IIT JEE Physics (1978-2018: 41 Years) Topic-wise Complete Solutions

# Vector Nature of Angular Momentum

## Introduction

Angular momentum of an object is given by $$\vec{L}=\sum m_i \vec{r}_i \times\vec{p}$$. For a rapidly spinning disk about its axes, it is very nearly $$\vec{L}=I\vec{\omega}$$ where $$I$$ is the momentum of inertia about the spin axis and $$\vec{\omega}$$ is the angular velocity vector.

The angular momentum can be changed by the torque $$\vec{\tau}=\sum \vec{r}_i\times \vec{F}$$. The equation is given by $$\vec{\tau}=\frac{\mathrm{d}\vec{L}}{\mathrm{d}t}$$.

The following experiment brings out the vector character of the angular momentum and how angular momentums add according to the vector addition rules. This experiment is also a very popular one, done with a rapidly spinning bicycle wheel held by a rope fixed at some place on the axis, away from the center of mass. We have made a compact and portable unit in place of bulky bicycle wheel.

## Apparatus

CD attached with an axel and a DC motor, 9 Volt battery, connector.

## Procedure

1. Look at the apparatus. See how the battery is connected to the motor both of which are fixed on the CD-Axel unit.
2. Hold the axel in your hand and connected the battery to the motor. The CD starts spinning at a high angular speed.
3. While holding the axel in one hand, stretch the string tied to the axel, in vertical position. Keeps the CD surface vertical using the hand holding the surface.
4. Now gently leave the axel, while you are firmly holding the string.

See how the spinning disk starts precessing about the vertical string acquiring orbital and spin motion. As seen from the above, is the orbital motion clockwise or anticlockwise?

## Discussion

Because of the rapidly spinning DC, the system has its angular momentum along the spin axis. The tension in the string exerts torque $$\vec{\tau}=\vec{R}\times\vec{F}$$ on the system and in a time cause a change in angular momentum $$\mathrm{d}\vec{L}=\vec{\tau}\mathrm{d}t$$. Check that direction of $$\vec{\tau}$$ and hence of $$\mathrm{d}\vec{L}$$ is horizontal and perpendicular to the spin axis. The changed angular momentum $$\vec{L}+\mathrm{d}\vec{L}$$ will be obtained by the vector addition rule and it will be tilted from the original direction of spin axis. The system therefore turns to align the spin axis with the new direction of angular momentum $$\vec{L}+\mathrm{d}\vec{L}$$. This continuous and the CD is set into orbit motion.