$A$ rough vertical board has an acceleration $a$ so that a $2 \ kg$ block pressing against it does not fall. The coefficient of friction between the block and the board should be

$A$ solid sphere of mass $2\, kg$ is resting inside a cube as shown in the figure. The cube is moving with a velocity $\vec v = (5t\,\hat i + 2t\,\hat j)\,m/s$. Here $t$ is the time in seconds. All surfaces are smooth. The sphere is at rest with respect to the cube. Find the total force $N$ exerted by the sphere on the cube. (Take $g = 10\, m/s^2$)

An object dropped in a stationary lift takes time $t_1$ to reach the floor. It takes time $t_2$ when the lift is moving up with constant acceleration. Then

$A$ box of mass $m \, kg$ is placed on the rear side of an open truck accelerating at $4 \, m/s^2$. The coefficient of friction between the box and the surface below it is $0.4$. The net acceleration of the box with respect to the truck is zero. The value of $m$ is $[g = 10 \, m/s^2]$.

$A$ smooth sphere of radius $R$ moves along a straight line with a constant acceleration $a = g$. $A$ particle is placed at the top of the sphere. It is released from there with zero velocity relative to the sphere. What will be its speed relative to the sphere as a function of the angle $\theta$ as the particle slides?

$A$ man is standing on a weighing machine placed in a lift. When stationary,his weight is recorded as $40 \, kg$. If the lift is accelerated upwards with an acceleration of $2 \, m/s^2$,then the weight recorded in the machine will be .......... $kg$ $(g = 10 \, m/s^2)$.