The upper half of an inclined plane of inclination $\theta$ is perfectly smooth,while the lower half is rough. $A$ block starting from rest at the top of the plane will again come to rest at the bottom if the coefficient of friction between the block and the lower half of the plane is given by:

If the net force on a system is zero,then:

$A$ disc of mass $100 \,g$ slides down from rest on an inclined plane of $30^{\circ}$ and comes to rest after travelling a distance of $1 \,m$ along the horizontal plane. If the coefficient of friction is $0.2$ for both inclined and horizontal planes, then the work done by the frictional force over the whole journey, approximately, is (Acceleration due to gravity, $g=10 \,ms^{-2}$) (in $\,J$)

$Assertion$: On a rainy day,it is difficult to drive a car or bus at high speed.
$Reason$: The value of the coefficient of friction is lowered due to the wetting of the surface.

Shown in the diagram is a system of two bodies,a block of mass $m$ and a disc of mass $4m$,held in equilibrium. If the string $3$ is burnt,find the acceleration of the disc. Neglect the masses of the pulleys $P$ and $Q$. The coefficient of friction between the block and the horizontal surface is $0.5$ and the friction between the disc and the string is zero. (Take $g = 10 \ m/s^2$)

$A$ block of mass $m$ is placed on a smooth horizontal surface. $A$ force making an angle $\theta$ with the horizontal starts acting on the block. The magnitude of the force is constant but its direction with the horizontal changes as $\theta = a + bs$,where $a$ and $b$ are constants and $s$ is the distance covered by the block. If $|F| = 2mb$,find the velocity of the block as a function of the angle $\theta$.