Question

A solid hemisphere rests in equilibrium on a rough ground and against a smooth wall. The curved surface touches the wall and the ground. The angle of inclination of the circular base to the horizontal is $30^{\circ }$ , The minimum coefficient of friction required between ground and hemisphere is $\mu $ . What is the value of $32\,\mu $ ?

Answer 1

First, make the free body diagram, of the solid hemisphere in equilibrium on rough ground and against the smooth wall,
then the equation from above FBD are,
$f=mg\times \frac{3 R}{8}\cos 60^{o}$
$N_{2} = f$
then normal reaction force,
$N_{1} = m g$
$f = \frac{3 m g}{16}$
$f \leq \mu N_{1}$
$\mu \geq \frac{3}{16}\Rightarrow 32\mu =32\times \frac{3}{16}=6$