In the system shown, all the surfaces are frictionless while pulley and string are massless. The mass of block $A$ is $2m$ and that of block $B$ is $m$. The acceleration of block $B$ immediately after system is released from rest is

A pendulum of mass $m$ hangs from a support fixed to a trolley. The direction of the string when the trolley rolls up a plane of inclination $\alpha$ with acceleration $a_0$ is

The acceleration of the block $B$ in figure, assuming the surfaces and the pulleys $P_1$and $P_2$ are all smooth, is

Two small rings $O$ and $O\text{'}$ are put on two vertical stationary rods $AB$ and $A\text{'}B\text{'}$, respectively. One end of an inextensible thread is tied at points $A\mathit{\text{'}}$. The thread passes through ring $O\text{'}$and its other end is tied to ring $O$. Assuming that ring $O\text{'}$ moves downwards at a constant velocity $v_1$, then velocity $v_2$ of the ring $O$, when $\angle AOO\text{'}=\alpha$, is

A sphere of mass $m$ is kept between two inclined walls, as shown in the figure. If the coefficient of friction between each wall and the sphere is zero, then the ratio of normal reaction $\left(\frac{N_1}{N_2}\right)$ offered by the walls $1$ and $2$ on the sphere will be

In the figure shown, all pulleys are massless and frictionless. The time taken by the ball to reach the upper end of the rod is