Question

A disc of mass $M$ & radius $R$ is placed a rough horizontal surface with its axis horizontal. A light rod of length '$2R$' is fixed to the disc at point '$A$' as shown in figure and a force $\frac{3}{2} M g$ is applied at the other end. If disc starts to roll without slipping find the value of " $10 \times \mu_{\min } "$ where $\mu_{\min }$ is minimum coefficient of friction $b / w$ disc & horizontal surface required for pure rolling-

Answer 1

$\tau$ at $\quad P =\frac{3 Mg }{2} \times 2 R$
$=3 MgR$
$3 MgR =\left(\frac{ MR ^{2}}{2}+ MR ^{2}\right) \alpha$
$\alpha=\frac{2 MgR }{ MR ^{2}}$
$\alpha=\frac{2 g }{ R }$
$a-\alpha R=0$
$ a =2 g $
$ f =M a $
$ f =2 M g$
$\frac{3}{2} Mg + Mg = N$
$N=\frac{5}{2} M g$
$f \leq \mu N$
$2 Mg \leq \mu \frac{5}{2} Mg$
$\mu \geq \frac{4}{5}$
$\mu_{\min }=\frac{4}{5}$
$10 \times \mu_{\min } \Rightarrow 10 \times \frac{4}{5}$
$ \Rightarrow 8$