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Q.

A dumbell consisting of two masses m each, connected by a light rigid rod of length l, falls on two pads of equal height, one steel and the other brass through a height h. The coefficients of restitution are e₁ and e₂ (e₁< e₂). To what maximum height will the centre of mass of the dumbell rise after bouncing off the pads?

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a

$\frac{h}{e_{1} + e_{2}}$

b

$\frac{{(e_{1} + e_{2})}^{2}}{4}$

c

${(e_{1} + e_{2})}^{2} \frac{h}{4}$

d

$\frac{4 h}{{(e_{1} + e_{2})}^{2}}$

answer is B.

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Detailed Solution

The velocity at which the dumbell strikes the mats is

$v = \sqrt{2 gh}$

After contact, the rebound velocities of the masses will be ${ve}_{1}, {ve}_{2}$

The centre of mass of the system after collision will move up with velocity at
$\frac{(e_{1} + e_{2}) \sqrt{2 gh}}{2}$

Max height reached by the COM will be
$\Rightarrow 0 = {[\frac{(e_{1} + e_{2})}{2} \sqrt{2 gh}]}^{2} - 2 gh'$

$h' = \frac{{(e_{1} + e_{2})}^{2} h}{4}$

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A dumbell consisting of two masses m each, connected by a light rigid rod of length l, falls on two pads of equal height, one steel and the other brass through a height h. The coefficients of restitution are e1 and e2 (e1 < e2). To what maximum height will the centre of mass of the dumbell rise after bouncing off the pads?