Angular Momentum

Summary

Angular momentum is used to describe how much force an object has when rotating. This type of momentum is not always conserved, nor is it an intrinsic property

An object can be moving linearly but still have angular momentum

Key equations:

Angular momentum:
$\vec{L} = \vec{r} \times \vec{p}$

Special case where a body is rotating symmetrically:
$\vec{L} = I ω$

Read Time

⏱ 2 mins

Definition

Angular momentum is used to describe how much "hard" it is to give an object rotation. It's similar to linear momentum in that it can be a conversed quantity, but is not an intrinsic property, nor is it always conserved. It depends on your reference frame and is only conserved if the net torque is zero.

Angular momentum does not ALWAYS mean spinning in a circle

Most situations of angular momentum involve an object spinning in a circle. You can also have angular momentum if an object has linear momentum and has a sideways component (not moving straight)

Mathematically, angular momentum is the cross product between linear momentum and the r displacement vector. The r displacement vector is the reference point, which can be arbitrarily chosen. Meaning certain reference frames will not have angular momentum.

\vec{L} = \vec{r} \times \vec{p}

Angular Momentum Diagram

!ang_4.png
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Example of angular momentum of a point like mass

Special Case For Rotating Rigid Bodies

Assumption

For this special case, assume the following:

The object is rotating symmetrically.
The origin at in the middle of the rotating object
The rotation can be described from rotational kinematics $v = ω \times r$

Use the diagram below as a visual aid of a rotating body rotating symmetrically.
!am_1.png

\begin{matrix} L = r \times p \\ L = r \times (m v) \\ L = m r \times (ω \times r) \\ L = m r^{2} ω \\ L = I ω \end{matrix}