Question

A thin uniform rod of length 'L' and mass 'M' is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is '$\omega$'. Its centre of mass rises to a maximum height of ______.

• A. $\frac{L^2 \omega^2}{2g}$
• B. $\frac{L \omega}{6g}$
• C. $\frac{L \omega}{2g}$
• D. $\frac{L^2 \omega^2}{6g}$

Hint:

For physical pendulums, potential energy $Mgh$ is ALWAYS calculated using the height change of the Center of Mass, not the physical end of the object. For a uniform rod, the COM is located at $L/2$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
A rigid uniform rod is acting as a physical pendulum, swinging from a pivot at its top end. We are given its maximum angular velocity (which occurs at the very bottom of the swing) and need to find the maximum vertical height achieved by its center of mass.

Step 2: Key Formula or Approach:
This requires the Principle of Conservation of Mechanical Energy.
Maximum Rotational Kinetic Energy at the bottom = Maximum Gravitational Potential Energy at the peak.
$$\frac{1}{2} I \omega^2 = Mgh$$
We must use the moment of inertia ($I$) for a rod rotating about its end.

Step 3: Detailed Explanation:
1. The moment of inertia of a uniform rod of mass $M$ and length $L$ pivoted exactly at its end is:
$$I = \frac{ML^2}{3}$$
2. The maximum kinetic energy at the lowest point is:
$$KE_{max} = \frac{1}{2} I \omega^2 = \frac{1}{2} \left( \frac{ML^2}{3} \right) \omega^2 = \frac{ML^2\omega^2}{6}$$
3. As the rod swings upward to its highest point, all this kinetic energy converts into gravitational potential energy. The potential energy is tracked by the vertical rise of its Center of Mass (COM), let's call this rise $h$.
$$PE_{max} = Mgh$$
4. Equating the two energies:
$$Mgh = \frac{ML^2\omega^2}{6}$$
The mass $M$ cancels out from both sides:
$$gh = \frac{L^2\omega^2}{6}$$
Isolate the height $h$:
$$h = \frac{L^2 \omega^2}{6g}$$

Step 4: Final Answer:
The maximum height risen by the center of mass is $\frac{L^2 \omega^2}{6g}$, matching option (d).