Total Angular Momentum: Difference between revisions
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Put this idea in historical context. Give the reader the Who, What, When, Where, and Why. | Put this idea in historical context. Give the reader the Who, What, When, Where, and Why. | ||
It is often convenient to consider the angular momentum of a collection of particles about their center of mass, because this simplifies the mathematics considerably. The angular momentum of a collection of particles is the sum of the angular momenta of each particle: | |||
<math>\mathbf{L}=\sum_i \mathbf{R}_i\times m_i \mathbf{V}_i</math> | |||
where <math>R_i</math> is the distance of particle i from the reference point, <math>m_i</math> is its mass, and <math>V_i</math> is its velocity. The center of mass is defined by: | |||
<math>\mathbf{R}=\frac{1}{M}\sum_i m_i \mathbf{R}_i</math> | |||
where <math>M</math> is the total mass of all the particles. | |||
If we define <math>\mathbf{r}_i</math> as the displacement of particle i from the center of mass, and <math>\mathbf{v}_i</math> as the velocity of particle i with respect to the center of mass, then we have | |||
<math>\mathbf{R}_i=\mathbf{R}+\mathbf{r}_i\,</math> and <math>\mathbf{V}_i=\mathbf{V}+\mathbf{v}_i\,</math> | |||
In this case, the total angular momentum is: | |||
<math>\mathbf{L}=\sum_i (\mathbf{R}+\mathbf{r}_i)\times m_i (\mathbf{V}+\mathbf{v}_i) = \left(\mathbf{R}\times M\mathbf{V}\right) + \left(\sum_i \mathbf{r}_i\times m_i \mathbf{v}_i\right)</math> | |||
== See also == | == See also == | ||
Revision as of 14:12, 25 November 2015
Work in Progress by Fatima Jamil
Total angular momentum can be expressed as 
. This page explains the breakdown of total angular momentum in these 2 components to help understand the difference between rotational angular momentum and translational angular momentum.
The Main Idea
It is conveniant to break apart total angular momentum for a multiparticle system into rotational angular momentum and translational angular momentum. The translational angular momentum is associated with a rotation of the center of mass about some point A. This differs for different choices of the location of point A. The rotational angular momentum is associated with a rotation about the center of mass. The rotational angular momentum is independent of the location of the point A and the motion of the center of mass.
A Mathematical Model
What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.
A Computational Model
How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript
Examples
Be sure to show all steps in your solution and include diagrams whenever possible
Simple
Middling
Difficult
Connectedness
- How is this topic connected to something that you are interested in?
- How is it connected to your major?
- Is there an interesting industrial application?
History
Put this idea in historical context. Give the reader the Who, What, When, Where, and Why. It is often convenient to consider the angular momentum of a collection of particles about their center of mass, because this simplifies the mathematics considerably. The angular momentum of a collection of particles is the sum of the angular momenta of each particle:
[math]\displaystyle{ \mathbf{L}=\sum_i \mathbf{R}_i\times m_i \mathbf{V}_i }[/math] where [math]\displaystyle{ R_i }[/math] is the distance of particle i from the reference point, [math]\displaystyle{ m_i }[/math] is its mass, and [math]\displaystyle{ V_i }[/math] is its velocity. The center of mass is defined by:
[math]\displaystyle{ \mathbf{R}=\frac{1}{M}\sum_i m_i \mathbf{R}_i }[/math] where [math]\displaystyle{ M }[/math] is the total mass of all the particles.
If we define [math]\displaystyle{ \mathbf{r}_i }[/math] as the displacement of particle i from the center of mass, and [math]\displaystyle{ \mathbf{v}_i }[/math] as the velocity of particle i with respect to the center of mass, then we have
[math]\displaystyle{ \mathbf{R}_i=\mathbf{R}+\mathbf{r}_i\, }[/math] and [math]\displaystyle{ \mathbf{V}_i=\mathbf{V}+\mathbf{v}_i\, }[/math] In this case, the total angular momentum is:
[math]\displaystyle{ \mathbf{L}=\sum_i (\mathbf{R}+\mathbf{r}_i)\times m_i (\mathbf{V}+\mathbf{v}_i) = \left(\mathbf{R}\times M\mathbf{V}\right) + \left(\sum_i \mathbf{r}_i\times m_i \mathbf{v}_i\right) }[/math]
See also
Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?
Further reading
Books, Articles or other print media on this topic
External links
Internet resources on this topic
References
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