angular momentum - 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.

Answer

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.