# Law of conservation of Angular momentum

The term conservation refers to something which doesn’t change. Angular momentum is the rotational analogue of linear momentum. In this article, Let’s study the law of conservation of angular momentum.

## What is Conservation of Angular Momentum?

It is the rotational analog of linear momentum, it is denoted by l, and angular momentum of a particle in rotational motion is defined as:

 $$\begin{array}{l}l = r~×~p\end{array}$$

This is a cross product of r ,i.e. the radius of the circle formed by the body in rotational motion, and p, i.e. the linear momentum of the body, the magnitude of a cross product of two vectors is always the product of their magnitude multiplied with the sine of the angle between them, therefore in the case of angular momentum the magnitude is given by,

 $$\begin{array}{l}l = r~p~sinθ\end{array}$$

### Torque and Angular Momentum Relationship

Relationship between torque and angular momentum can found as follows,

$$\begin{array}{l}\overrightarrow{l} = \overrightarrow{r}~×~\overrightarrow{p}\end{array}$$

Differentiating LHS and RHS,

$$\begin{array}{l}\frac{d\overrightarrow{l}}{dt} = \frac{d}{dt}(\overrightarrow{r}~×~\overrightarrow{p})\end{array}$$

By the property of differentiation on cross products, the above expression can be written as follows,
$$\begin{array}{l}\frac{d\overrightarrow{l}}{dt} = \frac{dr}{dt}~\times~\overrightarrow{p}~+~r~\times~\frac{d\overrightarrow{p}}{dt}\end{array}$$

$$\begin{array}{l}\frac{d\overrightarrow{r}}{dt}\,\, \textup{is change in displacement with time, therefore it is linear velocity}\,\,\overrightarrow{v}\end{array}$$

$$\begin{array}{l}\frac{d\overrightarrow{l}}{dt} = \overrightarrow{v}~\times~\overrightarrow{p}+~r~\times~\frac{d\overrightarrow{p}}{dt}\end{array}$$

$$\begin{array}{l}\overrightarrow{p}\,\, \textup{is linear momentum i.e. mass times velocity,}\end{array}$$

$$\begin{array}{l}\frac{d\overrightarrow{l}}{dt} = \overrightarrow{v}~\times~m\overrightarrow{v}~+~r~\times~\frac{d\overrightarrow{p}}{dt}\end{array}$$

Now notice the first term,
Velocity and linear momentum are in same direction so their cross product will be equal to zero.

$$\begin{array}{l}\frac{d\overrightarrow{p}}{dt}\textup{ is force,}\end{array}$$

$$\begin{array}{l}\frac{d\overrightarrow{l}}{dt} = \overrightarrow{r}~\times~\overrightarrow{F}\end{array}$$

$$\begin{array}{l}\textup{We know that}\,\, r~\times~f\,\, \textup{is torque, hence,}\end{array}$$

$$\begin{array}{l}\frac{d\overrightarrow{l}}{dt} = \overrightarrow{τ},\,\,\textup{torque}\end{array}$$

So rate of change of angular momentum is torque.

## Conservation of Angular Momentum – Calculation

Angular momentum of a system is conserved as long as there is no net external torque acting on the system, the earth has been rotating on its axis from the time the solar system was formed due to the law of conservation of angular momentum,

There are two ways to calculate the angular momentum of any object, if it is a point object in a rotation, then our angular momentum is equal to the radius times the linear momentum of the object,

$$\begin{array}{l}\overrightarrow{l} = \overrightarrow{r}~\times~\overrightarrow{p}\end{array}$$

If we have an extended object, like our earth, for example, the angular momentum is given by moment of inertia i.e. how much mass is in motions in the object and how far it is from the centre, times the angular velocity,

$$\begin{array}{l}\overrightarrow{l} =\overrightarrow{I}~\times~\overrightarrow{\omega}\end{array}$$

But in both case as long as there is no net force acting on it, the angular momentum before is equal to angular momentum after some given time, imagine rotating a ball tied to a long string, the angular momentum would be given by,

$$\begin{array}{l}\overrightarrow{l} = \overrightarrow{r}~\times~\overrightarrow{p} = \overrightarrow{r}~\times~m\overrightarrow{v}\end{array}$$

Now when we somehow decrease the radius of the ball by shortening the string while it is in rotation, the r will reduce, now according to the law of conservation of angular momentum L should remain the same, there is no way for mass to change, therefore vector v should increase, to keep the angular momentum constant, this is the proof for the conservation of angular momentum.

### Conservation of Angular Momentum Applications

Law of conservation of angular momentum has many applications, including:

• Electric generators,
• Aircraft engines, etc.

To learn more about the conservation of angular momentum and other related topics with the help of interactive video lessons, visit BYJU’S.

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Learn to solve problems based on the conservation of angular momentum by watching the video below.

## Frequently Asked Questions – FAQs

Q1

### What is the conservation of angular momentum?

For a system with no external torque, the angular momentum is constant.

Q2

### List a few applications of conservation of angular momentum?

A few applications of angular momentum are:

• Aircraft engine
• Electric generators
Q3

### If a particle moves in a way it is said to change its angular position relative to its reference axis, it is said to have angular momentum.?(Say True or false)

TRUE – If a particle moves in a way that its angular position changes relative to its reference axis, is said to have angular momentum.

Q4

### Consider an ice skater who begins to spin such that his arms are spread out as far as possible and is parallel to the ice. What happens to the angular velocity of the skater when he pulls his arms inwards and when he raises his arms vertically?

The angular velocity of the skater increases when he pulls his arms inwards since the moment of inertia is lowered. Angular velocity of the skater stays the same when he raises his arms vertically because the distribution of radius of mass does not change.

Q5

### Where can we find the center of mass of two particles with equal mass?

The center of mass of two particles with equal mass are found in the midway between them.

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