Question

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane
$\left(i\right)$ a ring of radius $R$,
$\left(ii\right)$a solid cylinder of radius $\frac{R}{2}$ and
$\left(iii\right)$ a solid sphere of radius $\frac{R}{4}$.
If in each case, the speed of the center of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is

• A. $4:3:2$
• B. $2:3:4$
• C. $10:15:7$
• D. $14:15:20$

Answer 1

The correct option is D $14:15:20$

Given:
$\left(i\right)$ a ring of radius $R$,
$\left(ii\right)$a solid cylinder of radius $\frac{R}{2}$ and
$\left(iii\right)$ a solid sphere of radius $\frac{R}{4}$

Energy conservation
$mgh=\frac{1}{2}m{v}_{cm}^2+\frac{1}{2}{I}_{cm}{\omega }^2$
$mgh=\frac{1}{2}m{v}_{cm}^2+\frac{1}{2}{I}_{cm}{\left(\frac{v_{cm}}{R}\right)}^2=\frac{1}{2}(m+\frac{I_{cm}}{R^2})v_{cm}^2$
$mgh=\frac{m}{2}[1+\frac{K^2}{R^2}]v_{cm}^2$
$\therefore h\propto [1+\frac{K^2}{R^2}]$

For ring $K=R$
$\Rightarrow h\propto 2$

For solid cylinder $K=\frac{R}{\sqrt{2}}$
$\Rightarrow h\propto \frac{3}{2}$

For solid sphere: $K=\sqrt{\frac{2}{5}}R$
$\Rightarrow h\propto \frac{7}{5}$

Ratio of heights $2:\frac{3}{2}:\frac{7}{5}\Rightarrow 20:15:14$
Hence option $\left(C\right)$ is correct.