First slide

Conservation of mechanical energy

Question

In the figure shown, the cart of mass 6m is initially at rest. A particle of mass m is attached to the end of the light rod which can rotate freely about A. If the rod is released from rest in a horizontal position shown, determine the velocity v_rel of the particle with respect to the cart when the rod is vertical.

Moderate

A

$\sqrt{\frac{14}{3} gl}$
B

$\sqrt{\frac{7}{6} gl}$
C

$\sqrt{\frac{7}{3} gl}$
D

$\sqrt{\frac{8}{3} gl}$

Solution

When the rod becomes vertical
${mv}_{1} = 6 {mv}_{2} \Rightarrow v_{1} = 6 v_{2}$ ….(i)
$\frac{1}{2} {mv}_{1}^{2} + \frac{1}{2} 6 {mv}_{2}^{2} = mgl$ …(ii)
Solve (i) and (ii) to get
$v_{rel} = v_{1} + v_{2} = \sqrt{\frac{7}{3} gl}$

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