Question

A non homogenous solid cylinder of radius $r$ is placed on rough inclined plane of angle of inclination ${60}^o$. The angle of repose corresponds to ${30}^o$. The maximum radius of gyration, in order to prevent slipping is :

• A. $\frac{r}{\sqrt{2}}$
• B. $\frac{r}{\sqrt{3}}$
• C. $\sqrt{\frac{2}{3}}r$
• D. $\frac{r}{2}$

Answer 1

The correct option is B $\frac{r}{\sqrt{3}}$

Let the friction be $f$,
$\mu =tan{30}^o$

$f=\mu mgcos{60}^o$

Lets say the Moment of inertia of system be $I$

$Fr=I\alpha =\mu mgcos{60}^o\times r$...(1)

$mgsin{60}^o-\mu mgcos{60}^o=m{a}_{cm}$ ...(2)

$a_{cm}=\alpha r$...(3)

on solving above 3 equation we get

$I=\frac{m{r}^2}{3}$

$I=m{k}^2=\frac{m{r}^2}{3}$

$k=\frac{r}{\sqrt{3}}$