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

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

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