Question

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)

• A. $\frac{m{v}^2}{L-nr}$
• B. $\frac{m{v}^2}{L-2nr}$
• C. $\frac{m{v}^2}{2(L-nr)}$
• D. $zero$

Answer 1

The correct option is B $\frac{m{v}^2}{L-2nr}$

Total length available for collision is $L-nr$
Assuming the collision be between beads be elastic, each bead comes to rest after collision and the collided beads moves with speed v.
Average time for a collision is $t_{avg}=\frac{L-2nr}{v}$ since a bead will collide with all other beads before returning to the same state( position and velocity ) from where it started.
The collision with a support will transfer momentum mv to the support.
The average force experienced by each support is $F=\frac{\Delta P}{\Delta T}=\frac{mv}{t_{avg}}=\frac{mv}{\frac{L-2nr}{v}}=\frac{m{v}^2}{L-2nr}$