$A$ lift is tied with thick iron ropes having mass $M$. The maximum acceleration of the lift is $a \ m/s^2$ and the maximum safe stress is $S \ N/m^2$. The minimum diameter of the rope is

$A$ pulley fixed to the ceiling carries a string with blocks of mass $m$ and $3m$ attached to its ends. The masses of the string and pulley are negligible. When the system is released,what is the acceleration of its centre of mass?

If $m_1 = 4m_2$ and the acceleration of $m_1$ is $a$,find the tension $T$ in the string.

One end of a light string is fixed to a clamp on the ground and the other end passes over a fixed frictionless pulley as shown in the figure. It makes an angle of $30^{\circ}$ with the ground. The clamp can tolerate a vertical force of $40 \,N$. If a monkey of mass $5 \,kg$ were to climb up the rope,then the maximum acceleration in the upward direction with which it can climb safely is $\left(g=10 \,ms^{-2}\right)$ (in $\,ms^{-2}$)

Find the acceleration of block $B$.

Two blocks $A$ and $B$ of masses $1.5 \ kg$ and $0.5 \ kg$ respectively are connected by a massless inextensible string passing over a frictionless pulley as shown in the figure. Block $A$ is lifted until block $B$ touches the ground and then block $A$ is released. The initial height of block $A$ is $80 \ cm$ when block $B$ just touches the ground. The maximum height reached by block $B$ from the ground after the block $A$ falls on the ground is (in $cm$)