A block of mass $10\mathrm{ }\mathrm{k}\mathrm{g}$ is in contact against the inner wall of a hollow cylindrical drum of radius$1 \mathrm{m}$. The coefficient of friction between the block and the inner wall of the cylinder is $0.1$. The minimum angular velocity needed for the cylinder to keep the block stationary when the cylinder is vertical and rotating about its axis will be  $\left(g=10\mathrm{m} {\mathrm{s}}^{-2}\right)$,

• A

  $\sqrt{10}\mathrm{r}\mathrm{a}\mathrm{d} {\mathrm{s}}^{-1}$
• B

  $\frac{10}{2\mathrm{\pi }}\mathrm{r}\mathrm{a}\mathrm{d} {\mathrm{s}}^{-1}$
• Correct answer
  C

  $10\mathrm{r}\mathrm{a}\mathrm{d} {\mathrm{s}}^{-1}$
• D

  $10\mathrm{\pi }\mathrm{r}\mathrm{a}\mathrm{d} {\mathrm{s}}^{-1}$

Solution

​​​​​​​
$N=m{\omega }^{2}R$
$f=mg$
For minimum angular velocity, 
$f=\mu N$
$mg=\mu ·m{\omega }^2\mathrm{R}$
$\omega =\sqrt{\frac{g}{\mu R}}=\sqrt{\frac{10}{0.1\times 1}}=10\mathrm{r}\mathrm{a}\mathrm{d} {\mathrm{s}}^{-1}$

Hence, the correct answer is option: 'C'.