A large number $(n)$ of identical beads, each of mass $m$ and radius $r$ are strung on a thin smooth rigid horizontal rod of length $L\, (L >> r)$ and are at rest at random positions. The rod is mounted between two rigid supports (see figure) . If one of the beads is now given a speed $v$, the average force experienced by each support after a long time is (assume all collisions are elastic)

There are three forces $\vec {F_1}$, $\vec {F_2}$ and $\vec {F_3}$ acting on a body, all acting on a point $P$ on the body. The body is found to move with uniform speed.

$(a)$ Show that the forces are coplanar.

$(b)$ Show that the torque acting on the body about any point due to these three forces is zero.

An object of mass $5\, kg$ is thrown vertically upwards from the ground. The air resistance produces a constant retarding force of $10\, N$ throughout the motion. The ratio of time of ascent to the time of descent will be equal to $\left[\right.$Use $g =10 \,ms ^{-2}]$

In the given arrangement, the normal force applied by block on the ground is

The $50\,kg$ homogeneous smooth sphere rests on the $30^{\circ}$ incline $A$ and bears against the smooth vertical wall $B$. Calculate the contact forces at $A$ and $B.$

Define $SI$ unit of force $N$. Define $CGS$ unit of force dyne.