Question

The block $m$ is falling down. The coefficient of friction between the two surfaces is $\mu$. Find minimum $F$ to prevent $m$ from falling down.

Answer 1

Here, $N_2$ decide the frictional force to prove falling of $m$.

because $N_1>N_2$

Let acceleration of whole system system is $a$

$a=\frac{F}{2m+m+2m}=\frac{F}{5m}$

for $2m$ mass block,

$N_2=2ma$

$N_2=2m\times \frac{F}{5m}=\frac{2F}{5}$

for $m$ mass block,

$2f=mg$

$2\mu N_2=mg$

$2\mu \times \frac{2F}{5}=mg$

$F=\frac{5mg}{4\mu }$

$2\mu mg=M(1-\mu )$

$M=\frac{2\mu mg}{(1-\mu )}$