# I don't understand the equation for angular momentum?

1. Oct 22, 2009

### kashiark

Why is it r*mv instead of just mv like "normal," linear momentum?

2. Oct 23, 2009

### Cleonis

To be of any use the definition of angular momentum must slot in with other areas of physics.

Look at the following comparison:
In linear motion kinetic energy is $\frac{1}{2} m v^2$
In the case of motion along a circle with radius r the following relation applies: $v = \omega r$ where $\omega$ is angular velocity.
In the expression for linear kinetic energy the 'v' can be substituted with $\omega r$ and you get:

$$\frac{1}{2} m \omega^2 r^2$$

Which is usually rearranged to group $mr^2$ together

$$\frac{1}{2} m r^2 \omega^2$$

The produkt of 'm' and 'r2' is called 'moment of inertia', it can be thought of as the rotational counterpart of linear inertia.

by defining angular momentum as $mr^2 \omega$ Kepler's law of areas is recovered; read http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Gravity/SecondLawDerivation.html" [Broken]

Cleonis

Last edited by a moderator: May 4, 2017
3. Oct 23, 2009

### kashiark

Ah, ok; I think I get it. Thanks!

4. Oct 23, 2009

### rcgldr

momentum is inertia x velocity.

For linear movement, inertia = mass, so linear momentum = mass x velocity.

For angular movment, angular momentum = ω I, where ω is rate of rotation, and I is the angular inertia. For a point mass, angular inertia is m r2. Other inertias are listed here:

http://en.wikipedia.org/wiki/List_of_moments_of_inertia

For a point mass I = m r2.

The linear velocity: v = ω r, so ω = v / r

Angular momentum for a point mass = ω I = (v/r) m r2 = r m v

Although angular momentum is conserved in a closed system, kinetic energy normally isn't if the radius is changed. This is because internal work is done to change the radius. During the transition, the object follows a spiral path, and the radial force includes a component of force in the direction of travel for the spiral path. The math for this is covered in this thread:

Last edited: Oct 23, 2009
5. Oct 23, 2009

### kashiark

Why is inertia mr²?

6. Oct 23, 2009

### Cleonis

Well, Jeff Reid referred to mr² as 'angular inertia'. A more common expression is 'moment of inertia'.

In physics we're looking for conserved quantities. We find that the quantity mv is conserved in collisions (and interactions in general).

Note that when Kepler's law of areas was formulated its close relationship to momentum wasn't immediately recognized. Newton showed that the area law follows logically from mechanics.

The area law is formulated geometrically, the counterpart of that in the form of a mathematical expression is that a quantity $m r^2 \omega$ is conserved. (where 'r' is the distance to the pivot point.)

The justification for defining the concept of 'moment of inertia' rests on the above: $m r^2 \omega$ (where 'r' is the distance to the pivot point) is conserved in collisions and interactions in general.

Cleonis