Physics Part Ii Chapter 14 Oscillations

If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?

• a²T²+ 4π²v²

• aT/x

• aT + 2πv

• aT/v 

While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of 18 cm during winter. Repeating the same experiment during summer, she measures the column length to be x cm for the second resonance.Then 

• 18 > x

• x >54

• 54 > x > 36 

• 36 > x > 18

The displacement of an object attached to a spring and executing simple harmonic motion is given by x = 2 × 10^-2 cos πt metres. The time at which the maximum speed first occurs

• 0.5 s

• 0.75 s

• 0.125 s

• 0.325 s

A point mass oscillates along the x-axis according to the law x = x₀ cos (ωt - π/4). If the acceleration of the particle is written as a = A cos (ωt + δ), then

• A = x₀ , δ = – π/4

• A = x₀ ω₂ , δ = π/4 

• A = x₀ ω₂, δ = –π/4 

• A = x₀ ω2, δ = 3π/4

Two springs, of force constants _k1 and k₂, are connected to a mass m as shown. The frequency of oscillation of the mass is f. If both k₁ and k₂ are made four times their original values, the frequency of oscillation becomes

• f/2

• f/4

• 4f

• 2f

A particle of mass m executes simple harmonic motion with amplitude ‘a’ and frequency ‘ν’. The average kinetic energy during its motion from the position of equilibrium to the end is

• π²m a² v²

• ma² v²/4

• 4π²ma²v²

• 2π²ma²v²

The maximum velocity of a particle, executing simple harmonic motion with an amplitude 7 mm, is 4.4 m/s. The period of oscillation is

• 100 s

• 0.01 s

• 10 s

• 0.1 s

Starting from the origin, a body oscillates simple harmonically with a period of 2 s. After what time will its kinetic energy be 75% of the total energy?

• 1/6s

• 1/12s

• 1/3s

• 1/4s

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency ω. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time

• at the highest position of the platform

• at the mean position of the platform

• for an amplitude of g/ω² 

• for an amplitude of g2/ ω² 

The function sin²(ωt) represents

• a periodic, but not simple harmonic motion with a period 2π/ω

• a periodic, but not simple harmonic motion with a period π/ω

• a simple harmonic motion with a period 2π/ω

• a simple harmonic motion with a period π/ω.

Two simple harmonic motions are represented by the equation y₁ = 0.1  and y₂ = 0.1 cosπt. The phase difference of the velocity of particle 1 w.r.t. the velocity of the particle 2 is

• −π/6

• π/3

• −π/3

• π/6

If a simple harmonic motion is represented by , its time period is its time period

• 2π/α

• 

• 2πα

• 

The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillation bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would

• first increase and then decrease to the original value.

• first decreased then increase to the original value.

• remain unchanged.

• increase towards a saturation value.

The bob of a simple pendulum executes simple harmonic motion in water with a period t, while the period of oscillation of the bob is t0 in the air. Neglecting frictional force of water and given that the density of the bob is (4/3) x 1000 ms^-1 . What relationship between t and t0 is true?

• t = t₀

• t = t₀/2

• t = 2t₀

• t = 4t₀

A particle at the end of a spring executes simple harmonic motion with a period t₁, while the corresponding period for another spring is t₂. If the period of oscillation with the two springs in series is t, then

• T = t₁ + t₂

• 
• 
• 

The total energy of particle, executing simple harmonic motion is

• ∝ x

• ∝ x²

• independent of x

• ∝ x1/2

A particle of mass m is attached to a spring (of spring constant k) and has a natural angular frequency ω₀. An external force F(t) proportional to cosωt (ω≠ω₀) is applied to the oscillator. The time displacement of the oscillator will be proportional to

In forced oscillation of a particle the amplitude is maximum for a frequency ω₁ of the force, while the energy is maximum for a frequency ω₂ of the force, then

• ω₁ = ω₂

• ω₁ > ω₂

• ω₁ < ω₂ when damping is small and ω₁ > ω₂ when damping is large

• ω₁ < ω₂

A particle executes linear simple harmonic motion with an amplitude of 3 cm. When the particle is at 2 cm from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is

A spring of force constant k is cut into lengths of ratio 1: 2 : 3. They are connected in series and the new force constant is k'. Then they are connected in parallel and force constant is k''. Then k' : k'' is

• 1:6

• 1:9

• 1:11

• 1:14

Two blocks A and B of masses 3m and m respectively are connected by a massless and inextensible string. The whole system is suspended by a massless spring as shown in the figure. The magnitudes of acceleration of A and B immediately after the string is cut, are respectively

• g,g/3

• g/3, g

• g,g

• g/3, g/3

A tuning fork is used to produce resonance in a glass tube. The length of the air column in this tube can be adjusted by a variable piston. At room temperature of 27ºC two successive resonances are produced at 20 cm and 73 cm of column length. If the frequency of the tuning fork is 320 Hz, the velocity of sound in air at 27ºC is

• 330 m/s

• 339 m/s

• 300 m/s

• 350 m/s

A pendulum is hung from the roof of a sufficiently high building and is moving freely to and fro like a simple harmonic oscillator. The acceleration of the bob of the pendulum is 20 m/s2 at a distance of 5 m from the mean position. The time period of oscillation is

• 2πs

• π s

• 1s

• 2s

A particular of mass m is executing oscillation about the origin on X- axis. Its potential energy is V(x) = K |x|³. Where K is a positive constant. If the amplitude of oscillation is a, then its time period T is proportional to

• $\frac{1}{\sqrt{a}}$

• a

• $\sqrt{a}$

• a^3/2

The displacement of a particle along the x-axis is given by x = a sin² ωt. The motion of the particle corresponds to 

• Simple harmonic motion of frequency ω/π

• Simple harmonic motion of frequency  3ω/2π

• non-simple harmonic motion

• Simple harmonic motion of frequency ω/2π

A simple pendulum of length l has a maximum angular displacement θ. The maximum kinetic energy of the bob is

• mgl (1-cosθ)

• 0.5 mgl

• mgl

• 0.5 mgl

Equations of motion in the same direction are given by

$$\begin{gathered} {\mathrm{y}}_1 =2\mathrm{\alpha } \sin \left(\mathrm{\omega t} -\mathrm{kx}\right) \\ {\mathrm{y}}_2 =2\mathrm{\alpha } \sin \left(\mathrm{\omega t} -\mathrm{kx}-\mathrm{\theta }\right) \end{gathered}$$

The amplitude of the medium particle will be

• $2\mathrm{\alpha } \mathrm{cos\theta }$

• $\sqrt{2}\mathrm{\alpha } \mathrm{cos\theta }$

• $4\mathrm{\alpha } \cos \frac{\mathrm{\theta }}{2}$

• $\sqrt{2}\mathrm{\alpha } \cos \frac{\mathrm{\theta }}{2}$

If the length of a pendulum is made 9 times and mass of the bob is made 4 times, then the value of time period becomes :

• 3 T

• 3/2T

• 4T

• 2T

A pendulum is undergoing SHM with frequency f What is the frequency of its kinetic energy ?

• $\frac{\mathrm{f}}{2}$

• 2 f

• 3 f

• 4 f

Two equal -ve charges -q are fixed at the point (O, a) and (O, a) on the y-axis. A positive charge Q is released from rest at the point (2a, 0) on the x-axis. The charge will :

• execute SHM about the origin

• move to the origin and remain at rest

• move to infinity

• execute oscillatory but not SHM

The circular motion of a particle with constant speed is

• simple harmonic but not periodic

• periodic and simple harmonic

• neither periodic nor simple harmonic

• periodic but not simple harmonic

A particle executing simple harmonic motion of amplitude 5 cm has maximum speed of 31.4 cm/s. The frequency of its oscillation is

• 3 Hz

• 2 Hz

• 4 Hz

• 1 Hz

A particle executes simple harmonic oscillation with an amplitude $\mathrm{\alpha }$ . The period of oscillation is T. The minimum time taken by the particle to travel half of the amplitude from the equilibrium position is

• $\frac{\mathrm{T}}{4}$

• $\frac{\mathrm{T}}{8}$

• $\frac{\mathrm{T}}{12}$

• $\frac{\mathrm{T}}{2}$

Given that $\mathrm{y} = \mathrm{A} \sin \left[\left( \frac{2\mathrm{\pi }}{\mathrm{\lambda }} \left(\mathrm{ct} - \mathrm{x}\right) \right)\right]$ , where y and x are measured in metre. Which of the following statements is true?

• The unit of λ is same as that of x and A

• The unit of λ is same as that of x but not of A

• The unit of c is same as that of $\frac{2\mathrm{\pi }}{\mathrm{\lambda }}$

• The unit of ( ct - x ) is same as that of $\frac{2\mathrm{\pi }}{\mathrm{\lambda }}$

Two simple pendulums of length 0.5 m and 20 m respectively are given small linear displacement in one direction at the same time. They will again be in the phase when the pendulum of shorter length has
completed x oscillations, where k is

• 1

• 3

• 2

• 5

A mass of 1 kg is suspended from a spring of force constant 400 N, executing SHM total energy of the body is 2J, then maximum acceleration of the spring will be

• 4 m/s²

• 40 m/s²

• 200 m/s²

• 400 m/s²

Vibrations of rope tied by two rigid ends shown by equation y = cos 2πt sin2πx, then minimum length of the rope will be

• 1 m

• $\frac{1}{2}\mathrm{m}$

• 5 m

• 2πm

The angular amplitude of a simple pendulum is θ₀. The maximum tension in its string will be

• mg (1 - θ₀)

• mg (1 + θ₀ )

• $\mathrm{mg} \left(1- {\mathrm{\theta }}_0^2\right)$

• $\mathrm{mg} \left(1 + {\mathrm{\theta }}_0^2\right)$

A particle is subjected to two simple harmonic motions along X-axis while other is along a line making angle 45° with the X-axis. The two motions are given by   x = x₀ sin ωt  and  s =s₀ sin ωt. The amplitude of resultant motion is 

• x₀ + s₀ + 2x₀s₀

• $\sqrt{{\mathrm{x}}_0^2 + {\mathrm{s}}_0^2}$

• $\sqrt{{\mathrm{x}}_0^2 - {\mathrm{s}}_0^2 + 2{\mathrm{x}}_0 {\mathrm{s}}_0}$

• ${\left[{\mathrm{x}}_0^2 + {\mathrm{s}}_0^2 + \sqrt{2} {\mathrm{x}}_0 {\mathrm{s}}_0\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

A pan with a set of weights is attached to a light spring. The period of vertical oscillation is 0.5s. When some additional weights are put in pan, then the period of oscillations increases by 0.1 s. The extension caused by the additional weight is

• 5.5 cm

• 3.8 cm

• 2.7 cm

• 1.3 cm

A particle is describing simple harmonic motion. If its velocities are v₁ and v₂ when the displacements from the mean position are y₁ and y₂ respectively, then its time period is

• $2\mathrm{\pi } \sqrt{\frac{{\mathrm{y}}_1^2 + {\mathrm{y}}_2^2}{{\mathrm{v}}_1^2 + {\mathrm{v}}_2^2}}$

• $2\mathrm{\pi } \sqrt{\frac{{\mathrm{v}}_2^2 - {\mathrm{v}}_1^2}{{\mathrm{y}}_1^2 - {\mathrm{y}}_2^2}}$

• $2\mathrm{\pi } \sqrt{\frac{{\mathrm{v}}_1^2 + {\mathrm{v}}_2^2}{{\mathrm{y}}_1^2 + {\mathrm{y}}_2^2}}$

• $2\mathrm{\pi } \sqrt{\frac{{\mathrm{y}}_1^2 - {\mathrm{y}}_2^2}{{\mathrm{v}}_2^2 - {\mathrm{v}}_1^2}}$

Assertion:  A particle in x-y plane is related by   x = $\mathrm{\alpha }$ sin t and  y =  $\mathrm{\alpha }$ (l - cos t), where $\mathrm{\alpha }$ and ω are constants, then the particle will have parabolic motion.

Reason:  A particle under the influence of two perpendicular velocities has parabolic motion.

• 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 body of mass 60 kg is suspended by means of three strings P Q and R as shown in the figure is in equilibrium. The tension in the string P is

• 130.9 g N

• 60 g N

• 50 g N

• 103.9 g N

The angular amplitude of simple pendulum is θ_o. The maximum tension in its string will be

• mg ( 1 $-$ θ₀ )

• mg ( 1 + cosθ₀ )

• mg ( 1 $-$ cos${\mathrm{\theta }}_0^2$ )

• mg ( 1 + cos${\mathrm{\theta }}_0^2$ )

The displacement of a particle executing SHM is given by y = 0.25 sin200t cm. The maximum speed of the particle is

• 200 cm s^-1

• 100 cm s^-1

• 50 cm s^-1

• 5.25 cm s^-1

A mass M is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes simple harmonic oscillations with a time period T. If the mass is increased by m, then the time period becomes $\left(\frac{5}{4} \mathrm{T}\right)$. The ratio of $\frac{\mathrm{m}}{\mathrm{M}}$ is

• 9/16

• 5/4

• 25/16

• 4/5

A wave is represented by the equation y= 0.5 sin(10t $-$ x) metre It is a travelling wave propagating along +x direction with velocity

• 10 m s^-1

• 20 m s^-1

• 5 m s^-1

• None of these

The frequency of oscillations of a mass m connected horizontally by a spring of spring constant k is 4 Hz. When the spring is replaced by two identical spring as shown in figure. Then the effective frequency is

• 4 $\sqrt{2}$

• 1.5

• 1.31

• 2$\sqrt{2}$

Assertion:  The graph of potential energy and kinetic energy of a particle in SHM with respect to position is a parabola. 

Reason:  Potential energy and kinetic energy do not vary linearly with position.

• 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.

Motion of an oscillating liquid in a U tube is

• periodic but not simple harmonic.

• non-periodic.

• simple harmonic and time period is independent of the density of the liquid.

• simple harmonic and time period is directly proportional to the density of the liquid.

The displacement x of a particle varies with time t as, $\mathrm{x} = {\mathrm{ae}}^{-\mathrm{\alpha t}} +\mathrm{b} {\mathrm{e}}^{ \mathrm{\beta t}}$ where a, b α and β are positive constants. The velocity of the particle will

• go on decreasing with time

• be independent of $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$

• drop to zero when $\mathrm{\alpha } = \mathrm{\beta }$

• go on increasing with time