# Resultant center of rotation due to multiple moments

Say, there are multiple moments (M1, M2, M3) acting on a body (of irregular shape) at points P1, P2, P3 respectively. The body is free to rotate about any point.

Now, which resultant center (the point which is not rotating at all, in the world frame atleast) does the body rotate about in this state?

Difficulties:

1. They are pure moments, and cant be expressed as forces, which is usually what I see in books
2. Resultant moment is simple to calculate using vectors, but it doesnt give the position of the resultant moment
3. Does a free body always rotate about its center? (In which case I am totally approaching it in a wrong way)
4. Does it work the same way as finding the center of mass for multiple point masses? This is my best guess, but I am not sure.

• If there are no forces, then the center of mass cannot accelerate. – BowlOfRed Aug 11 '15 at 0:41
• @BowlOfRed That seems Ok to me, indeed it shouldnt.But what about the center of rotation, which I intend to know? Will it not even rotate? – bendtherules Aug 11 '15 at 0:50
• If you apply a torque, then the angular momentum of the object will change. If it wasn't rotating ($L=0$), it will rotate. – BowlOfRed Aug 11 '15 at 0:52
• @BowlOfRed But again, which one is the center of rotation if I apply torque to it? It shouldnt accelerate, but from the world frame then it must look like its rotating about some point, right? – bendtherules Aug 11 '15 at 1:02
• Without external forces, the only possible point of rotation is the center of mass. The center of mass must be stationary in some inertial frame. – BowlOfRed Aug 11 '15 at 3:05

No difficulties at all.

If the net force applied on a body is zero then the center of mass is not going to move accelerate. This leads to the conclusion that the only motion allowed is a rotation about the center of mass.

For more details refer to: https://physics.stackexchange.com/a/81078/392

The relevant equations are:

$$\mathbf{F} = m \,\mathbf{a}_C$$ $$\mathbf{M}_C = I_C \mathbf{\alpha} + \mathbf{\omega} \times I_C \mathbf{\omega}$$

where $\bf F$ is the net force acting on a body (in your case it is zero), ${\bf M}_C$ is the net moment about the center of mass acting on the body, $m$ is the mass, $I_C$ is the mass moment of inertia about the center of mass, and $\omega$ and $\alpha$ are the rotational velocity and acceleration of the rigid body.

The full derivation of the equations of motion are here: https://physics.stackexchange.com/a/80449/392

If the net force $\mathbf{F}=0$ is zero then the acceleration of the center of mass is zero $\mathbf{a}_C=0$. Only a rotation about the center of mass can cause this condition. Otherwise the center of mass will accelerate.

• Thanks, that explained it. So, just a thought - You cant rotate a body about any point other than the COM using force and moment? – bendtherules Aug 11 '15 at 7:50
• You can with force. You can't with moment only. When a body rotates about a pin under the influence of a moment, it is the reaction forces of the pin that move the center of mass. – ja72 Aug 11 '15 at 15:27