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Motion In A Plane

Question
CBSEENPH11020545
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What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?

  • square root of 2 gR end root
  • square root of 3 gR end root
  • square root of 5 gR end root
  • square root of gR

Solution

C.

square root of 5 gR end root

The question is illustrated in the figure below,

Let, the tension at point A be TA.
Using Newton's second law, we have
straight T subscript straight A space minus space mg space equals space mv subscript straight c squared over straight R
Energy at point A = 1 half m v subscript o squared space space space space space space space space space... space left parenthesis i right parenthesis
Energy at point C is,
1 half m v subscript c squared space plus space m g space x space 2 R space space space space space space space space space space... space left parenthesis i i right parenthesis
At point C, using Newton's second law,
straight T subscript straight c space plus space mg space equals space mv subscript straight c squared over straight R
In order to complete a loop, Tc greater-than or slanted equal to space 0
So,
 mg space equals space mv subscript straight c squared over straight R rightwards double arrow space straight v subscript straight c space equals space square root of gR space space space space space space space space... space left parenthesis iii right parenthesis
From equation (i) and (ii)
Using the principle of conservation of energy,
1 half m v subscript o squared space equals space 1 half m v subscript c squared space plus space 2 space m g R rightwards double arrow space 1 half m v subscript o squared italic space italic equals italic space italic 1 over italic 2 m g R italic space italic plus italic space italic 2 m g R italic space x italic space italic 2 italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space open square brackets v subscript c italic space italic equals italic space square root of g R end root close square brackets italic rightwards double arrow italic space v subscript o to the power of italic 2 italic space italic equals italic space g R italic space italic plus italic space italic 4 g R italic rightwards double arrow italic space italic space v subscript o italic space italic equals italic space square root of italic 5 g R end root

Some More Questions From Motion in A Plane Chapter

Is a physical quantity, having a direction, necessarily a vector?

Name a quantity which is both scalar as well as vector.

Which of the followings is/are vector quantities: work, power, acceleration, mass, momentum?

Pick up the odd one from the following: force, linear momentum, mass, impulse.

Pick out the only vector quantities in the following list:
Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction and charge.

Pick out the two scalar quantities in the following list:
force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, reaction as per Newton’s third law, relative velocity.

What are polar vectors?

Give two examples of polar vectors.

What are axial vectors?

Give two examples of axial vectors.

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