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 $L ( L \gg r )$ and are at rest at random positions. The rod is mounted between two rigid supports. 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):
(1) $\frac { m v ^ { 2 } } { L - n r }$
(2) $\frac { m v ^ { 2 } } { L - 2 n r }$
(3) $\frac { m v ^ { 2 } } { 2 ( L - n r ) }$
(4) Zero

A proton of mass $m$ collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of $90 ^ { \circ }$ with respect to each other. The mass of unknown particle is:
(1) $\frac { \mathrm { m } } { \sqrt { 3 } }$
(2) $\frac { \mathrm { m } } { 2 }$
(3) 2 m
(4) m

A block of mass $\mathrm { m } = 1 \mathrm {~kg}$ slides with velocity $\mathrm { v } = 6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle $\theta$ before momentarily coming to rest. if the rod has mass $M = 2 \mathrm {~kg}$, and length $\ell = 1 \mathrm {~m}$, the value of $\theta$ is approximately (take $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)
(1) $63 ^ { \circ }$
(2) $55 ^ { \circ }$
(3) $69 ^ { \circ }$
(4) $49 ^ { \circ }$

Two particles of equal mass $m$ have respective initial velocities $u \hat { i }$ and $u \left( \frac { \hat { i } + \hat { j } } { 2 } \right)$. They collide completely inelastically. The energy lost in the process is:
(1) $\frac { 1 } { 3 } m u ^ { 2 }$
(2) $\frac { 1 } { 8 } m u ^ { 2 }$
(3) $\frac { 3 } { 4 } m u ^ { 2 }$
(4) $\sqrt { \frac { 2 } { 3 } } m u ^ { 2 }$

Eight equal drops of water are falling through air with a steady speed of $10 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$. If the drops coalesce, the new velocity is:-
(1) $16 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$
(2) $40 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$
(3) $5 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$
(4) $10 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$