- Finding angular acceleration of rotating Sphere. 3 methods 3 different answers
As shown in figure there's a homogeneous solid sphere. It is rotating about axis which is passing through point P directed perpendicular to the plane of paper. (In short like a pendulum).
I'm neglecting gravity and assuming a force F which is directed perpendicular to the string. (The string, if extended will pass though the centre of the sphere, also note that the force will be continuously changing its direction so as to always maintain 90° angle).
It's clear that the only external forces acting on the sphere are Tension and F (both perpendicular). The centre of mass moves in a circle (due to constraint by string).
Suppose I have to calculate angular acceleration of the sphere. There are three ways each yielding a different value. How is it possible?
Method 1: Centre of mass moves in circle about point P with tangential force F and radial force T. The angular acceleration of COM is simply Tangential Force (F) divided by product of distance OP and mass of body. (Only external forcees act on centre of mass, here external forces are F and Tension T)
Method 2: The axis passing through O is stationary. The torque about this axis is obvious. The moment of inertia of body about given axis is known. Using this the angular acceleration is found to be lesser than that found using method 1. (Please see uploaded picture at the end of question to see the exactly about the methods)
Method 3: If I consider an accelerating axis passing through centre of mass of sphere. The motion can be expressed as translation of COM + rotation about this axis. As this axis is accelerating the torque won't be equal to product of moment of inertia and angular acceleration (but I've found myself that if torque is taken about centre of mass then it is always true no matter where COM is accelerating or not, I'm going to show it how). To rectify this I'd apply pseudo force COM of sphere. But this pseudo force won't make any contribution to torque about this accelerating axis. Moreover there is no torque due to radial force T, in fact there is no torque at all! Something is definitely wrong, Maybe I've made a conceptual error. I've spend hours to find what am I doing wrong! Please help me!
EDIT: Why isn't someone replying? Is there something wrong with the question fell free to point out the mistake.