A carpet of mass m made of inextensible material is rolled along its length in the form of a cylinder of radius r and kept on a rough floor. The decrease in the potential energy of the system, when the carpet is unrolled to a radius `(r )/(2)` without sliding is (g = acceleration due to gravity)

Options

A. `(3)/(4)mgr`
B. `(5)/(8)mgr`
C. `(7)/(8)mgr`
D. `(1)/(2)mgr`

Answer

C

Explanation

Density `sigma=(M)/(pir^(2))=(M_(1))/(pi((r)/(2))^(2))`
or `M_(1)=(M)/(4)`
Potential energy `U_(1)=Mgr`
and `U_(2)=M_(1)g""(r)/(2)=(M)/(4)g""(r)/(2)=(Mgr)/(8)`
The decrease in the potential energy of the system
`DeltaU=U_(1)-U_(2)`
`=Mgr-(Mgr)/(8)`
`=(7Mgr)/(8)`