Question

A solid cylinder starts rolling from a height $h$ on an inclined plane. At some instant $t$, the ratio of its rotational $K.E.$ and the total $K.E.$ would be:

• A. $1:2$
• B. $1:3$
• C. $2:3$
• D. $1:1$

Answer 1

The correct option is B $1:3$

The cylinder has rotational $KE=K_1=\frac{1}{2}I{w}^2$

where $I=\frac{M{R}^2}{2}$ about $COM$ and $w=\frac{V}{R}$.

$\Rightarrow K_1=\frac{1}{2}\left(\frac{M{R}^2}{2}\right){\left(\frac{V}{R}\right)}^2=\frac{1}{4}M{V}^2$

If it has a velocity of $COM$, then translatory $KE$ is $K_2=\frac{1}{2}M{V}^2$

$\therefore$ Total $KE=K=K_1+K_2=\frac{2}{4}M{V}^2$

Required ratio

$\frac{K_1}{K}=\frac{1/4M{V}^2}{3/4M{V}^2}=\frac{1}{3}$