Question

Two blocks, $m_1=2kg$ and $m_2=4kg$, are connected with a light string that runs over a frictionless peg to a hanging block with a mass $M$ as shown in figure. The coefficient of sliding friction between block $m_2$ and the horizontal surface is ${\mu }_k=0.2$. The coefficient of static friction between the two blocks is $\mu s=0.4$. What is the maximum mass $M$ for the hanging block if the block $m_1$ is not to slip on block $m_2$ while $m_2$ is sliding over the surface?

Answer 1

Answer: 6.00

for not to slip $m_1$ pseudo force on $m_1$
should be equal to friction force $\left(f_s\right)$ on $m_1$.
$m_{1a}={\mu }_s{m}_{1g}$
$a={\mu }_{s}g$
$a=0.4\times 10$
$a=4m/s^2$
From fBD of $m_2$

Here $f_k={\mu }_k\left(m_1+m_2\right)g$
$T-f_k=\left(m_1+m_2\right)a$
$=0.2\times 6\times 10$
$T-12=6a=12N$
$T-12=6\times 4$
$T=36N$
For $M$
$Mg-T=Ma$
$10M=36+4M$
$6M=36$
$\Rightarrow M=6kg$