20.5 Conservation of angular momentum  (Page 4/5)

The statement of conservation of angular momentum for isolated body system can take advantage of the relation valid for the rigid body. Here,

$$\begin{array}{l}{L}_i=L_f\end{array}$$

We must, however, always keep in our mind that this form of conservation law is valid only for the rigid body for which angular momentum is measured about an axis.

Examples

We have all along emphasized the general nature of angular momentum. But when we think about examples of real world, which can be analyzed with the help of this conservation law - we realize that most of them are actually the rotational cases. This, however, does no reduce the importance of generality. The physical examples for general motion require complex analysis tool beyond the scope of this course and hence are not considered.

Here, some examples of rotational motion are given to illustrate conservation of angular momentum.

1: The revolution of planets around Sun

The planets like Earth move around Sun along an elliptical orbit. The gravitational pull provides the necessary centripetal force for the curved elliptical path of motion. This gravitational force, however, passes through center of mass of the Earth and the Sun. As such, it does not constitute a torque. Thus, no external torque is applied to the Earth - Sun system. We can, therefore, apply conservation of angular momentum to the system.

Earth revolving around sun

When the Earth comes closer to the Sun, the moment of inertia of the Earth about an axis through the center of mass of the Sun decreases. In order to conserve its angular momentum, it begins to orbit the Sun faster. Similarly, the Earth rotates slower when it is away from the Sun. All through Earth's revolution, following condition is met :

$$\begin{array}{l}{I}_i{\omega }_i=I_f{\omega }_f\end{array}$$

We should note here that we are considering rotation of the Earth about the Sun - not the rotation of Earth about its own axis of rotation. If we recall Kepler's law, we can see the convergence of results. This law states that the line joining Sun and Earth sweeps equal area in equal time and, thereby predicts greater tangential velocity (in turn, greater angular velocity), when closer to the Sun.

2: A person sitting on a turn table

A person sitting on a turn table can manipulate angular speed by changing moment of inertia about the axis of rotation without any external aid. In order to accentuate the effect, we consider that person is holding some weights in his outstretched hands. Let us consider that the system of the turntable and the person holding weights in the outstretched hands, is rotating about vertical axis at certain angular velocity in the beginning. The moment of inertia of the body and weights about the axis of rotation is :

A person sitting on a turn table

$$\begin{array}{l}I=\sum m_i{r_i}^2\end{array}$$

When the person folds his hands slowly, the moment of inertia about the axis decreases as the distribution of mass is closer to the axis of rotation. In order to conserve angular momentum, the turn table and person starts rotating at greater angular velocity.