A 1990 JEE problem

Classical Mechanics Level 3

A carpet of mass M made of inextensible material is rolled along its length in the form of a cylinder of radius R and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a negligible small push is given to it. Calculate the horizontal velocity of the axis of the cylinder part of the carpet when its radius is reduced to R/2.(TAKE THE VALUE OF g(gravitational acceleration) TO BE 10 m/s^2 AND THE RADIUS R TO BE 3 m).Give your answer up to two decimal places.


The answer is 11.83.

Ritwik Jain by ritwik jain , 18, India

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1 solution

Steven Chase
May 28, 2017

Find the difference in gravitational potential energy and equate it to the kinetic energy (translation + rotation):

U i − U f = M g R − M 4 g R 2 = 7 8 M g R 7 8 M g R = 1 2 M 4 v 2 + 1 2 ( M / 4 ) ( R / 2 ) 2 2 ( v ( R / 2 ) ) 2 7 8 ( 10 ) ( 3 ) = v 2 ( 1 8 + 1 16 ) v ≈ 11.83 \large{U_i - U_f = MgR - \frac{M}{4} g \frac{R}{2} = \frac{7}{8} MgR \\ \frac{7}{8} MgR = \frac{1}{2} \frac{M}{4} v^2 + \frac{1}{2} \frac{(M/4) (R/2)^2}{2} \Big(\frac{v}{(R/2)}\Big)^2 \\ \frac{7}{8} (10)(3) = v^2 \Big(\frac{1}{8} + \frac{1}{16}\Big) \\ v \approx \boxed{11.83}}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

How can we say that the energy is conserved here? What will happen to the energy when the complete carpet opens up?

Rohit Gupta - 4 years ago

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You are right. This problem is non-physical because it neglects the carpet thickness. It would be interesting to re-evaluate it without this oversimplification.

Steven Chase - 4 years ago

It neglects the gravitational potential energy of the carpet that has been laid down.

Steven Chase - 4 years ago

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