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 the figure). The objects are released from rest at the position shown. Then,the tension in the cord just after release is

Three blocks of different masses connected with inextensible strings are pulled by a force $F$ on a frictionless surface as shown in the figure. The ratio of tensions $T_1$ to $T_2$ is

$A$ monkey of mass $m$ is climbing a rope hanging from the roof with acceleration $a$. The coefficient of static friction between the body of the monkey and the rope is $\mu$. Find the direction and value of the friction force on the monkey.

$A$ monkey of mass $20 \, kg$ climbs up a rope which can withstand a maximum tension of $25 \, kgf$. What is the maximum acceleration with which the monkey can climb up? (Take $g = 10 \, m/s^2$)

$A$ small block of mass $m$ is placed on a wedge of mass $M$ as shown,which is initially at rest. All the surfaces are frictionless. The spring attached to the other end of the wedge has a force constant $k$. If $a'$ is the acceleration of $m$ relative to the wedge as it starts coming down and $A$ is the acceleration acquired by the wedge as the block starts coming down,then the maximum retardation of $M$ is:

Two blocks of masses $m$ and $2m$ kept on a frictionless horizontal surface are connected by a massless string. Two horizontal forces $F_1 = (4.2t) \text{ N}$ and $F_2 = (7.5t) \text{ N}$,where '$t$' is time in seconds,are acting on the system as shown in the figure. The time at which the tension in the string between the two blocks becomes $10.6 \text{ N}$ is $t$ seconds.