Question

Two blocks ($m$ and $M$) are arranged as shown. There is friction between ground and $M$ only and other surfaces are frictionless. The coefficient of friction between $M$ and ground is $0.25$. Find the maximum ratio of $m$ and $M\text{i.e.}\left(m/M\right)$ so that the system remains at rest

Answer 1

The FBDs are shown below

As system is at rest
For $m:T=mg$
For $M:T=f(\sum F_x=0)$
$N=T+Mg(\sum F_y=0$)
and, $f\le \mu N$
Hence, we have from above mentioned equations
$T\le \mu N\Rightarrow T\le \mu (m+M)g$
as $T=mg$
$\Rightarrow mg\le \mu (m+M)g$
$\Rightarrow m-\mu m\le \mu M$
$\Rightarrow m(1-\mu )\le \mu M$
$\Rightarrow \frac{m}{M}\le \frac{\mu }{1-\mu }\Rightarrow \frac{m}{M}\le \frac{0.25}{1-0.25}$
$\Rightarrow \frac{m}{M}\le \frac{1}{3}=0.33$
Hence, the maximum ratio is $0.33$