A large number ( n ) of identical beads, each of mass m and radius...

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 large number ( n ) of identical beads, each of mass m and radius r are strung on a thin smooth rigid horizontal rod

Options

A $\dfrac{m{v}^{2}}{L - nr}$
B $\dfrac{m{v}^{2}}{L - 2nr}$
C\dfrac{m{v}^{2}}{2 $\left(L - nr\right)}$
D $Zero$

Correct answer

B. $\dfrac{m{v}^{2}}{L - 2nr}$

Step-by-step solution

Average time for

1

collision to occur on

1

wall,~

= {t}_{avg} = \dfrac{averagedistancetravelled}{speed}= \dfrac{2\left(L - 2nr\right)}{v}

⇒ {t}_{avg} = \dfrac{2\left(L - 2nr\right)}{v}

. Change in momentum in~

1

collision

= 2mv

. Average force on each support,~

{f}_{avg} = \dfrac{2mv}{tavg} = \dfrac{2mv^{2}}{2\left(L - 2nr\right)}⇒ {f}_{avg} = \dfrac{mv^{2}}{L - 2nr}

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