Question

A horizontal rigid rod of length 1m is dropped from height h = 20 m, as shown in figure. The end of the rod collides with the table. If collision is perfectly elastic.

• A. Angular velocity of rod after collision is 60rad/sec
• B. Angular velocity of rod after collision is 30rad/sec
• C. Velocity of midpoint of rod just after collision is 15 m/s downward
• D. Velocity of midpoint of rod just after collision is 10 m/s downward

Answer 1

Answer: (B), (C)

Angular velocity ≈ 30 rad/s and Midpoint velocity = 15 m/s downward

Answer 2

We are given a 1 m long horizontal rod dropped from 20 m so that, just before impact, every point is moving downward with

$$v=\sqrt{2gh}=\sqrt{2\times9.8\times20}\approx 19.8\mbox{ m/s.}$$

When one end (say the left end, P) hits the table, a very short (impulsive) force reverses its vertical velocity from downward 19.8 m/s to upward 19.8 m/s (elastic collision). During this impulse the net external torque about P is zero so we can use conservation of angular momentum about P.

1. The moment of inertia of a rod about one end is

$$I_P=I_{cm}+M\Big(\frac{L}{2}\Big)^2=\frac{1}{12}ML^2+\frac{1}{4}ML^2=\frac{ML^2}{3}\,.$$

2. Just before impact, the entire rod is translating downward so that the angular momentum about P (taking the lever arm as $L/2$) is

$$L_i = Mv\Big(\frac{L}{2}\Big)= M\,(19.8)\frac{1}{2}\,.$$

3. Let the rod acquire an angular speed $\omega$ (about P) immediately after collision. Then

$$L_f = I_P \,\omega = \frac{ML^2}{3}\,\omega\,.$$

Conservation about P gives

$$M\,v\Big(\frac{L}{2}\Big)=\frac{ML^2}{3}\,\omega\quad\Longrightarrow\quad \omega=\frac{3v}{2L}\,.$$

With $v\approx19.8\,\mbox{m/s}$ and $L=1\,\mbox{m}$:

$$\omega\approx\frac{3\times19.8}{2}\approx29.7\,\mbox{rad/s}\,.$$

Rounded suitably, the angular velocity is about 30 rad/s.

4. The centre of mass, being at a distance $L/2$ from P, has a speed due to rotation:

$$v_{cm}=\omega\Big(\frac{L}{2}\Big)\approx30\times0.5=15\,\mbox{m/s}\,.$$

Its direction is downward (as can be checked using the geometry of the rotation).