$A$ frictionless wire $AB$ is fixed on a sphere of radius $R$. $A$ very small spherical ball slips on this wire. The time taken by this ball to slip from $A$ to $B$ is

Two masses $M_1$ and $M_2$ are arranged as shown in the figure. Let '$a$' be the magnitude of the acceleration of the system. If the mass of $M_1$ is doubled and that of $M_2$ is halved,then the new acceleration of the system is (Treat all surfaces as smooth; masses of pulley and rope are negligible).

Block $A$ of weight $100 \, N$ rests on a frictionless inclined plane of slope angle $30^{\circ}$. $A$ flexible cord attached to $A$ passes over a frictionless pulley and is connected to block $B$ of weight $w$. Find the weight $w$ for which the system is in equilibrium.

$A$ truck of mass $30,000 \ kg$ moves up an inclined plane of slope $1$ in $100$ at a speed of $30 \ km/h$. The power of the truck is ................. $kW$ (given $g = 10 \ m/s^2$).

Two wooden blocks are moving on a smooth horizontal surface such that the block of mass $m$ remains stationary with respect to the block of mass $M$ as shown in the figure. The magnitude of force $P$ is:

As shown in the figure,two spheres are released from point $A$ along paths $AB$ and $AC$ respectively. If both surfaces are smooth,find the time taken by both spheres to reach the bottom of the incline. (Take $g = 10 \, m/s^2$)