Physics - Rotation Question with Solution | TestHub

Let$\mu$be the friction coefficient between sphere and each wall. Free body diagram of sphere is



Net force on the sphere in horizontal diarection is zero.
$\therefore N_1\cos {60}^{\mathrm{o}}+\mu N_2\cos {60}^{\mathrm{o}}=N_2\cos {30}^{\mathrm{o}}+\mu N_1\cos {30}^{\mathrm{o}}$
$\Rightarrow N_1+\mu N_2=\sqrt{3}\left(N_2+\mu N_1\right)$
$\Rightarrow N_1\left(1-\sqrt{3}\mu \right)=N_2\left(\sqrt{3}-\mu \right)$
$\Rightarrow \frac{N_1}{N_2}=\frac{\sqrt{3}-\mu }{1-\sqrt{3}\mu }$
Substituting$\mu =\frac{1}{3}$we get,
$\Rightarrow \frac{N_1}{N_2}=\frac{\sqrt{3}-\frac{1}{3}}{1-\frac{\sqrt{3}}{3}}=\frac{3\sqrt{3}-1}{3-\sqrt{3}}=1+\frac{4}{\sqrt{3}}$
$\Rightarrow \frac{f_1}{f_2}=\frac{\mu N_1}{\mu N_2}=1+\frac{4}{\sqrt{3}}$