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

Let sphere rolls down with acceleration a'. Writing force and torque equations for sphere

$\begin{array}{l}N=\frac{mg}{2} .....\left(i\right)\\ mg-f=m{a}^{\mathrm{\prime }} ......\left(ii\right)\\ fR=\left(\frac{2}{5}m{R}^2\right)\cdot \alpha \\ f=\frac{2}{5}mR\alpha .......\left(iii\right)\\ \alpha R=a^{\mathrm{\prime }} .......\left(iv\right)\end{array}$

$\begin{array}{l}\Rightarrow f=\frac{2}{5}m{a}^{\mathrm{\prime }}\text{ or }{a}^{\mathrm{\prime }}=\frac{5}{2}\frac{f}{m} ........\left(v\right)\\ \text{ from (ii) and (v) }\\ mg-f=\frac{5}{2}f\Rightarrow mg=\frac{7}{2}f\text{ or }f=\frac{2}{7}mg\\ \text{ But, }f\le N\text{ hence }\frac{2}{7}mg\le \frac{\mu mg}{2}\Rightarrow \mu \ge \frac{4}{7}\end{array}$