# 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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## The Figure Skater Example

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.

## Formal Statement

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.

A helicopter-manufacturing industry, at their annual meeting, proposed to build the helicopters at a cheaper rate keeping its purpose and efficiency the same as it was before. The company has prepared two models of helicopters. Plan A has a helicopter with two propellers, and Plan B has a helicopter with one propeller. You are given the task of selecting the best one of them. Which one will you select?

Try my World of Physics to solve many problems like this one.

## Similarity to Newton's Second Law

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.

## Relation to Gyroscopes and Falling Cats

## Conservation of Angular Momentum in Planetary Systems

**Cite as:**Conservation of Angular Momentum.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/conservation-of-angular-momentum/