First slide

Rolling motion

Question

The following figure shows two situations in which a uniform round rigid body is released from rest from the positions shown, such that it is just able to loop the loop without leaving contact with the track. Assuming that radius of the track is large in comparison to the radius of round body, the ratio h₁/h₂

Moderate

A

must be greater than 1
B

must be less than 1
C

must be equal to 1
D

can be greater than or less than 1, depending on the moment of inertia of the round body.

Solution

Case-1 : ${mgh}_{1} = (KE)_{trans} = K_{trans} = \frac{1}{2} {mv}^{2}$
Whole gravitational potential energy is converted into translational K.E. only.
Case-2 : ${mgh}_{2} = K_{trans} + K_{rot} = \frac{1}{2} {mv}^{2} + \frac{1}{2} I_{cm} ω^{2}$
Some part of gravitational potential energy will be converted into rotational K.E. also.
Hence in second case, greater height is required.

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