Question

A rough horizontal plate rotates with an angular velocity $\omega$ about a fixed vertical axis. A particle of mass $m$ lies on the plate at a distance $\frac{5a}{4}$ from this axis. The coefficient of friction between the plate and the particle is $\frac{1}{3}$. The largest value of ${\omega }^2$ for which the particle will continue to be at rest on the revolving plate is

• A. $\frac{g}{3a}$
• B. $\frac{4g}{5a}$
• C. $\frac{4g}{9a}$
• D. $\frac{4g}{15a}$

Answer 1

The correct option is D $\frac{4g}{15a}$

Since friction provides necessary centripetal force
$f_s=\frac{m{v}^2}{r}=mr{\omega }^2$

But $f_s={\mu }_{s}R={\mu }_{s}mg$ (limiting case, when $\omega$ is maximum)
$\Rightarrow {\mu }_{s}mg=mr{\omega }^2$
$\Rightarrow {\mu }_{s}g=r{\omega }^2$
$\Rightarrow {\omega }_{max}^2=\frac{{\mu }_{s}g}{r}=\frac{\left(\frac{1}{3}\right)\times g}{\frac{5a}{4}}=\frac{4g}{15a}$