$A$ light string fixed at one end to a clamp on the ground passes over a fixed pulley and hangs on the other side. It makes an angle of $30^{\circ}$ with the ground. $A$ monkey of mass $5 \ kg$ climbs up the rope. The clamp can tolerate a vertical force of $40 \ N$ only. The maximum acceleration in the upward direction with which the monkey can climb safely is ............ $m/s^2$ (neglect friction and take $g = 10 \ m/s^2$).

In the system shown in the figure,the pulleys and strings are ideal. Find the acceleration of $m_1$ with respect to $m_2$ $(m_1 = 2\ kg, m_2 = 2\ kg)$.

In the figure,the mass of a ball is $\frac{9}{5}$ times the mass of the rod. The length of the rod is $1 \, m$. The initial level of the ball is the same as the lower end of the rod. Find the time taken by the ball to reach the upper end of the rod. (in $s$)

Two men of unequal masses hold on to the two sections of a light rope passing over a smooth light pulley. Which of the following are possible?

Two weights of mass $m_{1}$ and $m_{2} (> m_{1})$ are joined by an inextensible string of negligible mass passing over a fixed frictionless pulley. The magnitude of the acceleration of the loads is

Two particles of mass $m$ each are tied at the ends of a light string of length $2a$. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance $a$ from the center $P$ (as shown in the figure). Now,the mid-point of the string is pulled vertically upwards with a small but constant force $F$. As a result,the particles move towards each other on the surface. The magnitude of acceleration,when the separation between them becomes $2x$ is