Rotation about the center of mass and spin angular momentum

Click For Summary

Discussion Overview

The discussion revolves around the concept of angular momentum related to rotation about the center of mass, specifically addressing a theorem proposed by Prof. Walter Lewin regarding its independence from the choice of coordinate axes. Participants explore the need for a mathematical proof of this theorem, referencing various physics texts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant requests a mathematical proof of the theorem that angular momentum due to rotation about the center of mass is universal and independent of the choice of coordinate axes.
  • Another participant suggests consulting specific pages in Kleppner & Kolenkow and David Morin's books for the proof.
  • A participant expresses frustration that the referenced pages do not contain the proof and emphasizes the need for careful reading of their request.
  • A later reply corrects the page numbers for Kleppner & Kolenkow and discusses the reasoning provided in the texts, particularly the independence of rotational motion about the center of mass from translational motion.
  • This participant elaborates on the mathematical relationships involved, arguing that definitions relative to the center of mass should not change with the choice of coordinate system.
  • Another participant expresses gratitude for the clarification provided in the discussion.

Areas of Agreement / Disagreement

The discussion reflects a lack of consensus on the existence of a formal proof for the theorem in question, with some participants providing references while others challenge the adequacy of those references. The nature of the theorem itself is debated, particularly regarding its implications and the reasoning behind it.

Contextual Notes

Participants note specific pages in textbooks that may contain relevant information, but there is uncertainty regarding the completeness of the proofs provided. The discussion also highlights the dependence on definitions and the implications of shifting coordinate systems.

sokrates
Messages
481
Reaction score
2
I needed to refresh my classical physics knowledge and I was going through Prof. Walter Lewin's physics videos at ocw.mit.edu and at some point he proposed the following theorem without proof:

"The angular momentum due to rotation about the center of mass is universal and does not depend on the relative choice of coordinate axis, unlike, say orbital angular momentum"

It might be really trivial, but I need to see a mathematical proof of this statement, can anyone help?
 
Physics news on Phys.org
You can find proof in Kleppner & Kolenkow (pp. 262-4) or David Morin's book (pp. 380-1).
 
I took the time to find both books with great effort, and I checked the pages you refer me to.

None of them has the proof to the theorem I asked.

I wish you had taken the time to more carefully read what I asked.
 
Regarding Kleppner & Kolenkow, the correct pages are 260-2, not 262-4, so it seems it's my mistake. However, on p. 263 they state and reason quite clearly that "rotational motion about the center of mass depends only on the torque about the center of mass, independent of the translational motion ... "

Morin proves on page 381 that \mathbf{L}=M(\mathbf{R} \times \mathbf{V}) + \mathbf{L}_{CM}, where \mathbf{L}_{CM}=\int \mathbf{r'} \times (\mathbf{\omega} \times \mathbf{r'}) dm.

He states clearly that r' and omega' are both measured relative to the CM. How can something defined relative to the CM be dependent on the origin? Just take a moment to visualize this - you can move your frame of reference around, yet your CM won't budge, right? How can anything defined relative to the CM change when some other arbitrary coordinate system is moved around?

If you need to see this formally, then note he r' = R - r, where R is the CM coordinate and r the particle's position. Moving your frame of reference moves both R and r by the same amount and this cancels out: r' = (R+a) - (r+a) = R-a, independent of a. The particle's velocity in the CM frame (v' = omega' cross r' = V - v) is also independent of shifting the origin of your frame of reference for the same reason.

-----
Assaf
http://www.physicallyincorrect.com"
 
Last edited by a moderator:
This was extremely helpful. Thank you very much
 

Similar threads

Undergrad Axial angular momentum calculation
  • · Replies 30 ·
2
Replies
30
Views
4K
Undergrad Effects of External Torques on the Angular Momentum of Asymmetric Bodies
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Question about a Rotating system with mass moving inwards
  • · Replies 19 ·
Replies
19
Views
3K
Undergrad Conservation of angular momentum in the iceskater example
  • · Replies 10 ·
Replies
10
Views
1K
Undergrad How Does Angular Momentum Apply to Earth and Spinning Bike Wheels?
  • · Replies 36 ·
2
Replies
36
Views
4K
Graduate Conceptual questions about Angular Momentum Conservation and torque
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Solving for the trajectory of the center of mass
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Energy of spinning objects as axis of rotation moves
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Orbital and spin angular velocity?
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Does the spin angular momentum count?
  • · Replies 10 ·
Replies
10
Views
2K