Question

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 $\frac{r}{2}$ without sliding is ($g=$ acceleration due to gravity)

• A. $\frac{3}{4}mgr$
• B. $\frac{5}{8}mgr$
• C. $\frac{7}{8}mgr$
• D. $\frac{1}{2}mgr$
• E. $\frac{1}{8}mgr$

Answer 1

Answer: (C)

$\frac{7}{8}mgr$

Density $\sigma =\frac{M}{\pi r^2}=\frac{M_1}{\pi {\left(\frac{r}{2}\right)}^2}$
or $M_1=\frac{M}{4}$
Potential energy $U_1=Mgr$
and $U_2=M_{1}g\frac{r}{2}=\frac{M}{4}g\frac{r}{2}=\frac{Mgr}{8}$
The decrease in the potential energy of the system
$\Delta U=U_1-U_2$
$=Mgr-\frac{Mgr}{8}$
$=\frac{7Mgr}{8}$