How Does the Velocity of a Carpet's Axis Change as It Unrolls?

[AlchemistK]

Homework Statement

A carpet of mass M made of in-extensible material is rolled along its length in the form of a cylinder of radius R an 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 cylindrical part of the carpet when its radius reduces to R/2

The Attempt at a Solution

I tried conserving energy since friction won't be doing any work, my answer came out to be √(4gR/3)

The answer in the book is √(14gR/3). Is anyone getting this answer?

(Also, will angular momentum be conserved? I'm thinking yes.)

 

[tiny-tim]

Hi AlchemistK!

The answer in the book is √(14gR/3). Is anyone getting this answer?

Yup!

Show us what you did. ​

 

[LawrenceC]

Hint: Careful with kinetic energy considerations.

 

[AlchemistK]

x.x I got the answer, I made the mistake of assuming that since the radius dropped by half, even the mass dropped by half, while in fact it drops by a factor of 4.
Thanks for your help! I'd never have even tried it again if didn't get to know that the answer was correct.