For the system shown in the figure,if $M_1 = 10 \ kg$,$M_2 = 5 \ kg$,$\theta = 30^\circ$,and $g = 10 \ m/s^2$,find the tension in the string in $N$.

$A$ football of radius $R$ is kept on a hole of radius $r$ (where the diameter of the hole is $2r$ and $r < R$) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted,making an angle $\theta$ with the horizontal as shown in the figure. The maximum value of $\theta$ such that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale) -

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).

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:

$A$ smooth inclined plane is inclined at an angle $\theta$ with the horizontal. $A$ body starts from rest and slides down the inclined surface. The time taken by it to reach the bottom is

$A$ body starts falling freely from height $H$ and hits an inclined plane in its path at height $h$ from the ground. As a result of this perfectly elastic impact,the direction of the velocity of the body becomes horizontal. The value of $\frac{H}{h}$ for which the body will take the maximum time to reach the ground is . . . . . . .