Conservation of Angular Momentum

In angular kinematics, the conservation of angular momentum refers to the tendency of a system to preserve its rotational momentum in the absence of an external torque.

For a circular orbit, the formula for angular momentum is (mass)$\times$(velocity)$\times$(radius of the circle):

$$\text{(angular momentum)}=m \times v \times r.$$

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What makes the figure skater skate so fast and smooth? The reason is because she is an expert in conserving angular momentum. Let's see what happens when angular momentum is conserved:

$$(\text{Angular Momentum})=(\text{Moment of Inertia}) \times (\text{Angular velocity}).$$

In the case of the figure skater, the axis of rotation is the vertical axis of her body, so she decreases the moment of inertia by bringing her hands and legs close to her body. From the formula of angular momentum, we see that the moment of inertia is inversely proportional to the angular velocity. As a result, by bringing a part of her body closer to the axis of rotation, her moment of inertia decreases. This leads to an increase in her angular velocity, since the angular momentum (the quantity on LHS) is constant. This conservation of angular momentum explains the speed and smoothness of her skating.

If no net external torque acts on a system, the total angular momentum of the system remains constant. When net external torque acting on an object is zero, the total initial angular momentum $l_0$ is equal to total final angular momentum $l_f$:

$$\tau_{ext}=0\implies \dfrac{dL}{dt}=0 \implies L=\text{constant} \implies l_0=l_f.$$

When angular momentum $L$ is expressed in vector form, the components of $L$—$L_x,L_y,L_z$ along the $x$-, $y$-, $z$-axes, respectively—remain as some constants.

Newton's second law gives us this idea:

The net external force acting on an object is equal to the rate of change of its linear momentum.

However, the rotational analogue of the above law can be stated as follows:

A change in angular momentum is proportional to the applied torque and occurs about the same axis as that torque. The change in the angular momentum of the body is directly proportional to the torque acting on it for some time. If the torque is zero, then the angular momentum is conserved. A system dealing with constant angular momentum is a closed system, and in a closed system there is no requirement of external influence in the form of torque.

Thus, Newton's second law deals with the relation between net external force and rate of change of linear momentum. Conservation of angular momentum deals with external influence of torque and change in angular momentum. In a way, the two concepts are similar.