Question

A block is in contact with the inner wall of a hollow cylindrical drum as shown in figure. The minimum angular velocity (in $\text{rad/s}$) needed for the cylinder to keep the block stationary, when the cylinder is vertical and rotating about its axis will be

Answer 1

Let us suppose angular velocity needed for the cylinder to keep the block stationary is $\omega$.
FBD and figure as per question -


For block, on applying $\sum F=ma$ along the vertical.
$f-mg=0$ [ for vertical equilibrium ]
$\Rightarrow f=mg$
$\Rightarrow \mu N=mg$
$\Rightarrow N=\frac{mg}{\mu }$ .........(1)
[considering limiting friction for minimum angular velocity]

Similarly, for horizontal direction
$N=F_c$
$\Rightarrow N=m{\omega }^{2}r$ [ for circular motion ]
$\Rightarrow \frac{mg}{\mu }=m{\omega }^{2}r$ [ from (1) ]
$\Rightarrow \omega =\sqrt{\frac{g}{r\mu }}$
$\Rightarrow \omega =\sqrt{\frac{10}{2\times 0.2}}$ [from figure]
$=5\text{rad/s}$