Which is the axis of rotation?

Question

This should be simple, but it keeps bothering me. If a rigid body has no fixed axis, and a torque (defined relative to a point $A$) is applied, it will rotate around $A$. But often I can also calculate the torque relative to another point $B$ (which often seems to be non-zero too). So does this mean that the rigid body will have an angular acceleration about both axises? This seems a bit strange to me.

(For a fixed axis I assume that a rotation around any axis (other than the fixed axis) is impossible, because there will always be zero torque around those axes.)

Answer 1

Given a rigid body, a basic result in mechanics is that there exists a family of time-dependent rotations $R(t)$ such that for a given a fixed point $\vec x_0(t)$ in the body and for each point $\vec x_\alpha(t)$ in the body, one has $$ \vec x_\alpha(t) = \vec x_0(t) + R(t)(\vec x_\alpha(0) - \vec x_0(0)) $$ In words, this equation says that the motion of a rigid body can be described by a translation of any chosen fixed point, plus a rotation about that point. For each time $t$, the axis of rotation of $R(t)$ defines the axis of rotation of the rigid body. Notice that this rotation does not depend on the reference point fixed in the body that we choose. We could pick either the center of mass, or any other point in the body, but the description will remain the same. So, as far as I can tell, the question of "which point the body is rotating about" does not have a unique answer; the answer depends on how you choose to describe its motion, namely which fixed point $\vec x_0(t)$ you have chosen.

Cheers!

Answer 2

It will not rotate around A, since the centrifugal forces will make the body rotate about its center of mass. You can move the torque, it is not fixed to a given point.

Answer 3

In an inertial frame, Rotation is always about the Centre of Mass or Barycentre, If no constraints such a fulcrum's equivalent is present

If the choice of frame is not a matter of concern, You can take a different axis and you will get the exact same analysis if you account for pseudo forces and moment caused by that pseudo force.

Take C to be the centre of mass. B being another point and describe angular kinematic variables $\omega$ and $\alpha$

Now B goes with linear acceleration $\alpha r_B$.

Take B to be FIXED, ie apply $-\alpha r dm$ Everywhere in the body

this will give an effective torque which if you divide by I at B, will give you same $\alpha$ as before.

Now you can take any axis, but not forget pseudo force. At non-inertial frame rotation happens around Centre of Mass though.

Answer 4

There is another issue that must be clarified before making any definite conclusion. The question is:"Is rigid body in free state or is it constrained"? This question must be asked when dealing with rotors constrained by gravity in the bearings, constituting a rotor/ bearing system. Then next question arises:"Does this system has a resonance (natural motion) at particular angular velocity"? The experiments showed that rotor will rotate around the axis(natural motion), at which torque is applied, and mass axis (of the eccentric rotor) will precess around torque (forced centroidal axis). Above the resonance velocity, natural tendency of rotating body is to rotate around its mass axis, and torque applied axis will precess now around rotor mass axis. When this happens on rotating machines, it generates "vibrations" at constraints , i.e. the bearings.