Question

A uniform solid sphere rolls down a vertical surface without sliding. If the vertical surface moves with an acceleration $a=\frac{g}{2}$, If the minimum coefficient of friction between the sphere and vertical surfaces so as to prevent relative sliding is found to be $\frac{n}{7}$. The value of $n$ is ____.

Answer 1

Answer: 4

Let sphere rolls down with acceleration $a^{\prime }$, writing force and torque equations for sphere

$N=\frac{mg}{2}...(i)$
$mg-f=m{a}^{\prime }...(ii)$
$fR=\left(\frac{2}{5}m{R}^2\right)\cdot \alpha$
$f=\frac{2}{5}mR\alpha ...(iii)$
$\alpha R=a^{\prime }$ ...(iv)
$\Rightarrow f=\frac{2}{5}m{a}^{\prime }$
or $a^{\prime }=\frac{5}{2}\frac{f}{m}...(v)$
from (ii) and (v)
$mg-f=\frac{5}{2}f$
$\Rightarrow mg=\frac{7}{2}f$
or $f=\frac{2}{7}mg$
But, $f\le N$
hence $\frac{2}{7}mg\le \frac{\mu mg}{2}$
$\Rightarrow \mu \ge \frac{4}{7}$