A uniform rod of mass m and length L is hinged at one end and free to rotate in the horizontal plane. All the surface are smooth. A particle of same mass m collides with the rod perpendicular to the l

Question

A uniform rod of mass m and length L is hinged at one end and free to rotate in the horizontal plane. All the surface are smooth. A particle of same mass m collides with the rod perpendicular to the length of rod with a speed V 0 . The coefficient of restitution for the collision is e = 1 2 . If hinge reaction during the collision is zero then the value of x is:

TestHub · Accepted Answer

Angular momentum conservation about the hinge

$m v_{0} x = m v_{1} x + \frac{m L^{2}}{3} ω$

$v_{0} x = v_{1} x + \frac{ω L^{2}}{3}$ ...(i)

$e = \frac{ω x - v_{1}}{v_{0}} = \frac{1}{2}$ ...(ii)

Impulse of hinge

$J = m ω \frac{L}{2} + m v_{1} - m v_{0}$

$J = m ω \frac{L}{2} - m \frac{ω L^{2}}{3 x}$

$J = 0$

$x = \frac{2 L}{3}$

It is independent of 'e'

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