A painter of mass $M$ stands on a platform of mass $m$ and pulls himself up by two ropes which hang over pulley as shown in the figure. He pulls each rope with force $F$ and moves upward with a uniform acceleration $a$. Find $a$, neglecting the fact that no one could do this for long time. 

A bead of mass $m$ is attached to one end of a spring of natural length $R$ and spring constant $K=\frac{(\sqrt{3}+1)mg}{R}$. The other end of the spring is fixed at a point $A$ on a smooth vertical ring of radius $R$ as shown in the figure. The normal reaction at $B$ just after it is released to move is 

Two objects $A$ and $B$, each of mass $m$, are connected by a light inextensible string. They are restricted to move on a frictionless ring of radius $R$ in a vertical plane (as shown in figure). The objects are released from rest at the position shown. Then the tension in the cord just after release is

If block $A$ is moving with an acceleration of $5 \mathrm{m} {\mathrm{s}}^{-2}$, the acceleration of $B$ with respect to ground is, 

Two uniform solid cylinders $A$ and $B$ each of mass $1 kg$ are connected by a spring of constant $200 {\mathrm{Nm}}^{-1}$ at their axles and are placed on a fixed wedge as shown in the figure. There is no friction between cylinders and wedge. The angle made by the line $AB$ with the horizontal, in equilibrium, is

The velocity of point $A$ on the rod is $2 \mathrm{m} {\mathrm{s}}^{-1}$ (leftwards) at the instant shown in the figure. The velocity of the point $B$ on the rod at this instant is

A fixed $U-$shaped smooth wire has a semi-circular bending between $A$ and $B$ as shown in figure. $A$ bead of mass $m$ moving with uniform speed $v$ through the wire enters the semicircular bend at $A$ and leaves at $B$. The average force exerted by the bead on the part $AB$ of the wire is 

A block of mass $m_1$ lies on the top of fixed wedge as shown in figure $\left(a\right)$and another block of mass $m_2$ lies on top of wedge which is free to move as shown in figure $\left(b\right)$. At time $t=0$, both the blocks are released from rest from a vertical height $h$ above the respective horizontal surface on which the wedge is placed as shown. There is no friction between block and wedge in both the figures. Let $T_1$and $T_2$ be the time taken by the blocks, respectively, to just reach the horizontal surface. Then