Question

Find the maximum value of mass of block $C$ in kg so that neither $A$ nor $B$ moves, ($m_A=100\text{kg},m_B=200\text{kg}$). Pulleys and strings are massless. Coefficient of friction between $A$ and $B$ and between $B$ and horizontal surface is $\mu =0.4$.

(Take $g=10{\text{ms}}^{-2}$)

Answer 1

Maximum friction that can be obtained between $A$ and $B$ is
$f_1=\mu m_{A}g=\left(0.4\right)\left(100\right)\left(10\right)=400\text{N}$
and maximum friction between $B$ and ground is
$f_2=\mu (m_A+m_B)g=\left(0.4\right)(100+200)\left(10\right)=1200\text{N}$
Drawing diagrams of $A,B$ and $C$ in limiting case,

Equilibrium of $A$ gives $T_1=f_1=400\text{N}$ ... (i)
Equilibrium of $B$ gives
$2T_1+f_1+f_2=T_2\Rightarrow T_2=2\left(400\right)+400+1200=2400\text{N}$ ... (ii)
and equilibrium of $C$ gives
$m_{C}g=T_2\Rightarrow 10m_C=2400$
$\therefore m_C=240\text{kg}$