Question

A hollow cylinder with inner radius R, outer radius 2R and mass M is rolling with speed of its axis v. $\frac{10+x}{16}M{v}^2$.

Find $x$.

Answer 1

For the condition of rolling $v=2R\omega$

MI of the cylinder is same as that of a disc.
Mass of the cylinder: $M=\pi \left(\right(2R{)}^2-R^2)=3\pi R^2$
MI of the hollow cylinder $I=\frac{M_{outer}(2R{)}^2}{2}-\frac{M_{inner}{R}^2}{2}=\pi (2R{)}^2\frac{(2R{)}^2}{2}-\pi R^2\frac{R^2}{2}=\frac{15}{2}\pi R^4=\frac{5}{2}M{R}^2$
KE: $K=K_{tr}+K_{rot}=\frac{1}{2}M{V}^2+\frac{1}{2}I{\omega }^2$
$=\frac{1}{2}M{V}^2+\frac{1}{2}\times \frac{5}{2}\times M{R}^2(\frac{V}{2R}{)}^2$
$=\frac{1}{2}M{V}^2+\frac{5}{16}M{V}^2$
$=\frac{13}{16}M{V}^2$
$=\frac{13}{16}M{V}^2$
$=\frac{x}{16}M{V}^2$
$\Rightarrow x=13$