Question

A carpet of mass M, made of an extensible material is rolled along its length in the form of a cylinder of radius R and kept on a rough floor. If the carpet is unrolled, without sliding to a radius R/2, the decrease in potential energy is

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

Answer 1

Answer: (B)

$\frac{7}{8}MgR$

Let $M_1$ be the mass of the carpet of radius $\frac{r}{2}$

Density of carpet remains the same for radius of $r$ and $\frac{r}{2}$.

$\rho =\frac{M}{\pi r^2}=\frac{M_1}{\pi (\frac{r}{2}{)}^2}⟹M_1=\frac{M}{4}$

Initial Potential energy, $U_1=Mgr$

Final Potential energy, $U_2=M_{1}g\frac{r}{2}=\frac{Mgr}{8}$

The decrease in the potential energy of the system

$\therefore \Delta U=U_1-U_2=Mgr-\frac{Mgr}{8}=\frac{7Mgr}{8}$

Hence, option $\left(B\right)$ is correct.