$A$ very broad elevator is going up vertically with a constant acceleration of $2\,m/s^2$. At the instant when its velocity is $4\,m/s$,a ball is projected from the floor of the lift with a speed of $4\,m/s$ relative to the floor at an elevation of $30^{\circ}$. The time taken by the ball to return to the floor is $..............\,s$ $(g=10\,m/s^2)$.

$A$ boy having a mass equal to $40 \, kg$ is standing in an elevator. The force felt by the feet of the boy (apparent weight) will be greatest when the elevator $(g = 9.8 \, m/s^2)$:

$A$ person sitting inside an elevator performs a weighing experiment with an object of mass $50 \ kg$. Suppose that the variation of the height $y$ (in $m$) of the elevator,from the ground,with time $t$ (in $s$) is given by $y = 8[1 + \sin(\frac{2 \pi t}{T})]$,where $T = 40 \pi \ s$. Taking acceleration due to gravity,$g = 10 \ m/s^2$,the maximum variation of the object's weight (in $N$) as observed in the experiment is $.....$ .

$A$ man of mass $70 \; kg$ stands on a weighing scale in a lift which is moving:
$(a)$ Upwards with a uniform speed of $10 \; m s^{-1}$.
$(b)$ Downwards with a uniform acceleration of $5 \; m s^{-2}$.
$(c)$ Upwards with a uniform acceleration of $5 \; m s^{-2}$.
What would be the readings on the scale in each case?
$(d)$ What would be the reading if the lift mechanism failed and it hurtled down freely under gravity?

$A$ plank is moving in a horizontal direction with a constant acceleration $a \hat{i}$. $A$ uniform rough cubical block of side $l$ rests on the plank and is at rest relative to the plank. Let the centre of mass of the block be at $(0, l/2)$ at a given instant. If $a = g/10$,then the normal reaction exerted by the plank on the block at that instant acts at

$A$ 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?