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)

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 (r/2) without sliding is (g= acceleration due to gravity) (A) (3/4)mgr (B) (5/8)mgr (C) (7/8)mgr (D) (1/2)mgr

Tardigrade Admin · Accepted Answer

The centre of mass of the whole carpet is originally at a height R above the floor. When the carpet unrolls itself and has a radius R/2, the centre of mass is at a height R/2. The mass left over unrolled is

\frac{M π ( R /2 ) ^{2}}{π R ^{2}} = \frac{M}{4}

Decrease in potential energy

M g R - (\frac{M}{4}) g (\frac{R}{2}) = \frac{7}{8} M g R

Density

σ = \frac{M}{π r _{2}} = \frac{M _{1}}{π ( \frac{r}{2} ) ^{2}}

or

M_{1} = \frac{M}{4}

Potential energy

U_{1} =

M g r
and

U_{2} = M_{1} g \frac{r}{2} = \frac{M}{4} g \frac{r}{2}

The decrease in the potential energy of the system

Δ U = U_{1} - U_{2}

= M g r - \frac{M g r}{8}

= \frac{7 M g r}{8}

$\frac{3}{4} m g r$

$\frac{5}{8} m g r$

$\frac{7}{8} m g r$

$\frac{1}{2} m g r$

$\frac{1}{8} m g r$

Q. 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 2r​ without sliding is (g= acceleration due to gravity)

43​mgr

85​mgr

87​mgr

21​mgr

81​mgr

$\frac{3}{4} m g r$

$\frac{5}{8} m g r$

$\frac{7}{8} m g r$

$\frac{1}{2} m g r$

$\frac{1}{8} m g r$