Question

A spherical shell of radius $R$ is rolling down an incline of inclination $\theta$ without slipping as shown in figure. Find the minimum value of coefficient of friction.

• A. $\frac{2}{3}\tan \theta$
• B. $\frac{2}{7}\tan \theta$
• C. $\frac{2}{5}\tan \theta$
• D. $\tan \theta$

Answer 1

The correct option is C $\frac{2}{5}\tan \theta$

Forces acting on spherical shell shown in figure below,
Equation of the translation motion of spherical shell gives,
$mg\sin \theta -f=ma$ ...$\left(1\right)$
where, $m=$ mass of spherical shell
$f=$ friction
$a=$ acceleration
Eaquation of rotational motion of spherical shell gives,
${\tau }_0=f\times R$
$\Rightarrow I_0\alpha =fR$
$\Rightarrow \frac{2}{3}m{R}^2\left(\frac{a}{R}\right)=fR$
$\Rightarrow a=\frac{3f}{2m}$
Putting the value of $a$ in eq $\left(1\right)$
$mg\sin \theta -f=m\times \frac{3f}{2m}$
$\Rightarrow f=\frac{2}{5}mg\sin \theta$
We know that,
$f_R\le f_{s,max}$
$\Rightarrow \frac{2}{5}mg\sin \theta \le \mu N=\mu mg\cos \theta$
$\Rightarrow \mu \ge \frac{2}{5}\tan \theta$
For minimum coefficient of friction
${\mu }_{min}=\frac{2}{5}\tan \theta$

Why this question? Tips: Enhance the concept of rotational motion and translational motion simultaneously.