Question

Find the minimum coefficient of friction between the cylindrical shell and inclined plane as shown in the figure, so that the shell will perform pure rolling.

• A. $\frac{1}{2}\sin \theta$
• B. $\frac{1}{2}\cos \theta$
• C. $\frac{1}{2}\tan \theta$
• D. $\frac{1}{2}\cos \theta$

Answer 1

The correct option is C $\frac{1}{2}\tan \theta$


Let $f$ be the friction force and $N$ be the normal force acting on the shell.
For translation motion,
$-f+Mg\sin \theta =Ma....\left(I\right)$
[$a$ is the acceleration of shell in downward direction]
For rotational motion,
${\tau }_0=fR$
$\Rightarrow I_c\alpha =fR$
$\Rightarrow M{R}^2\left(\frac{a}{R}\right)=fR$
$\Rightarrow f=Ma....\left(II\right)$
[MOI of cylindrical shell $=M{R}^2$]
From eq (I) & (II),
$-f+Mg\sin \theta =f$
$\Rightarrow f=\frac{Mg}{2}\sin \theta$
Since ${\mu }_s\ge {\mu }_k\ge \mu$,
$\Rightarrow f_{s\left(max\right)}\ge f_k\ge f$
$\left[\begin{array}{c}{f}_s=\text{Static friction}\\ f_k=\text{kinetic friction}\end{array}\right]$
$\Rightarrow \mu Mg\cos \theta \ge \frac{1}{2}Mg\sin \theta$
or $\mu \ge \frac{1}{2}\tan \theta$