Question

In the figure shown mass of both, the spherical body and block is $m$. Moment of inertia of the spherical body about centre of mass is $2m{R}^2$. The spherical body rolls on horizontal surface. There is no slipping at any surfaces in contact. The ratio of kinetic energy of the spherical body to that of block is

• A. $3/4$
• B. $1/3$
• C. $2/3$
• D. $1/2$

Answer 1

The correct option is C $2/3$

Let velocity of spherical body be v' and that of block be v.

As there is no slipping at B, $⟹v^{\prime }=2Rw$

Also there is no slipping at A, thus $v^{\prime }+Rw=v$

$⟹v=3Rw$ and $v^{\prime }=\frac{2}{3}v$

Now K.E of spherical body, $E_s=\frac{1}{2}m(v^{\prime }{)}^2+\frac{1}{2}I{w}^2$

$E_s=\frac{1}{2}m(v^{\prime }{)}^2+\frac{1}{2}2m{R}^2{w}^2$

$E_s=\frac{1}{2}m(v^{\prime }{)}^2+\frac{1}{4}m(v^{\prime }{)}^2=\frac{3}{4}m(v^{\prime }{)}^2$

$E_s=\frac{3}{4}m(\frac{2}{3}v{)}^2=\frac{1}{3}m{v}^2$

K.E of block, $E_b=\frac{1}{2}m{v}^2$

$\frac{E_s}{E_b}=\frac{2}{3}$