined plane is,
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037697-3.png)
The limiting friction between the block and the inclined plane is,
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037697-5.png)
Here, we can see that the limiting friction is greater than the force that tends to slide the body.
Therefore the body will be a rest and force of friction on the block is 9.8N
= 370
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037699.png)
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037699-3.png)
and
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037703.png)
in the above equation, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037703-5.png)
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037703-8.png)
[ Given] 
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037708.png)
...(1) 
is,
...(2)
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037708-5.png)
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037708-7.png)
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037710-2.png)
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037710-3.png)
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037710-4.png)
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037710-5.png)
, is the coefficient of friction of rough inclined plane.
be the length of the inclined plane and μ be the coefficient of friction between the block and inclined plane.
...(1)
A man weighs 70 kg. He stands on a weighing scale in a lift which is moving:
(a) upwards with uniform speed of 10m/s,
(b) downwards with uniform acceleration of 5 m/s2 and
(c) upwards with uniform acceleration of 5 m/s2.
What would be the reading on the scale in each case?
(d) What would be the reading, if the lift mechanism failed and it hurtled down freely under gravity?
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037718-2.png)
and displacement is zero.
Relation between instantaneous speed and angular speed of a particle,
Relation between instantaneous linear velocity and angular velocity is,
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037745.png)
Relation between instantaneous tangential acceleration and angular acceleration is,
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037746.png)
?
The force required to revolve the body of mass 'm' in circular orbit of radius 'r' with speed 'v'
the work done by the centripetal force is zero because the initial and the final points are the same.
The optimum speed = 
where,
r is radius of track and 
is the angle of banking.
What is the optimum speed at flat rough road?
The optimum speed = ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037776.png)
where,
r is radius of track and 
is coefficient of friction.
Length of the string = l
Minimum speed given to the bob=
With this speed, the bob may complete the loop in the vertical plane.
is connected to a particle of mass m and the other to a small peg on a smooth horizontal table. If the particle moves in a horizontal circle with speed v, then what is the net force on the particle directed towards the center?
Length of the string = 
Mass of the particle = m
Particle moves with a velocity v
Net force acting = 
Angular displacement is defined as the angle described by particle about the axis of rotation. It is represented by 
.
Angular velocity is defined as the rate of change of angular position.
It is represented by 
Mathematically it is given by, 
Angular acceleration is the rate of change of angular velocity.
It is represented by a.
Mathematically it is, 
Which is greater - angular velocity of the earth around its own axis or angular velocity of the earth around the sun?
, is non-zero.
The minute hand of a watch, which points towards 6, is turned by 
Where will it come?
When the minute hand turns by 
the minute hand advances by one unit.
Therefore, on turning through ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037800-2.png)
the minute hand advances by 4 units.
Thus the minute hand will come at 10.
Hour's hand completes one revolution in 12 hours.
Angular velocity of hour's hand is,
Second’s hand completes one revolution in 60 seconds.
Angular velocity of second’s hand is,
∴ ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037804-2.png)
, is the required ratio.
.
.
be the angular velocity at any instant. 
We know,
Angular velocity, 
or 
For uniform angular accelerated motion, 
Hence, proved.
Derive the relation 
for uniform angular accelerated motion.
We know,
Angular acceleration,
Now, on integrating both sides, we get
= 0.6 rad/s2. Find the angular speed of the particle after it turns through 24 revolutions.
Given, a particle starts from rest.
That is, initial angular velocity, 
= 0 rad/s
Angular acceleration, 
We know, 
i.e., 
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#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037832-7.png)
That is, angular speed of the particle is, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037832-8.png)
Here, we have
Initial angular velocity, 
Velocity attained after some time, 
Now, using the formula,
∴ 
, is the angular acceleration of the wheel.
We have,
Initial angular velocity, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037847.png)
Now, using the formula,
∴ 
Also,
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037847-6.png)
∴ ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037847-8.png)
. What is the change in velocity when the particle describes an angle of 90°?
. 
be the velocity of particle at P and 
be the velocity of particle at Q after revolving by angle 
is 
Here,
Angular speed, 
Initial speed, 
Time of travel, t = 20s
By using the relation 
we have
The angular displacement of flywheel is,
Here, we have
and t = 10s
∴ 
Given,
Attained angular velocity, 
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037868-3.png)
Using the formula,
![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037868.png)
or 
∴ ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037868-4.png)
Given,
Initial speed, 
Attained speed, 
Time of travel, 
Now, using the formula,
or 
∴ 
Also, we have 
That is,
= 240 rev
Here,
We have the formula,
That is,
∴ 
Now 
∴ 
So, 10 seconds elapsed from the moment the fan was switched off to the moment it stopeed.
and angular displacement 
is very small, therefore 
...(1) 
is perpendicular to plane of paper in outward direction, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037898-10.png)
is along 
in the plane of paper and 
is in the plane of paper perpendicular to both 
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#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037898-14.png)
, is the required relation. 
and at time t+∆t, it is ω+∆ω. 
Let a particle be revolving in a circle of radius r.
Let ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037904.png)
be its position vector and ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037904-1.png)
be its angular velocity at any instant.
The linear velocity of particle at that instant is,
The linear acceleration of object is,
The magnitude of net acceleration of particle is,
Total acceleration makes an angle θ with radial acceleration and is given by, 
Derive an expression for centripetal acceleration experienced by a particle revolving in circular orbit of radius r with constant angular speed ω.
.
The direction of centripetal force is always perpendicular to the direction of instantaneous velocity.
The instantaneous displacement is in the direction of instantaneous velocity, therefore instantaneous displacement of particle is perpendicular to centripetal force.
Hence work done by centripetal force is zero.
A stone tied to the end of a string in a horizontal plane is whirled round in a circle and suddenly the string breaks. Which of the following correctly describes the trajectory of stone after the string breaks:
(a) the stone jerks radially outwards,
(b) the stone flies off tangentially from the instant the string breaks,
(c) the stone flies off at an angle with the tangent whose magnitude depends on the speed of the particle?
For a person in the laboratory frame, this statement is wrong because w.r.t. observer in laboratory frame, the particle is in uniform circular motion which is an accelerated motion. The acceleration of the particle is acting along the radius towards center. Therefore, force is required to revolve the particle in circle known as centripetal force and cannot be zero.
But with respect to the observer in the frame rotating along with the particle, the particle appears to be at rest. Hence, the net force experienced by particle must be zero.
Therefore, the observer in this frame finds that some forces must act on the particle equal and opposite to centripetal force which keep the particle in equilibrium.
Therefore, the statement makes sense only with respect to the observer in the frame rotating along with particle i.e. non-inertial frame.
Read the following statements and pick up the true and false statements:
(a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre.
(b) The change in velocity vector of a particle in uniform circular motion in one complete cycle is zero vector.
(c) The magnitude of the change in acceleration vector in uniform circular motion over half cycle is zero.
be the angular velocity of the disc. 
, is the angular velocity when the body will just slip.
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037918-3.png)
A body is revolving in a circular orbit of radius r with constant speed v. If at any instant its velocity is 
then after what time the velocity will change by ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037920-1.png)
?
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037920-2.png)
be the velocity of body at any instant t and after time 
its velocity becomes 
Therefore change in velocity is,
Given,
Length of the string, l = 80cm
No. of revolutions made by the stone in 25 seconds = 14
angular speed, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037921.png)
Therefore acceleration of stone is,
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037921-3.png)
Here, motion of stone is circular motion. Therefore, acceleration of stone at any instant is directed towards the centre along the radius.
Given that, 
Speed of the aircraft, 
Now the centripetal acceleration is, 
The centripetal acceleration relative to acceleration due to gravity is,
Given,
Mass of the stone, m = 0.3 kg
Length of the string, l = 1m
Angular velocity,
Therefore tension in the string,
Let 
be the maximum angular speed of stone with which it can be whirled around.
The centrifugal force on the stone should not increase above maximum tension 200N. Otherwise the string will break.
∴ 
A long playing record revolves with a speed of 
and has a radius of 15 cm. Two coins are placed at 4 cm and 14 cm away from the centre of the record. If the coefficient of friction between the coins and the record is 0.15, which of the two coins will revolve with the record?
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037927-2.png)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037927-5.png)
Given here,
Speed of the cyclist, v = 27 km/h = 7.5m/s
Radius of the circular tun on the road, r = 80 m
Therefore the centripetal acceleration of the cyclist at the moment when velocity is 7.5 m/s is,
When brakes are applied, the speed is decreased at the rate of 0.5m/s every second.
Therefore the tangential acceleration is,
Therefore the total acceleration of the cyclist is,
Let m be the mass of the car.
While crossing the over bridge, the centrifugal force on the car at the highest point is 
Therefore net force exerted by car on the bridge is given by,
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037932-1.png)
in inward direction as shown in figure above. 
with the vertical. 
...(1) 
...(2) 
given by 
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037936-4.png)
, is the angle that the cyclist makes with the vertical.
When the outer part of the road is raised little above the inner part, to take the turn along the circular track, it is called banking of road.
The angle that the inclined track makes with horizontal is called angle of banking.
Consider a vehicle of weight mg mving around a banked curved track of radius r and inclination θ with speed v. 
The vehicle is under the action of following forces:
(i) The weight mg acting vertically downward.
(ii) The reaction R of the track to the vehicle is acting perpendicular to banked track in upward direction.
The components of R is resolved into it's components.
The vertical component Rcosθ of normal reaction R will balance the weight mg of the vehicle.
The horizontal component R sinθ will provide the necessary centripetal force to the vehicle.
i.e. 
...(1)
and
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#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037940-2.png)
...(2)
Dividing (2) and (1),
The above equation gives the angle of banking.
'
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#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037941-5.png)
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#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037942.png)
...(1) 
therefore velocity at highest point is,
...(2) ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037942-6.png)
in equation (1), we get 
, is the minimum velcity given to the mass at the lowest point.
A body of mass m is tied with string and rotated in vertical circle of radius r. The body is given the velocity u at lowest point 
What is the tension at the highest point and does the difference in the tensions at the lowest and the highest point depend upon u?
Tension in the string at any point is given by,
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037943-1.png)
At highest point, 
Therefore,
At the lowest point, 
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037943-7.png)
∴ 
The difference in the tension at the highest and the lowest point does not depend upon velocity at the lowest point and is constant, equal to 6 mg.
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#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037944-5.png)
, is the least horizontal velocity that is given to the mass. 
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#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037947.png)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037948-1.png)
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037948-6.png)
, is the required minimum height. Given,
∴ 
We have,
Velocity of the aircraft, v = 720 km/hr = 200 m/s
∴ Radius of the loop, 
= 15232.86 m
= 15.23 km
The maximum speed of car is given by,
Here,
Limiting friction, 
Radius of the track, r = 30 m
Acceleration due to gravity, 
∴
Maximum speed, ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037950-3.png)
Given,
Radius of the circular railway track, r = 200m
Speed of the train, 
We know, the angle of banking is,
, is the angle of banking.
...(2)
, is the required speed of the body to be in equilibrium.
A conical pendulum is a small heavy mass attached with massless, flexible and inextensible string suspended from a rigid support and the mass is a constraint to whirl in a horizontal circle with constant speed.
Consider a conical pendulum consisting of point heavy mass m attached with massless, flexible and inextensible string of length l.
Let the pendulum be suspended from a rigid support S and the mass be whirled with constant speed v in a horizontal circle of radius r, such that the string makes an angle θ with the vertical.
The different forces acting on the mass are:
(i) Weight mg in vertically downward direction.
(ii) Tension T in the string along AS.
Resolve the components of Tension T, as shown in figure.
The component Tcosθ balances the weight mg of the mass and component T sinθ provides the necessary centripetal force to whirl the mass.
Thus, 
. Find the radius of circle in which block revolves and the angular position of block with vertical at equilibrium position while bowl is rotating.
Consider a car of mass m taking the turn on banked rough circular track of radius r.
Let 
be the angle of banking and μ be the coefficient of friction.
Let u be the velocity of car so that there is no wear and tear.
Different forces acting on car are:
(i) Weight of car, acting vertically downwards.
(ii) Reaction R on car, normal to track.
(iii) Frictional force f = μR, down the inclined plane.
(iv) Centrifugal force ![<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre>](/application/zrc/images/qvar/PHEN11037955.png)
, radially outwards.
Resolving the rectangular component of different forces.
Since there is no wear and tear on the car, therefore, net force in vertical and horizontal direction is zero.
That is, 
Given that,
Radius of the road, 
Tangent of the angle of banking of road is,
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Now the upper limit of the velocity is,
So, upper limit of the velocity of road is, v = 14.36 m/s
Given that, 
The maximum safe speed on the flat road is,
The flat track is not suitable for speed 20m/s.
Let for safe turn bank the road at angle 
Thus,
, is the angle of banking required.
= 37o, with the vertical.
Given here,
m1 = 11 kg; m2 = 11.5 kg
Therefore, 
Here,
Mass, m = 10 kg; t = 8 sec
After the force stops acting, the body moves with a uniform velocity given by,
v = 
Initial velocity of the body, u = 0
Using equation of motion,
v = u + at
16 = 0 + a x 8
a = 2 m/s2
Therefore,
Force, F = ma = 10 
= 20 N
Impulse imparted to each ball = change in momentum of each ball.
= 2 mv
= 2 x (50 x 10-3) x 10
= 1 kg m/s
A rocket of initial mass 6000 kg ejects mass at a constant rate of 16 kh/s with constant relative speed of 11 km/s. What is the acceleration of the rocket one minute after blast?
Here,
mo = 6000 kg
u = 11 km/s = 11000 m/s
Acceleration,a = ?
Time taken, t = =1 min = 60 sec
Mass left after 1 minute = mo - (dm/dt) t
= 6000 - ( 16 x 60)
= 5040 kg
Using the formula,
F = m. (dv/dt)= u . (dm/dt), we get
a = dv/dt = 
Here,
m1 = 50 kg, m2 = 150 kg
Force applied, F = ?
Tension in the string, T = ?
Acceleration in both the bodies, a = 5 m/s2
Therefore,
F = (m1 + m2) a
F = (50 +150) 5 = 1000 N
Also, T = m2a= 150 x 5 = 750 N
a hammer weighing 1 kg moving with a speed of 20 m/s strikes the head of a nail driving it 10 cm into a wall. Neglecting the mass of the nail, calculate a) the acceleration during the impact, b) the time interval during impact, c) impulse.
Here,
m = 1 kg, u = 20 m/s
s = 10 cm = 0.1 m
v = 0, a = ?
a) As,
v2 = u2 + 2as
b) v = u + at 
c) Impulse = F. t
= m (v-u)
= 1 (0-20)
= -20 N.s.
In an X-ray machine, an electron is subjected to a force of 10-23 N. In how much time will the electron cover a distance of 0.1 m. Take mass of electron as 10-30 kg.
Given,
Force, F = 10-23 N, t = ?
s = 0.1 m, m = 10-30 kg
Acceleration, a = 
Using the equation of motion, 
A curved road of diameter 1.8 km is banked so that no friction is required at a speed of 30 m/s. What is the banking angle?
Diameter, 2 r = 1.8 km = 1800 m
r = 1800/2 = 900 m
Speed, v = 30 m/s
Banking angle, 
= ?
We know that, 
When the brakes of a bike are on, its wheels cannot rotate. They will simply skid. Thus, rolling friction will be converted into sliding friction which is comparatively larger. Hence, it becomes difficult to move the bike.
When angular speed and linear speed are equal, the radius of the path is 1 m.
The direction of velocity at any instant is the direction in which the body tends to move if there is no external force acting on it.
The body in a circular path moves along the tangent drawn to the circle at a point.
Hence, the tangent drawn to the circle at any point gives the direction of the velocity of the body at that point.
Let, n bullets be fired in one second.
To stop the tiger, momentum of bullets = momentum of tiger.
n x ( 50 x 10-3) x 150 = 60 x 10
n = 
A car travels on a flat, circular track of radius 200 m at 30 m/s and has a centripetal acceleration = 4.5 m/s2. a) If the mass of the car is 1000 kg, what frictional force is required to provide the acceleration? b) If the coefficient of static friction is 0.8, what is the maximum speed at which the car can circle the track?
=4500 N 
A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble,
(a) during its upward motion,
(b) during its downward motion,
(c) at the highest point where it is momentarily at rest. Do your answers change if the pebble was thrown at an angle of 45° with the horizontal direction?
Ignore air resistance.
Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg,
(a) just after it is dropped from the window of a stationary train,
(b) just after it is dropped from the window of a train running at a constant velocity of 36 km/h,
(c) just after it is dropped from the window of a train accelerating with 1 m s-2,
(d) lying on the floor of a train which is accelerating with 1 m s-2, the stone being at rest relative to the train.
Neglect air resistance throughout.
a) We have here,
Mass, m = 0.1 kg
Acceleration due to gravity, a = +g = 10 m/s2
Net force, F = ma = 0.1 × 10 = 1.0 N, acting in vertically downward direction.
b) When the train is running at a constant velocity, its acceleration = 0.
Due to this motion, no force is acting on the stone.
Therefore,
Force on the stone, F = weight of stone = mg
= 0.1 × 10 = 1.0 N
This force acts in the vertically downward direction.
c) Acceleration of the train, a = 1 m s-2
Additional force, F' = ma = 0.1 × 1 = 0.1 N, acts on the stone in the horizontal direction.
But once the stone is dropped from the train, F' becomes zero.
Net force on the stone, F = mg = 0.1 × 10
= 1.0 N, acting vertically downwards.
d) When the stone is lying on the train, its acceleration is same as that of the train.
Therefore, the force acting on stone, F = ma
= 0.1 × 1 = 0.1 N
This force is along the horizontal direction of motion of the train.
In each case, the weight of the stone is being balanced by the normal reaction.
i) T is the correct answer.
A particle connected to a string is revolving in a circular orbit around the center.
For rotation, the centripetal force is provided by the tension produced in the string.
Therefore, the net force produced on the particle is the tension, T.
That is, 
where,
F is the net force acting on the particle.
Retarding Force, F = - 50 N
Mass of the body, m = 20 kg
Initial velocity of the body, u = 15 m/s
Final velocity of the body, v = 0
Acceleration produced in the body can be calculated as:
F = ma
-50 = 20 x a
a = 
Using the first equation of motion,
v = u + at
Time (t) taken by the body to come to rest can be calculated as,
Therefore,
t = 
Mass of the body, m = 5 kg
The given situation can be represented as follows:
The resultant of two forces is given by,
R = 
is the angle made by R with the force of 8 N.
Therefore, 
The negative sign indicates that θ is in the clockwise direction with respect to the force of magnitude 8 N.
According to Newton’s second law of motion
Acceleration (a) of the body is given as,
F = ma
Therefore,
a =
a" t2 Given,
Initial velocity of the truck, u = 0
Acceleration, a = 2 m/s2
Time, t = 10 s
a)
According to the equation of motion, we have
v = u + at
= 0 + 2x10
= 20 m/s
The final velocity of the truck and hence, of the stone is 20 m/s.
At t = 11 s, the horizontal component (vx) of velocity, in the absence of air resistance, remains unchanged.
i.e., vx = 20 m/s
The vertical component (vy) of velocity of the stone is given by the first equation of motion as,
vy = u + ay δt
where,
δt = 11 – 10 = 1 s, and
ay = g = 10 m/s2
∴ vy = 0 + 10 × 1
= 10 m/s
The resultant velocity (v) of the stone is given as:
Let, 
be the angle made by the resultant velocity with the horizontal component of velocity vx,
Therefore, 
b)
When the stone is dropped from the truck, the horizontal force acting on it becomes zero. However, the stone continues to move under the influence of gravity. Hence, the acceleration of the stone is 10 m/s2 and it acts vertically downward.
A man of mass 70 kg stands on a weighing scale in a lift which is moving,
(a) upwards with a uniform speed of 10 m s–1,
(b) downwards with a uniform acceleration of 5 m s–2,
(c) upwards with a uniform acceleration of 5 m s–2.
What would be the readings on the scale in each case?
(d) What would be the reading if the lift mechanism failed and it hurtled down freely under gravity?
= 35 kg
= 0 kg 
m/s
- 0)
m/s 
) = -3 kg m/s
m/s 
- 0) = 3 kg m/s
) = -3 kg m/s 
= 
Mass of each ball, m = 0.05 kg
Initial velocity of the ball = 6 m/s
Magnitude of the initial momentum of the ball, p1 = 0.05 x 6
= 0.3 kg m/s
After collision, the balls change their directions of motion without changing the magnitudes of their velocity.
Final momentum of each ball, pf = - 0.3 kg m/s
Impulse imparted to each ball = pf - pi = -0.3 -0.3 = -0.6 kg m/s
The negative sign indicates that the impulses imparted to the balls are opposite in direction.
The given question can be illustrated as in the following fig. below.
Here,
AO = incident path of the ball,
OB = Path followed by ball after deflection,
< AOB = Angle between the incident and deflected paths of the ball = 45o
Therefore,
∠AOP = ∠BOP = 22.5° = θ
Initial velocity of the ball = Final velocity of the ball = v
On resolving the component of velocity along v, we have
Horizontal component of the initial velocity = vcos θ, along RO
Vertical component of the initial velocity = vsin θ, along PO
Horizontal component of the final velocity = vcos θ, along OS
Vertical component of the final velocity = v sin θ, along OP
The horizontal components of velocities suffer no change.
The vertical components of velocities are in the opposite directions.
So, change in linear momentum of the ball gives us the impulse which is imparted to the ball.
That is,
Impulse = mvCosθ - (-mvCosθ)
= 2mvCosθ
Mass of the ball, m = 0.15 kg
Velocity of the ball, v = 54 km/h
= 15 m/s
Therefore,
Impulse = 2 x 0.15 x 15 cos 22.5o
= 4.16 kg m/s
A stone of mass 0.25 kg tied to the end of a string is whirled round in a circle of radius 1.5 m with a speed of 40 rev./min in a horizontal plane. What is the tension in the string? What is the maximum speed with which the stone can be whirled around if the string can withstand a maximum tension of 200 N?
40 / 60 = 2 / 3 rps
= 2πn 
)2
| Lowest Point | Highest Point | |
| a | mg – T1 | mg + T2 |
| b | mg + T1 | mg – T2 |
| c | mg + T1 – (mv12) / R | mg – T2 + (mv12) / R |
| d | mg – T1 – (mv12) / R | mg + T2 + (mv12) / R |
...(i)
...(ii)Given,
Mass of the helicopter, mh = 1000 kg
Mass of the crew and passengers, mp = 300 kg
Total mass of the system, m = 1300 kg
Acceleration of the helicopter, a = 15 m/s2
Using Newton’s second law of motion, the reaction force R,
R – mpg = ma
= mp(g + a)
= 300 (10 + 15)
= 300 × 25
= 7500 N
The reaction force will be directed upwards, the helicopter is accelerating vertically upwards.
According to Newton’s third law of motion, the force on the floor by the crew and passengers = 7500 N, directed downward.
(b)
Using Newton’s second law of motion, the reaction force R’ experienced by the helicopter can be calculated as,
R' - mg = ma
= m(g + a)
= 1300 (10 + 15)
= 1300 × 25
= 32500 N
The reaction force experienced by the helicopter from the surrounding air is acting upward. Hence, as per Newton’s third law of motion, the action of the rotor on the surrounding air will be 32500 N, directed downward.
(c) The force on the helicopter due to the surrounding air is 32500 N, directed upwards.
Given,
Mass of the block, m = 25 kg
Mass of the man, M = 50 kg
Acceleration due to gravity, g = 10 m/s2
Force applied on the block, F = 25 × 10
= 250 N
Weight of the man, W = 50 × 10 = 500 N
Case (a): When the man lifts the block directly
In this case, the man applies a force in the upward direction.
This increases his apparent weight.
Therefore,
Action on the floor by the man = 250 + 500 = 750 N
Case (b): When the man lifts the block using a pulley
In this case, the man applies a force in the downward direction. This decreases his apparent weight.
Therefore,
Action on the floor by the man = 500 – 250
= 250 N
If the floor can yield to a normal force of 700 N, then the man should adopt the second method to easily lift the block by applying lesser force.
s from the start. 
)2 
rev/min, and has a radius of 15 cm. Two coins are placed at 4 cm and 14 cm away from the centre of the record. If the coefficient of friction between the coins and the record is 0.15, which of the coins will revolve with the record?
rev/min
rev/s
. What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for ω = 
? Neglect friction.
ω2 = 2g / R ...(vi) 
= g / RCos θ Given in the figure are two blocks A and B of weight 20 N and 100 N respectively. These are being pressed against a wall by a force F as shown. If the coefficient of friction between the blocks is 0.1 and between block B and the wall is 0.15, the frictional force applied by the wall on block B is
100N
80 N
120 N
150 N
C.
120 N
In the vertical direction, weight are balanced by frictional forces.
As the blocks are in equilibrium balance forces are in horizontal and vertical direction.
For the system of blocks (A+B)
F = N
For block A, fA = 20 N and for block B.
fB = fA +100 = 120 N
A mass ‘m’ is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?
2g/3
g/2
5g/6
g
B.
g/2
For the mass m,
mg-T = ma
As we know, a = Rα ... (i)
So, mg-T = mRα
Torque about centre of pully
T x R = mR2α ...... (ii)
From Eqs. (i) and (ii), we get
a = g/2
Hence, the acceleration of the mass of a body fall is g/2.
A thin liquid film formed between a U-shaped wire and a light slider supports a weight of 1.5 x10–2N (see figure). The length of the slider is 30 cm and its weight negligible. The surface tension of the liquid film is
0.0125 Nm-1
0.1 Nm-1
0.05 Nm-1
0.025 Nm-1
D.
0.025 Nm-1
⇒F = 2Tl = w
⇒2T(0.3) = 1.5 x 10–2
⇒T = 2.5 x 10–2 N/m
A liquid in a beaker has temperature θ(t) at time t and θ0 is temperature of surroundings, then according to Newton's law of cooling the correct graph between loge (θ – θ0) and t is
A.
According to Newton's law of cooling.
Hence the plot of ln(θ – θ0) vs t should be a straight line with negative slope.
Work done in increasing the size of a soap bubble from a radius of 3 cm to 5 cm is nearly. (Surface tension of soap solution = 0.03 Nm-1)
4π mJ
0.2π mJ
2π mJ
0.4π mJ
D.
0.4π mJ
Work done = Change in surface energy
⇒ W = 2T x 4π (R22-R12)
= 2 x 0.03 x 4π [ (5)2-(3)2] x 10-4 J
= 0.4 π mJ
A pulley of radius 2m is rotated about its axis by a force F = (20t - 5t2) Newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg-m2, the number of rotations made by the pulley before its direction of motion if reversed is
more than 3 but less than 6
more than 6 but less than 9
more than 9
less than
A.
more than 3 but less than 6
To reverse the direction
STATEMENT – 1
Two particles moving in the same direction do not lose all their energy in a completely inelastic collision.
STATEMENT – 2
Principle of conservation of momentum holds true for all kinds of collisions.
The statement I is True, Statement II is False.
The statement I is True, Statement II is True; Statement II is a correct explanation for Statement I.
The statement I is True, Statement II is True; Statement II is not the correct explanation for Statement I.
Statement I is False, Statement II is
B.
The statement I is True, Statement II is True; Statement II is a correct explanation for Statement I.
Two fixed frictionless inclined planes making an angle 30° and 60° with the vertical are shown in the figure. Two blocks A and B are placed on the two planes. What is the relative vertical acceleration of A with respect to B?
4.9 ms-2 in horizontal direction
9.8 ms-2 in vertical direction
zero
4.9 ms-2 in vertical direction
D.
4.9 ms-2 in vertical direction
For the motion of block along inclined plane
mg sin θ =ma
a = g sin θ
where a is along the inclined plane.
The vertical component of acceleration is g sin2θ
Therefore, the relative vertical acceleration of A with respect to B is
g (sin260 - sin230) = g/2 = 4.9 ms-2 (in vertical direction)
A body of mass m = 3.513 kg is moving along the x-axis with a speed of 5.00 ms−1.The magnitude of its momentum is recorded as
17.6 kg ms−1
17.565 kg ms−1
17.56 kg ms−1
17.57 kg ms−1
A.
17.6 kg ms−1
P = mv = 3.513 × 5.00 ≈ 17.6
Out of the following pair, which one does NOT have identical dimensions is
angular momentum and Planck’s constant
impulse and momentum
moment of inertia and moment of a force
work and torque
C.
moment of inertia and moment of a force
A body A of mass M while falling vertically downwards under gravity breaks into two parts; a body B of mass 1/3 M and a body C of mass 2/3 M. The centre of mass of bodies B and C taken together shifts compared to that of body A towards
depends on height of breaking
does not shift
body C
body B
B.
does not shift
No horizontal external force is acting
∴ = acm =0
since vcm= 0
∴∆xcm = 0
A mass ‘m’ moves with a velocity v and collides inelastically with another identical mass. After collision, the 1st mass moves with velocity v/√3 in a direction perpendicular to the initial direction of motion. Find the speed of the 2nd mass after collision
v
√3 v
2v/√3
v/√3
C.
2v/√3
mv= mv1 cos = θ
A ball is released from the top of a tower of height h metres. It takes T seconds to reach the ground. What is the position of the ball in T/3 seconds?
h/9 metres from the ground
7h/9 metres from the ground
8h/9 metres from the ground
17h/18 metres from the ground.
C.
8h/9 metres from the ground
An automobile travelling with speed of 60 km/h, can brake to stop within a distance of 20 cm. If the car is going twice as fast, i.e 120 km/h, the stopping distance will be
20 m
40 m
60 m
80 m
D.
80 m
Third equation of motion gives
v2 = u2 + 2as ⇒ 2
s ∝ u (since v = 0)
where a = retardation of body in both the cases
therefore, 
.... (i)
Here, s1 = 20 m, u1 = 60 km/h, u2 = 120 km/h. Putting the given values in eq. (i), we get
A machine gun fires a bullet of mass 40 g with a velocity 1200 ms−1. The man holding it can exert a maximum force of 144 N on the gun. How many bullets can he fire per second at the most?
one
Four
Two
Three
D.
Three
The force exerted by machine gun on man's hand in firing a bullet = change in momentum per second on a bullet or rate of change of momentum
The force exerted by man on machine gun = 144 N Hence, number of bullets fired =144/48 = 3
2.0
4.0
1.6
2.4
A.
2.0
Let the mass of block be m.
Frictional force in rest position
F = mg sin 30
It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is pd; while for its similar collision with carbon nucleus at rest, fractional loss of energy is pc. The values of pd and pc are respectively :
(0,1)
(.89,.28)
(.28,.89)
(0,0)
B.
(.89,.28)
For collision of a neutron with deuterium:
Applying conservation of momentum:
mv + 0 = mv1 + 2mv2 .....(i)
v2 -v1 = v ...... (ii)
Therefore, Collision is elastic, e = 1
From equ (i) and equ (ii) v1 = -v/3
Now, for the collision of neutron with carbon nucleus
Applying conservation of momentum
mv + 0 = mv1 + 12mv2 ....; (iii)
v = v2-v1 ....(iv)
A car is negotiating a curved road of radius R. The road is banked at angle 
. The coefficient of friction between the tyres of the car and the road is 
. The maximum safe velocity on this road is,
A.
A car is negotiating a curved road of radius R. The road is banked at angle 
and the coefficient of friction between the tyres of car and the road is 
.
The given situation is illustrated as:
In the case of vertical equilibrium,
N cos 
= mg + f1 sin 
mg = N cos 
... (i)
In the case of horizontal equilibrium,
... (ii)
Dividing Eqns. (i) and (ii), we get
A particle of mass m is driven by a machine that delivers a constant power K watts. If the particle starts from rest, the force on the particle at time t is
A.
As the machine delivers a constant power
So F, v =constant = k (watts)
Three blocks A, B, and C of masses 4 kg, 2 kg and 1 kg respectively, are in contact on a 14 N is applied to the 4 kg block, then the contact force between A and B is
2 N
6 N
8 N
18 N
B.
6 N
Given, mA = 4 kg
mB = 2 kg
=> mC =1 kg
So total mass (M) = 4+2+1 = 7 kg
Now, F = Ma
14 = 7a
a=2 m/s2
F-F' = 4a
F' = 14-4x2
F' = 6N
A block A of mass m1 rests on a horizontal table. A light string connected to it passes over a frictionless pulley at the edge of the table and from its other end, another block B of mass m2 is suspended. The coefficient of kinetic friction between the block and the table is 
. When the block A is sliding on the table, the tension in the string is
C.
FBD of block A,
T -m1a = fk ..... (i)
FBD of block B
A car of mass m is moving on a level circular track of radius R. If 
represents the static friction between the road and tyres of the car, the maximum speed of the car in circular motion is given by
D.
In this condition, centripetal force is equal to static frictional force between road and tyres,
so
A system consists of three masses m1, m2 and m3 connected by a string passing over a pulley P. The mass m1 hangs freely and m2 and m3 are on a rough horizontal table (the coefficient of friction= 
). the pulley is frictionless and of negligible mass. The downward acceleration of mass m1 is,
C.
First of all consider the forces on the blocks,
For the 1st block,
mg - T1 = m x a ... (i)
Let us consider second and third block as a system,
Then,
T1 - 
= 2m x a
mg (
) = 3m x a
The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle 
without slipping and slipping down the incline without rolling is,
5:7
2:3
2:5
7:5
A.
5:7
A solid sphere rolling without slipping down an inclined plane is,
We have in this case,
Two sphere A and B of masses m1 and m2 respectively collide. A is at rest initially and B is moving with velocity v along the x -axis. After the collision, B has a velocity v/2 in a direction perpendicular to the original direction. The mass A moves after collision in the direction
same as that of B
opposite to that of B
C.
Here, Pi = m2vi +m2 x 0
The upper half of an inclined plane of the inclination 
is perfectly smooth while lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom, if the coefficient of friction between the block and lower half of the plane is given by,
C.
. L = 
mg cos 
An explosion breaks a rock into three parts in a horizontal plane. Two of them go off at right angles to each other. The first part of mass 1 kg moves with a speed of 12 m/s and the second part of mass 2 kg moves with second part of mass 2 kg moves with 8 m/s speed. If the third part flies off with 4 m/s speed, then its mass is,
3 kg
5 kg
7 kg
17 kg
B.
5 kg
We have,
p1 + p2 + p3 = 0 [ because p = mv]
Therefore,
1 x 12 i + 2 x 8 j + p3 = 0
Now. p3 = m3 v3
This implies,
A person of mass 62 kg is inside of a lift of mass 940 kg and presses the button on control panel. The lift starts moving upwards with an acceleration 1.0 m/s2. If g = 10 m/s2 ,the tension in the supporting cable is
9680 N
11000 N
1200 N
8600 N
B.
11000 N
A planet moving along an elliptical orbit is closer to the sun at a distance r1 and farthest away at a distance of r2. if v1 and v2 are the linear velocities at these points respectively, then the ratio v1/v2 is
r2/r1
(r2/r1)2
r1 /r2
(r1/r2)2
A.
r2/r1
From the law of conservation of angular momentum
mr1v1 = mr2v2
r1v1 = r2v2
v1/v2 = r2/r1
Force F on a particle moving in a straight line varies with distance d as shown in the figure. The work done on the particle during its displacement of 12 m is
21 J
26 J
13 J
18 J
C.
13 J
Work done = Area under (F - x) graph
= 2 x (7-3) + 
Uniform magnetic field acting along AB. If the magnetic force on the arm BC Is F, the force on the arm AC is
A.
-F
Two stone of masses m and 2m are whirled in horizontal circles, the heavier one in a radius r/2 and the lighter one in radius r. The tangential speed of lighter one in radius r. The tangential speed of lighter stone is n times that of the value of heavier stone when they experience same centripetal forces. The value of n is
2
3
4
1
A.
2
Given, that two stones of masses m and 2m are whirled in horizontal circles, the heavier one in a radius r/2 and lighter one in radius r as shown in a figure.
As lighter stone is n times that of the value of heavier stone when they experience same centripetal forces, we get
(Fc)heavier = (Fc)lighter
Two identical piano wires kept under the same tension T have a fundamental frequency of 600 Hz. The fractional increase in the tension of one of the wires which will lead to occurrence of 6 beats/s when both the wires oscillate together would be
0.02
0.03
0.04
0.01
A.
0.02
According to law of tension, the frequency of the string varies directly as the square root of its tension
A mass m moving horizontally (along the x -axis) with velocity v collides and sticks to mass of 3m moving vertically upward (along the y -axis) with velocity 2v. The final velocity of the combination
A.
From the law of conservation of linear momentum
mv + (3m)(2v) = (4m)v'
mv i + 6mv j = 4 mv'
A block of mass m is in contact with the cart C as shown in the figure
The coefficient of static friction between the block and the cart is μ. The acceleration α of the cart that will prevent the block from falling satisfies
C.
When a cart moves with some acceleration toward right then a pseudo force (mα) acts on block towards left. The force (mα) is action force by a block on the cart. Now, block will remain static w.r.t cart if frictional force 
A ball moving with velocity 2 ms-1 collides head on with another stationary ball of double the mass. If the coefficient of restitution is 0.5, then their velocities (in ms-1) after collision will be
0,1
1,1
1,0.5
0,2
A.
0,1
If two bodies collide head on with coefficient of restitution
A man of 50 kg mass standing in a gravity free space at height of 10 m above the floor. He throws a stone of 0.5 kg mass downwards with a speed 2 ms-1. When the stone reaches the floor, the distance of the man above the floor will be
9.9 m
10.1
10 m
20 m
B.
10.1
m r = constant
The distance of the man above the floor (total height) = 10+0.1 = 10.1
An explosion blows a rock into three parts. Two parts go off at right angles to each other. These two are 1 kg first part moving with a velocity of 12 ms-1 and 2 kg second part moving with a velocity of 8 ms-1.If the third part flies off with a velocity of 4 ms-,its mass would be
5 kg
7 kg
17 kg
3 kg
A.
5 kg
apply the law of conservation of linear momentum.
momentum of first part = 1 x 12 = 12 kg ms-1
Momentum of the second part = 2 x 8 = 16 kg ms-1 '
Resultant monmentum
= [(12)2 +(16)2]1/2 = 20 kg ms-1
The third part should also have the same momentum.
The mass of a lift is 2000 kg. When the tension in the supporting cable is 28000 N, then its acceleration is
30 ms-2 downward
4 ms-2 upwards
4 ms-2 downwards
14 ms-2 upwards
B.
4 ms-2 upwards
Apparent weight > actual weight, then the lift is accelerating upward.
Lift is accelerating upward at the rate of a
Hence, equation of motion is written as
R - mg = ma
28000-20000 = 2000a
a = 8000/2000 = 4 ms-2 upwards.
A body under the action of a force,
acquires an acceleration of 1 ms-2. The mass of this body must be
20 kg
10 kg
D.
According to Newton's second law of motion, force = mass x acceleration.
Here, 
A closed loop PQRS carrying a current is placed in a uniform magnetic field. If the magnetic forces on segments PS, SR and RQ are F1, F2 and F3 respectively and are in the plane of the paper and along the directions shown, the force on the segment of QP is
F3 - F1-F2
F3 - F2 + F1
B.
Three forces acting on a body are shown in the figure. To have the resultant force only along the y-direction, the magnitude of the minimum additional force needed is
0.5 N
1.5 N
B.
1.5 N
Minimum additional force needed
F = - (Fresultant)x
A shell of mass 200 g is ejected from a gun of mass 4 g by an explosion that generates 1.05 kJ of energy. The initial velocity of the shell is
100 ms-1
80 ms-1
40 ms-1
120 ms-1
A.
100 ms-1
In the given problem conservation of linear momentum and energy hold good.
Conservation of momentum yields.
m1v1 + m2v2 = 0
4v1 + 0.2 v2 = 0
Conservation of energy yields
Sand is being dropped on a conveyor belt at the rate of M kg/s. The force necessary to keep the belt moving with a constant velocity of v m/s will be
Mv newton
2 Mv newton
Mv/2 Newton
zero
A.
Mv newton
Force required,
as velocity v is constant, hence
F= Mv newton
A block B is pushed momentarily along a horizontal surface with an initial velocity v. if μ is the coefficient of sliding friction between B and the surface, block B will come to rest after a time:
v/gμ
gμ/v
g/v
v/g
A.
v/gμ
Block B will come to rest, if force applied to it will vanish due to frictional force acting between block B and surface, ie,
force applied = frictional force
i.e., μmg = ma
μmg = m (v/t)
t =v/μg
300 J of work is done in sliding a 2 kg block up an inclined plane of height 10 m. Taking =10 m/s2, work done against friction is
200 J
100 J
Zero
1000 J
B.
100 J
Net work done in sliding a body up to a height h on inclined plane
= Work done against gravitational force + Work done against frictional force
A 0.5 kg ball moving with a speed of 12 m/s strikes a hard wall at an angle of 30° with the wall. It is reflected with the same speed and at the same angle. If the ball is in contact with the wall for 0.25 s, the average force acting on the wall is
48 N
24 N
12 N
96 N
B.
24 N
The vector 
represents the momentum of the object before the collision, and the vector 
that after the collision. The vector 
represents the change in momentum of the object 
. 
As the magnitudes of 
are equal, the components of 
along the wall are equal and in the same direction, while those perpendicular to the wall are equal and opposite. Thus, the change in momentum is due only to the change in direction of the perpendicular components.
Hence, 
Its time rate will appear in the form of average force acting on the wall.
The moment of inertia of a uniform circular disc of radius R and mass M about an axis touching the disc at its diameter and normal to the disc is
C.
The moment of inertia about an axis passing through centre of mass of disc and perpendicular to its plane is
where M is the mass of disc and R its radius. According to theorem of parallel axis, MI of circular disc about an axis touching the disc at it diameter and normal to the disc is
A uniform rod of length l and mass m is free to rotate in a vertical plane about A. The rod initially in horizontal position is released. The initial angular acceleration of the rod is: (Moment of inertia of rod about A is 
)
A.
The moment of inertia of the uniform rod about an axis through the end and perpendicular to length is
where m is mass of rod and l its length.
Torque 
acting on centre of gravity of rod is given by
or 
An arrangement of three parallel straight wires placed perpendicular to plane of paper carrying same current ‘I’ along the same direction is shown in Fig. Magnitude of force per unit length on the middle wire ‘B’ is given by
D.
Force between BC and AB will be same in magnitude.
A body initially at rest and sliding along a frictionless track from a height h (as shown in the figure) just completes a vertical circle of diameter AB = D. The height h is equal to
3D/2
D
5D/4
7D/5
C.
5D/4
As track is frictionless, so total mechanical energy will remain constant,
Which one of the following statements is incorrect?
Rolling friction is smaller than sliding friction.
Limiting value of static friction is directly proportional to normal reaction.
Coefficient of sliding friction has dimensions of length.
Frictional force opposes the relative motion.
C.
Coefficient of sliding friction has dimensions of length.
Coefficient of friction or sliding friction has no dimension.
A moving block having mass m, collides with another stationary block having mass 4m. The lighter block comes to rest after collision. When the initial velocity of the lighter block is v, then the value of coefficient of restitution (e) will be
0.5
0.25
0.4
0.8
B.
0.25
According to law of conservation of linear momentum,
mv + 4m x 0 = 4mv' + 0
A block of mass m is placed on a smooth inclined wedge ABC of inclination θ as shown in the figure. The wedge is given an acceleration 'a' towards the right. The relation between a and θ for the block to remain stationary on the wedge is
a = g tan θ
a = g cos θ
C.
a = g tan θ
In non-inertial frame,
N sin θ = ma ...(i)
N cos θ = mg... (ii)
tan θ = a/g
a = g tan θ
A cylindrical tube of uniform cross-sectional area A is fitted with two air tight frictionless pistons. The pistons are connected to each other by a metallic wire. Initially, the pressure of the gas is p0 and temperature is T0, atmospheric pressure is also p0. Now, the temperature of the gas is increased to 2T0, the tension of the wire will be
2p0A
p0A
p0A/2
4p0A
B.
p0A
The volume of the gas is constant i.e., V = constant
∴ p ∝ T i.e pressure will be doubled if the temperature is doubled.
Let F be the tension in the wire. Then, the equilibrium of anyone pipes gives.
A skier starts from rest at point A and slides down the hill without turning or breaking. The friction coefficient is μ. When he stops at point B, his horizontal displacement is S. what is the height difference between points A and B?
(The velocity of the skier is small so that the additional pressure on the snow due to the curvature can be neglected. Neglect also the friction of air and the dependence of μ on the velocity of the skier.)
h = μS
h = μ/S
h = 2μS
h = μS2
A.
h = μS
According to the question, the condition is shown in the figure
For a sufficiently safe-horizontal displacement △S can be considered straight. If the corresponding length of path element is △L, the friction force is given by μmg(△S/△L ).△L = μmg△S
Adding up, we find that along the whole path the total work done by the friction force is μmgs. By energy conservation, this must equal the decrease mgh in potential energy of skier.
Hence, h = μS
A bicycle wheel rolls without slipping on the horizontal floor. Which one of the following is true about the motion of points on the rim of the wheel, relative to the axis at the wheel's centre?
Points near the top move faster than points near the bottom
Points near the bottom move faster than points near the top
All points on the rim move with the same speed
All points have the velocity vectors that are pointing in the radial direction towards the centre of the wheel
A.
Points near the top move faster than points near the bottom
We have, VA = 2v sinθ/2
Hence, the velocity of a point on rim increases with θ for 0° < θ < 180° and decreases with θ for 180° < θ < 30°.
The upper half of an inclined plane of inclination θ is perfectly smooth while the lower half rough. A block starting from rest at the top of the plane will again come to rest at the bottom if the coefficient of friction between the block and the lower half of the plane is given by
μ = 2 tanθ
μ = tanθ
μ = 2/(tan θ)
μ = 1/ tan θ
A.
μ = 2 tanθ
suppose the length of plane is L. When block descends the plane, rise in kinetic energy = fall in potential energy = mg L sinθ i
Work done by friction = μ x (reaction) x distance
0 + μ (mg cos θ) x (L/2)
= μ (mg cos θ) x (L/2)
Now, work done = change in KE
mgL sin θ = μ (mg cos θ) x (L/2)
⇒ tan θ = μ/2
μ = 2 tan θ
Two masses 10 kg and 20 kg respectively are connected by a massless springs as shown in the figure. A force of 200 N acts on the 20 kg mass. At the instant shown is a figure the 10 kg mass has an acceleration of 12 m/s2. The value of the acceleration of 20 kg mass is
4 m/s2
10 m/s2
20 m/s2
30 m/s2
A.
4 m/s2
The equation of motion of m1 = 10 kg mass is
F1 = m1a1 = 10 x 12 = 120 N
Force on 10 kg mass is 120 N to the right. As action and reaction are equal and opposite, the reaction force F- on 20 kg mass F = 120 N to the left.
therefore, equation of motion of mass m2 = 20 kg is
200 - F = 20 a2
200-120 = 20a2
80 = 20a2
a2 = 80 /20 = 4 m/s2
A cylinder rolls up an inclined plane, reaches some height and then rolls down (without slipping throuhgout these motions). The directions of the frictional force acting on the cylinder are
Up the incline, while ascending and down the incline while descending
Up the incline, while ascending as well as descending
down the incline, while ascending and up the incline while descending
Down the incline while ascending as well as descending.
B.
Up the incline, while ascending as well as descending
It is obvious that during ascending, a retarding i.e. anticlockwise moment is required. I should be remembered that torque due to friction has the same sense the angular acceleration.
A long block A of mass M is at rest on a smooth horizontal surface. A small block B of mass M/2 is placed on A at one end and projected along A with same velocity v. The coefficient of friction between the block is μ. Then the acceleration of blocks A and B before reaching a common velocity will be respectively
B.
The force causing the motion of A is a frictional force between A and B,
So, acceleration of A
Block B experiences frictions force towards the left.
MBaB = μMBg ⇒aB = μg towards left
State Hooke's Law.
Hooked law states that for a small force, Stress is directly proportional to strain. the constant of proportionality is the modulus of elasticity
State Boyle's Law
Boyle’s law is a gas law which states that the pressure exerted by a gas (of a given mass, kept at a constant temperature) is inversely proportional to the volume occupied by it. In other words, the pressure and volume of a gas are inversely proportional to each other as long as the temperature and the quantity of gas are kept constant. Boyle’s law was put forward by the Anglo-Irish chemist Robert Boyle in the year 1662.
For a gas, the relationship between volume and pressure (at constant mass and temperature) can be expressed mathematically as follows.
P ∝ (1/V)
Where P is the pressure exerted by the gas and V is the volume occupied by it. This proportionality can be converted into an equation by adding a constant, k.
P = k*(1/V) ⇒ PV = k
The pressure v/s volume curve for a fixed amount of gas kept at constant temperature is illustrated below.
It can be observed that a straight line is obtained when the pressure exerted by the gas (P) is taken on the Y-axis and the inverse of the volume occupied by the gas (1/V) is taken on the X-axis.
Action and reaction forces do not balance each other why?
They do not balance each other as the action and reaction forces acts on two different bodies.
A cone filled with water is revolved in a vertical circle of radius 4 m and the water does not fall down. What must be the maximum period of revolution?
2 s
4 s
1 s
6 s
B.
4 s
When a cone filled with water is revolved in a vertical circle, then the velocity at the highest point is given by
A.
Two weights w1 and w2 are suspended to the two strings on a frictionless pulley. When the pulley is pulled up with an acceleration g, then the tension in the string is :
A.
Equation of motion for first weight
This is a frictionless and inextensible pulley.
for first weight
T - m1g = m1 ( a - g)
⇒ T - 2m1 g = m1a
For second weight
m2g -T = m2 ( a - g)
2 m2 g - T = m2 a
on solving equations (1) & (2)
A bomb of mass 30 kg at rest explodes into two pieces of masses 18 kg and 12 kg. The velocity of 18 kg mass is 6 ms'.The kinetic energy of the other mass is
256 J
486 J
524 J
324 J
B.
486 J
The linear momentum of the exploding part will remain conserved.
Applying coservation of linear momentum,
m1 u1 =m2 u2
Here, m1 = 18kg, m2 = 12kg
u1 = 6ms-1, u2 = ?
Thus kinetic energy of 12 kg mass
A drum of radius R and mass M, rolls down without slipping along an inclined plane of angle θ. The frictional force
converts translational energy to rotational energy
dissipates energy as heat
decreases the rotational motion
decreases the rotational and translational motion
A.
converts translational energy to rotational energy
When a body rolls down without slipping along an inclined plane of inclination θ, it rotates about a horizontal axis through its centre of mass and also its centre of mass moves. Therefore, rolling motion may be regarded as a rotational motion about an axis through its centre of mass plus a translational motion of the centre of mass. As it rolls down, it suffers loss in gravitational potential energy provided translational energy due to frictional force is converted into rotational energy.
A block B is pushed momentarily along a horizontal surface with an initial velocity v. If µ is the coefficient of sliding friction between B and the surface, block B will come to rest after a time
A.
Block B will come to rest, if force applied to it will vanish due to frictional force acting between block B and surface, ie,
force applied = frictional force
μmg = m
where μ - constant of proportionality
Diwali rocket is ejecting 50 g of gases/s at a velocity of 400 m/s. The accelerating force on the rocket will be
22 dyne
20 N
20 dyne
100 N
B.
20 N
The accelerating force on the rocket
= upward thrust =
Given:-
,
ν = 400 m/s
Accelerating force = 50 × 10 × 400
= 20 N
A weightless thread can bear tension upto 37 N. A stone of mass 500 g is tied to it and revolved in a circular path of radius 4 min a vertical plane. If g = 10 ms-2, then the maximum angular velocity the stone will be
2 rad s-1
4 rad s-1
8 rad s-1
16 rad s-1
B.
4 rad s-1
According to second law, the force f provding the acceleration is
F =
where m is the mass of the body. This force directed towards the centre is called the centripetal force. For a stone rotated in a circle by a string, the centripetal force is provided by the string.
Maximum tension in the thread is given by
Tmax = mg +
⇒ Tmax = mg + mrω2 ( )
ω2 =
Given
Tmax = 37 N, m = 500 g = 0.5 kg,
g = 10 ms-2, r = 4 m
∴
⇒ ω2 = 16
⇒ ω = 4 rad s-1
A bullet of mass 20 g and moving with 600 m/s collides with a block of mass 4 kg hanging with the string. What is velocity of bullet when it comes out of block, if block rises to height 0.2 m after collision?
200 m/s
150 m/s
400 m/s
300 m/s
A.
200 m/s
Linear momentum is a product of the mass (m) of an object and the velocity (v) of the object.
According to conservation of linear momentum
m1 v1 = m1 v + m2 v2
where v1 is the velocity of the bullet before the collision, v is the velocity of bullet after the collision and v2 is the velocity of the block.
∴ 0.02 × 600 = 0.02 v + 4 v2
We have
=
v2 = 2 m/s
∴ 0.02 × 600 = 0.02 v + 4 × 2
⇒ 0.02 v = 12 - 8
⇒ v =
⇒ v = 200 m/s
Voltage in the secondary coil of a transformer does not depend upon
frequency of the source
voltage in the primary coil
ratio of number of turns in the two coils
Both (b) and (c)
A.
frequency of the source
= Turns ratio
Ratio of number of turns in the two coils results induced emf.
Voltage in the primary coil also affects the induced emf.
A shell of mass 20 kg at rest explodes into two fragments whose masses are in the ratio 2 : 3. The smaller fragment moves with a velocity of 6 m/s. The kinetic energy of the larger fragment is
96 J
216 J
144 J
360 J
A.
96 J
Total mass of the shell = 20 kg
Ratio of the masses of the fragments are 8 kg and 12 kg
Now according to the conservation of momentum
m1 v1 = m2 v2
∴ 8 × 6 = 12 × v
v (velocity of the larger fragment) = 4 m/s
Kinetic energy =
=
Kinetic energy = 96 J
An electric bulb has a rated power of 50 W at 100 V. If it is used on an AC source 200 V, 50 Hz, a choke has to be used in series with it. This choke should have an inductance of
0.1 mH
1 mH
0.1 mH
1.1 H
D.
1.1 H
Power P = I × V
= V R × V
P = V2 R
Resistance of bulb
R =
R =
= 200 Ω
Current through bulb
(I) =
=
I = 0.5 A
In a circuit containing inductive reactance (XL ) and resistance (R ), impedance (Z) of the circuit is
Z = ...... (i)
Z =
Z = 400 Ω
Now,
= ( 400 )2 - ( 200 )2
=
L = 1.1 H
Three concurrent co-planar forces 1 N, 2 N and 3 N acting along different directions on a body
can keep the body in equilibrium if 2 N and 3 N act at right angle
can keep the body in equilibrium if 1 N and 2 N act at right angle
cannot keep the body in equilibrium
can keep the body in equilibrium in 1 N and 3 N act at an acute angle
C.
cannot keep the body in equilibrium
If we keep 1 N and 2 N forces act in the same direction then these are balanced by 3 N force, but this is against statement of question.
Hence, options (c) is correct.
Two rectangular blocks A and B of masses 2 kg and 3 kg respectively are connected by a spring of constant 10.8 Nm-1 and are placed on a frictionless horizontal surface. The block A was given an initial velocity of 0.15 ms-1 in the direction shown in the figure. The maximum compression of the spring during the motion is
0.01 m
0.02 m
0.05 m
0.03 m
C.
0.05 m
As the block A moves with velocity 0.15 ms-1 it compresses the spring which pushes B towards right. A goes on compressing the spring till the velocity acquired by B becomes equal to the velocity of A, i.e., 0.15 ms-1.Let this velocity be v.
Now, spring is in a state of maximum compression. Let x be the maximum compression at this stage.
According to the law of conservation of linear momentum, we get
⇒ v =
⇒ =
v = 0.06 ms-1
According to the law of conservation of energy
0.0225 - 0.009 =
0.0135 =
⇒ x =
⇒ =
x = 0.05 m
A boy on a cycle pedals around a circle of 20 metres radius at a speed of 20 m/s. The combined mass of the body and the cycle makes with the vertical so that it may not fall is (g = 9.8 m/s2 )
60.25o
63.90o
26.12o
30.00o
B.
63.90o
A body that travels an equal distance in equal amounts of time along a circular path has a constant speed but not constant velocity. This is because velocity is a vector and thus it has magnitude as well as direction.
v = ( Rg tanθ )1/2
=
tan θ = 63.70o
tanθ ≈ 63.90o
A car of mass 1000 kg moves on a circular track of radius 40 m. If the coefficient of friction is 1.28. The maximum velocity with which the car can be moved, is
22.4 m/s
112 m/s
m/s
1000 m/s
A.
22.4 m/s
A car moves in a circular track so it perform circular motion.
According to second law the force providing this acceleration is
fc =
But according to static friction
fs ≤ μs N
f =
Given:- μ = 1.28
where μ is the coefficient of friction
R = 40 m
The maximum velocity
vmax =
=
vmax = 22.4 m/s
A moving body of mass m and velocity 3 km/h collides with a rest body ofmass 2 m and stick to it. Now the combined mass starts to move. What will be the combined velocity?
4 km/h
1 km/h
2 km/h
3 km/h
B.
1 km/h
The law of conservation of momentum states that for two objects colliding in an isolated system, the total momentum before and after collision is equal.
Applying law of conservation of momentum
m1 v1 = MV
[ m2 v2 = 0 because v2 = 0 and M = m1 + m2 , V = final velocity
m × 3 = ( m + 2m ) v1
So, v1 = 1 km/h
A block B is pushed momentarily along a horizontal surface with an initial velocity v. If is the coefficient of vlicling friction between B and the surface, block B will come to rest after a time
A.
Block will come to rest, if force applied on it. It will vanish due to frictional force acting between block Band surface, i.e.,
Force applied = Frictional force
⇒ μ mg = ma
⇒ μ mg =
⇒ t =
A block slides down on an incline of angle 30° with an acceleration , Find the kinetic 4 friction coefficient.
0.6
C.
Consider the situation
Let the mass of block be m.
The equation of forces
mg sin 30o - f =
f = .....( since sin30o = )
f =
Also, N = mg cos 30° = mg
As the block is slipping on the incline, friction is
μn =
=
μn =
A pendulum having a bob of mass m is hanging in a ship sailing along the equator form east to west. When the strip is stationary with respect to water, the tension in the string is T0, The difference between T0 and
earth attraction on the bob, is
mω2 R
D.
mω2 R
The attraction of earth on the bob = Weight of bob = mg
Speed of ship due to rotation of the earth
Velocity = angular velocity × radius
v = ω R
Also due to the rotation of the earth, a centripetal force acts on the bob = mω2 R on earth
The net force
T0 = mg + mω2 R
⇒ T0 - mg = mω2 R
A ball of mass m hits the floor with a speed v making an angle of incidence θ with the normal. The coefficient of restitution is e. The speed of reflected ball and the angle of reflection of the ball will be
v' = v, θ = θ'
v' = , θ = 2θ'
v' = 2v, θ = 2θ'
v' =
A.
v' = v, θ = θ'
The coefficient of restitution is a number which indicates how much kinetic energy ( energy of motion ) remains after collision of two object.
The parallel component of velocity of the ball remains unchanged. This gives
v' sinθ = v sinθ .... (i)
For the components normal to the floor, the velocity of separation = v' cosθ
Hence v' cosθ' = e v cosθ .....(ii)
From equations (i) and (ii)
v' =
and tanθ' =
For elastic collision, e = 1, so that
θ' = θ and v' = v
A particle slides on surface of a fixed smooth sphere starting from top most point. The angle rotated by the radius through the particle, when it leaves contact with the sphere, is
B.
See the diagram
Let the velocity be v when the body leaves the surface. From free body diagram,
⇒ v2 = R g cosθ
Again from work-energy principle
change in KE = work done
⇒
v2 = 2gR ( 1 - cosθ)
From Eqs. (i) and (ii)
Rg cosθ = 2gR ( 1 - cosθ )
⇒ 3 gR cosθ = 2gR
⇒
A monkey of mass 15 kg is climbing on a rope with one end fixed to the ceiling. If it wishes to go up with an acceleration 1 m/s, how much force should it apply to the rope if rope is 5 m long and the monkey starts from rest?
150 N
> 160 N
165 N
150 < T ≤ 160 N
C.
165 N
The mass of monkey = 15 kg
Acceleration a = 1 m/s2
For the motion of the monkey T [ 15 g + 15 (1) ]
Hence, T is tension in the string T = 15 g + 15
T = 15 ( 10 + 1)
= 15 × 11
T = 165 N
The monkey should apply 165 N force to the rope.
A body weighing 8 g when placed in one pan and 18 g when placed on the other pan of a false balance. If the beam is horizontal when both the pans are empty, then the true weight of the body is
13 g
9 g
22 g
12 g
D.
12 g
Consider the diagram in balance
8x = wy
⇒ ...(i)
Also, wx = 18 y
⇒ ....(ii)
Dividing Eq. (i) and Eq. (ii) we get
⇒ 1 =
⇒ w =
w = 12 g
Universal time is based on
rotational effect of the earth about its axis
vibrations of cesium atom
orbital motion of the earth around the sun
oscillation of quartz crystal
A.
rotational effect of the earth about its axis
Time defined in terms of the rotation of the earth is called Universal Time (UT).
A liquid of density 800 kg/m3 is filled in a cylindrical vessel upto a height of 3 m. This cylindrical vessel stands on a horizontal plane. There is a circular hole on the side of the vessel. What should be the minimum diameter of the hole to move the vessel on the
floor, if plug is removed. Take the coefficient of friction between the bottom of the vessel and the plane as 0.5 and total mass of vessel plus vessel as 95 kg.
0.107 m
0.053 m
0.206 m
0.535 m
A.
0.107 m
Given:-
ρ = 800 kg/m3
h = 3 m
μ = 0.5
m = 95 kg
dmin = ?
Let area of hole be a
∴ Reaction force F = ρ av2
= ρa. 2gh [v = ]
and fmax = μN
= μ mg
μ= coefficient of friction
⇒ F ≥ fmax
⇒ 2 ρ agh ≥ μ mg
⇒ a ≥
=
a = 0.009
r2 ≥ 0.009
r ≥
rmin ≥ 0.0535 m
dmin = 2 rmin
= 2 × 0.0535
dmin = 0.107 m
The figure shows a horizontal force F acting on a block of mass M on an inclined plane (angle θ). What is the normal reaction N on the block?
mg sinθ + F cosθ
mg sinθ Fcosθ
mg cosθ F sinθ
mg cosθ + F sinθ
D.
mg cosθ + F sinθ
From the figure, the normal reaction on the block is
N = mg cosθ + F sinθ
Assertion: Pseudo force is an imaginary force which is recognised only by a non-inertial observer to explain the physical situation according to Newton's laws.
Reason: Pseudo force has no physical origin, i.e., it is not caused by one of the basic interactions in nature. It does not exist in the action-reaction pair.
If both assertion and reason are true and reason is the correct explanation of assertion.
If both assertion and reason are true but reason is not the correct explanation of assertion.
If assertion is true but reason is false.
If both assertion and reason are false.
B.
If both assertion and reason are true but reason is not the correct explanation of assertion.
Pseudo force is an imaginary force which is recognised only by a non-inertial observer to explain the physical situation according to Newton's laws. The magnitude of this force F is equal to the mass m of the object and acceleration α of the frame of reference. The direction of force is opposite to the direction of acceleration.
F =
Assertion: When a charged particle moves in a circular path. It produces electromagnetic wave.
Reason: Charged particle has acceleration
If both assertion and reason are true and reason is the correct explanation of assertion.
If both assertion and reason are true but reason is not the correct explanation of assertion.
If assertion is true but reason is false.
If both assertion and reason are false
A.
If both assertion and reason are true and reason is the correct explanation of assertion.
Magnetic forces can cause charged particles to move in circular or spiral paths. Particle accelerators keep protons following circular paths and magnetic force. The magnetic force is perpendicular to the velocity and so velocity changes in direction but not in magnitude.
Accelerated charges radiate electromagnetic waves.
The radii of circular paths of two particles of same mass are in ratio 6 : 8 then what will be velocities ratio if they have a constant centripetal force?
D.
The radii of circular paths of two particles of same mass m1 = m2 and ratio is given by
= 1
Centripetal force is a force that acts on a moving body in a circular path and is directed towards the centre around which the body is moving.
⇒
=
∴
Assertion: Use of ball bearing between two moving parts of a machine is a common practice.
Reason: Ball bearing reduces rolling friction and provides good stability.
If both assertion and reason are true and reason is the correct explanation of assertion.
If both assertion and reason are true but reason is not the correct explanation of assertion.
If assertion is true but reason is false.
If both assertion and reason are false.
A.
If both assertion and reason are true and reason is the correct explanation of assertion.
A bearing is a machine element that constrains relative motion to only the desired motion, and reduces friction between moving parts. The simplest form of bearing, the plain bearing, consist of a shaft rotating in a hole.
Assertion: Centripetal force is always required for motion in curved path.
Reason: On a banked curved track, vertical component of normal reaction provides the necessary centripetal force.
If both assertion and reason are true and reason is the correct explanation of assertion.
If both assertion and reason are true but reason is not the correct explanation of assertion.
If assertion is true but reason is false.
If both assertion and reason are false.
C.
If assertion is true but reason is false.
While taking a turn on banked road, the necessary centripetal force is provided by the horizontal component of normal reaction and vertical component balances the weight of the vehicle. Also roads are usually banked for average speed of vehicles passing over them. However, ifthe speed of a vehicle is somewhat less or more than this, the self adjusting static friction will operate between the tyres and the road.
Sponsor Area
Sponsor Area
meson is the field particle of which interaction? 
point P (R, θ) on the circle of radius R is (Here θ is measured from the x–axis)
