Angular Momentum of Multiparticle Systems: Difference between revisions

Latest revision as of 03:33, 5 December 2015

By Jonathan Peraza (jonperaza)

Main Idea

The angular momentum principle may be extended to a multiparticle system to provide various insights, such as an explanation for the counter-intuitive behavior of gyroscpoes. This page details how one could derive the angular momentum principle of a multiparticle system and provides a few examples to illustrate the concept.

Mathematical Model

The derivation of the angular momentum principle of a multiparticle system is most easily understood when following a simple example, in this case we will work with a three particle system:

The angular momentum principle is written for each individual particle relative to location A, with position vectors r and external force vectors F.

The individual equation is then summed. Note that, due to the reciprocation of forces, the internal forces f cancel out.

The equation is now rewritten with the right side, representing net torque, as τ_net,A

Computational Model

Glowscript Trinket Example

Glowscript Example

Examples

Simple

Question:

What is the total angular momentum L_tot of a system of particles with individual angular momenta L₁ = <0,0,10>, L₂ = <0,0,-5>, & L₃ = <0,0,15>?

Answer:

L_tot = <0,0,20>

Difficult

Each of the six particles in the figure have identical mass m and moves in the x-y plane with the same speed v as indicated in the diagram. Take the location of particle 5 as the reference point for all angular momentum calculations when answering the following questions.Be sure to express your answer as a vector.

What is the momentum of the individual particles?

Consider the particles above as a single multiparticle system. With respect to particle 5, what is the total angular momentum of the system? What is the translational angular momentum? What is the rotational angular momentum?

Connectedness

This concept is highly interesting because of its part in describing highly complex systems and interactions on various scales, as multi-component systems exist on all scales from quantum to cosmic. Being in MSE, it may be of interest to investigate materials as multi-component systems, and to apply the angular momentum principle in this manner.

History

See: History of the Angular Momentum Principle

References

Matter & Interactions, Vol. I: Modern Mechanics, 4nd Edition by R. Chabay & B. Sherwood (John Wiley & Sons 2015)

@@ Line 1: / Line 1: @@
-The [http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle angular momentum principle] may be extended to a multiparticle system to provide various insights.
+By Jonathan Peraza (jonperaza)
-==The Main Idea==
+==Main Idea==
+[[The Angular Momentum Principle|The angular momentum principle]] may be extended to a multiparticle system to provide various insights, such as an explanation for the counter-intuitive behavior of [[Gyroscopes|gyroscpoes]]. This page details how one could derive the angular momentum principle of a multiparticle system and provides a few examples to illustrate the concept.
+===Mathematical Model===
-===A Mathematical Model===
 The derivation of the angular momentum principle of a multiparticle system is most easily understood when following a simple example, in this case we will work with a three particle system:
@@ Line 23: / Line 23: @@
 ::::[[File:Angular_Momentum_Principle_for_a_Multiparticle_System.jpg]]
-===A Computational Model===
+===Computational Model===
+[[File:Vpython screenshot.jpg|none|thumb|500px|VPython Screenshot]]
 [https://trinket.io/glowscript/88ddf42892 Glowscript Trinket Example]
+[http://www.glowscript.org/#/user/Jonathan_Peraza/folder/Public/program/AngularMomentumofMultiparticleSystem Glowscript Example]
 ==Examples==
-Be sure to show all steps in your solution and include diagrams whenever possible
+===Simple===
+'''Question:'''
-===Simple===
+::What is the total angular momentum ''L''<sub>tot</sub> of a system of particles with individual angular momenta ''L''<sub>1</sub> = <0,0,10>, ''L''<sub>2</sub> = <0,0,-5>, & ''L''<sub>3</sub> = <0,0,15>?
+'''Answer:'''
+::''L''<sub>tot</sub> = <0,0,20>
 ===Difficult===
@@ Line 39: / Line 51: @@
 ::::[[File: Angular_Momentum_of_Multiparticle_System_Difficult_Example_Image1.jpg]]
-:::What is the momentum of the individual particles?
+::What is the momentum of the individual particles?
+::::[[File:Angular_Momentum_of_Multiparticle_System_Difficult_Example_Solution_Part1.jpg]]
+::Consider the particles above as a single multiparticle system. With respect to particle 5, what is the total angular momentum of the system? What is the translational angular momentum? What is the rotational angular momentum?
+::::[[File:Angular_Momentum_of_Multiparticle_System_Difficult_Example_Solution_Part2.jpg]]
 ==Connectedness==
-#How is this topic connected to something that you are interested in?
-#How is it connected to your major?
+This concept is highly interesting because of its part in describing highly complex systems and interactions on various scales, as multi-component systems exist on all scales from quantum to cosmic. Being in MSE, it may be of interest to investigate materials as multi-component systems, and to apply the angular momentum principle in this manner.
-#Is there an interesting industrial application?
 ==History==
-Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
+[[The Angular Momentum Principle#History|''See: History of the Angular Momentum Principle'']]
 == 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?
+[[The Angular Momentum Principle]]
-===Further reading===
+[[Multi-particle Analysis of Momentum]]
-Books, Articles or other print media on this topic
+===External links===
-===External links===
+[http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:mp_multi The Momentum Principle in Multi-Particle Systems (Video)]
-[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
+[http://farside.ph.utexas.edu/teaching/301/lectures/node120.html Angular momentum of a multi-component system]
+[http://www.damtp.cam.ac.uk/user/stcs/courses/dynamics/lecturenotes/section5.pdf Chapter 5: Systems of particles, Dynamics Lecture Notes, University of Cambridge Department of Applied Mathematics and Theoretical Physics]
+[http://web.mit.edu/8.01t/www/materials/modules/chapter19.pdf Chapter 19 Angular Momentum, MIT Course Materials]
+[http://www.rakeshkapoor.us/ClassNotes/classnotes.php?notes=TorqueAndAngularMomentum&index=1 Torque and Angular Momentum, General Physics Class Notes, Dr. Rakesh Kapoor, M.Sc., Ph.D. Former Faculty-University of Alabama at Birmingham]
 ==References==
-This section contains the the references you used while writing this page
+Matter & Interactions, Vol. I: Modern Mechanics, 4nd Edition by R. Chabay & B. Sherwood (John Wiley & Sons 2015)
-[[Category:Which Category did you place this in?]]
+[[Category:Angular Momentum]]