Forest UCM Ch3 AngMom

Definition of Angular Momentum

The angular momentum of a single particle is defined as

[math]\vec \ell = \vec r \times \vec p[/math]

An coordinate must be defined in order to express the vectors for the particles position and momentum. The resulting angular momentum is defined with respect to the origin (rotation point) of the particle.

Torque

If I take the derivative of angular momentum with respect to time I get

[math]\vec{\dot \ell} = \frac{d}{dt} \left ( \vec r \times \vec p \right )[/math]

[math]= \left ( \vec \dot r \times \vec p \right ) + \left ( \vec r \times \vec \dot p \right )[/math]

[math] \left ( \vec \dot r \times \vec p \right )= \left ( \frac{1}{m} \vec p \times \vec p \right ) =0 [/math] cross product of parallel vectors

[math] \left ( \vec r \times \vec \dot p \right )=\left ( \vec r \times \vec F \right )= \vec \mathcal T[/math] Definition of Torque

[math]\vec \mathcal T =\vec{\dot \ell} [/math] Newton's second law for angular motion

Kepler's second Law

Kepler's second Law

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

The above law is the result of the conservation of angular momentum when two bodies are attracted to eachother by a central force.

Forest_UCM_MnAM#Angular_Momentum