# Looking for proof that a free rigid body will rotate about its CM

• scoomer
In summary, the center of mass (CM) moves in a straight line, and the body rotates about an axis through its CM.
scoomer
Can anyone recommend a textbook or web paper that proves that, in general, a free rigid body will act in the following way when the body is subjected to a force (or impulse) that is torque producing:
a) The center of mass (CM) moves in a straight line.
b) The body rotates about an axis through its CM.
(By "free" I mean unconstrained.)

It seems intuitive and many textbooks state the above. But the ones I've looked at don't seem to present a proof.

Thanks.

scoomer said:
Can anyone recommend a textbook or web paper that proves that, in general, a free rigid body will act in the following way when the body is subjected to a force (or impulse) that is torque producing:
a) The center of mass (CM) moves in a straight line.
b) The body rotates about an axis through its CM.
(By "free" I mean unconstrained.)
It follows from the conservation of total linear momentum, if you consider that the body consist of many small masses, each with it's own linear momentum. I don't have a formal proof, but that is the line it would along.

I also missed this very important concepts from mechanics books. So I worked it out by myself. I define coordinates (all relevant coordinates are vectors):
$$r_i=r_c+r_i'$$
$$v_i=v_c+v_i'=v_c+\omega\times r_i'$$
I suppose it is possible to show that the necessary vector $\omega$ corresponds to the angular velocity about just any axis. I use center of mass coordinates since later I will need $\sum m_ir_i'=0$.
-----------------------------
Now
$$\sum F_i=\sum m_ia_i$$
$$\sum F_i=\frac{\mathrm{d}}{\mathrm{d}t}\sum m_iv_i$$
$$\sum F_i^\text{ext}=\frac{\mathrm{d}}{\mathrm{d}t}\left((\sum m_i)v_c\right)$$
$$F^\text{ext}=Ma_c$$
On the LHS I used that internal forces cancel $F_{ij}=-F_{ji}$ and on the RHS I used that $\sum m_iv_i'=\omega\times\sum m_i r_i'=0$. Now you have the linear law of motion.
--------------------------
For the angular part
$$\frac{\mathrm{d}}{\mathrm{d}t}\sum m_ir_i'\times v_i'=\sum m_i r_i'\times a_i'=\sum m_i r_i'\times (a_i-a_c)=\sum r_i'\times F_i=\sum r_i'\times F_i^\text{ext}$$
The internal forces cancel again since force are along the line of action $r_i\times F_{ij}+r_j\times F_{ji}=\Delta r\times F_{ij}=0$. So, the torque from external forces about the center of mass is equal to the angular momentum about the center of mass.

With some additional ideas one can show that therefore in the absence of external forces the center of mass velocity and the angular velocity are constant.

Last edited:
thank you, that was very helpful

Actually my derivation doesn't specifically address the rigid body (so I don't need $\omega$). It is for a general set of interacting particles.

That's where I believe conservation of angular momentum comes from. At least I haven't met anyone who convincingly can use Noethers theorem, without including lots of hidden assumptions

what?!
has my reply been deleted again..?
anyway... can you please prov the second law of Newton for rotation?
thanks :D

## 1. How do you define a free rigid body?

A free rigid body is an object that is not under the influence of any external forces or torques. This means that the body is not constrained in any way and is able to move and rotate freely.

## 2. What is the center of mass of a free rigid body?

The center of mass of a free rigid body is the point where the entire mass of the body can be considered to be concentrated. It is the average position of all the mass in the body.

## 3. How do you prove that a free rigid body will rotate about its center of mass?

The proof for this is based on the principle of conservation of angular momentum, which states that the total angular momentum of a system remains constant if no external torques act on it. Since a free rigid body has no external torques acting on it, its angular momentum about its center of mass will remain constant, causing it to rotate about that point.

## 4. Can a free rigid body rotate about any other point besides its center of mass?

No, a free rigid body can only rotate about its center of mass. This is because the center of mass is the unique point where all the mass of the body can be considered to be concentrated, and thus the only point where the body can rotate without any external torques acting on it.

## 5. What are some real-life examples of free rigid bodies rotating about their center of mass?

One example is a spinning top, where the center of mass remains fixed while the top rotates around it. Another example is a diving board, where the board rotates around its center of mass as a diver jumps off of it. In both cases, the external forces and torques acting on the bodies are negligible, allowing them to rotate freely about their center of mass.

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