Rotation about the center of mass and spin angular momentum

In summary, the theorem states that 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.
  • #1
sokrates
483
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?
 
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  • #2
You can find proof in Kleppner & Kolenkow (pp. 262-4) or David Morin's book (pp. 380-1).
 
  • #3
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.
 
  • #4
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 [itex]\mathbf{L}=M(\mathbf{R} \times \mathbf{V}) + \mathbf{L}_{CM}[/itex], where [itex]\mathbf{L}_{CM}=\int \mathbf{r'} \times (\mathbf{\omega} \times \mathbf{r'}) dm[/itex].

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" [Broken]
 
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  • #5
This was extremely helpful. Thank you very much
 

What is rotation about the center of mass?

Rotation about the center of mass is a type of rotational motion where an object rotates around an imaginary point called the center of mass. This point is the average position of all the mass in the object.

What is the difference between rotation about the center of mass and rotation about a fixed point?

The main difference is that in rotation about the center of mass, the object rotates around the center of mass which stays fixed, while in rotation about a fixed point, the object rotates around a specific point in space.

How is spin angular momentum related to rotation about the center of mass?

Spin angular momentum is a measure of the rotational motion of an object around its own axis. In rotation about the center of mass, the object rotates around an imaginary axis passing through the center of mass, so the spin angular momentum is directly related to this type of rotation.

What are the applications of rotation about the center of mass in physics?

Rotation about the center of mass is a fundamental concept in physics and has various applications. It is used to explain the motion of objects such as planets, stars, and satellites. It is also essential in understanding the behavior of rigid bodies and their stability.

How does the distribution of mass affect rotation about the center of mass?

The distribution of mass in an object can affect its rotation about the center of mass. Objects with a more concentrated mass towards the center of mass will have a faster rotation, while objects with a more spread out mass will have a slower rotation. This is because the mass distribution affects the moment of inertia, which is a measure of an object's resistance to rotation.

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