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oblique collision of a rigid rod with a plane surface

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A uniform rod AB of length L is released from rest with AB inclined at angle 60° with horizontal, with the closer end of the rod to the ground, at a height 49m from it. Assuming the collision of the rod with the ground to be perfectly elastic, calculate the height the centre of mass of the rod will rebound after impact (assume ground to be frictionless).

asked Jan 1 in Physics Problems by Harsh (110 points)
edited Jan 3 by sammy gerbil
What have you tried? What difficulty are you having with this problem?
I calculated the velocity of the rod at the lowest point. After that, I know that the ground is going to apply an impulse in the vertically upward direction, but I am not quite able to get the answer
How have you made use of the impulse? What calculation have you made?

1 Answer

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The normal contact impulse $J$ between the rod and plane has two effects : it changes the velocity of the CM of the rod, and it makes the rod rotate about its CM.

The impulsive force is related to the change in momentum of the CM of the rod, and the impulsive torque to the change in angular momentum of the rod, as follows : $$J=m(v_0+v_1)$$ $$J\frac{L}{2}\cos\theta=I\omega$$ Here $v_0, v_1$ are the vertical velocities of the CM before and after the collision, and $I=\frac{1}{12}mL^2$ is the moment of inertia of the rod about its CM.

The rod is not rotating initially. After the collision, the velocity of the end A which collided with the ground is $v_2'=\frac{L}{2}\omega$ relative to the CM of the rod, directed perpendicular to the rod, ie at an angle of $\theta=60^{\circ}$ to the vertical. The CM of the rod is moving upwards with speed $v_1$ so the vertical component of the velocity of A relative to the ground after collision is $$v_2=v_2'\cos\theta+v_1$$

The elasticity of the collision is known ($e=1$) so the Law of Restitution can be applied to the relative velocities of approach and separation at the point of contact : $$v_2=ev_0$$

From the above equations you can eliminate $v_2, v_2', \omega$ to find the rebound speed $v_1$ of the CM of the rod and thereby the height to which the CM rises after collision.

answered Jan 2 by sammy gerbil (28,448 points)
edited Jan 3 by sammy gerbil
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