Question

A solid sphere, a hollow sphere and a disc, all having the same mass and radius, are placed at the top of an incline and released. The friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling. The smallest kinetic energy at the bottom of the incline will be achieved by

• A. the solid sphere
• B. the hollow sphere
• C. the disc
• D. all will achieve the same kinetic energy

Answer 1

The correct option is B the hollow sphere

Work done by friction force, $W_f=\int f{v}_{A}dt$

$W_f=\left(\mu mgcos\theta \right)\int (v-rw)dt$ .............(1)

$a=gsin\theta -\mu gcos\theta$

$v=0+at=$ $(gsin\theta -\mu gcos\theta )t$ .........(a)

Also $fr=I\alpha$

$w=0+\alpha t=\frac{fr}{I}t$ ............(b)

$⟹$ $W_f=f\int$ $\left[\right(gsin\theta -\mu gcos\theta )t$ $-\frac{f{r}^2}{I}t]dt$

$W_f=$ $\left[\right(gsin\theta -\mu gcos\theta )$ $-\frac{f{r}^2}{I}]\frac{f{t}^2}{2}$

Work -energy theorem, $mgh-W_f=K.E$

$⟹mgh+\left[\right(-gsin\theta +\mu gcos\theta )$ $+\frac{f{r}^2}{I}]\frac{f{t}^2}{2}=K.E$

$K.E$ is minimum for which $I$ is maximum.

Hence $K.E$ is the smallest for hollow sphere.