Rotating-chair experiment: Moment Of Inertia

Introduction (100 words)

Demonstrate that the moment of inertia (rotational analogue of mass) depends on how mass is distributed about the rotation axis, and show this experimentally using a rotating chair and moveable masses. Use conservation of angular momentum to quantify the change.

Precautions

1. Sit centered on the chair; keep feet off the floor while spinning (or rest slightly but not apply torque).
2. Use light, steady holds on the masses to reduce wobble.
3. Avoid sudden jerks or throwing weights — safety first.

Materials required

• Rotating stool or swivel chair that can rotate freely (low friction).
• Two identical small masses (dumbbells, heavy cans, or weights) with handles so student can hold them at different radii. (Mass each $m$ known.)

Science behind it (200 words)

The purpose of this experiment is to show that the moment of inertia, the rotational equivalent of mass, depends on how mass is distributed around the axis of rotation. You can demonstrate this using a freely rotating chair and two equal weights.

Sit on the chair holding one weight in each hand close to your chest. A helper gives you a gentle spin, and you measure your angular speed by timing how long it takes to complete several rotations. Next, extend your arms outward with the weights and measure the new angular speed. You should notice that when the masses are farther away from the axis your rotation slows down, while when you pull them in close you spin faster. This happens because angular momentum, defined as the product of moment of inertia and angular velocity, remains conserved when there is no external torque. Since the moment of inertia of a point mass is proportional to the square of its distance from the axis, increasing that distance greatly increases inertia, and the only way to keep the product constant is for angular speed to drop.

A simple calculation illustrates this. Two weights of mass \(m\) held at radius \(r\) contribute \(I = 2mr^2\). If your initial angular speed is \(\omega_i\) with arms in and later \(\omega_f\) with arms out, conservation of angular momentum tells us \[ (I_{\text{body}} + 2mr_{\text{in}}^2)\,\omega_i = (I_{\text{body}} + 2mr_{\text{out}}^2)\,\omega_f \] By measuring the angular speeds you can even solve for the effective inertia of the chair and person.