This is a concept which is bothering me a lot. I'm sure there are other duplicates to this question but on seeing a few such duplicates, I haven't yet obtained an answer.
My question is, for a rigid body undergoing rotational motion, Can I find ' Total Torque about any line on the body and equate it to the moment of inertia of the body about that line multiplied by angular acceleration and say without doubt that that angular accelerarion is the angular acceleration with which the whole body rotates ? ' In other words, ' Will the CALCULATED angular acceleration about different lines be the same'. I do know that angular acceleration is the same at all points in a rigid body. However, when we proved $\tau=I\alpha$ to hold in rigid body we did such that τ and I were taken about AXIS OF ROTATION and finally after proof we said $\tau=I\alpha$ , where I and τ are calculated about AXIS OF ROTATION; and $\alpha$ is the angular accelerarion of all all the particles of the rigid body and hence it is the angular acceleration of the body. Will this $\tau=I\alpha$ equation still hold good and will it give me the same $\alpha$ if I change $\tau$ and $I$ by calculating it about a different line?
Now I considered a sphere and pure rolling on the ground with friction acting at contact along with an external force F on another part of the rigid body. Working with axis of rotation as Centre of mass or working with axis of rotation as contact point/instantaneous axis of rotation gave me the same result when I solved a problem. Now in any general case how many points can I take as axis of rotation? Or is there only one point which i can take which is the actual axis of rotation and along with it in cases of pure rolling I can take even instantaneous axis of rotation?