# What is the distinction between work done by torque and force about an axis other than centre of mass?

Suppose a point particle is undergoing circular motion about a fixed point such that a tangential force is accelerating its speed continuously. Now I believe that I can calculate the work done on this particle either by Force or by Torque.

However, for a rigid body, when I calculate work, I calculate it as:

1. Work done by Torque in the frame of centre of mass to cause angular displacement added to the work done by the force in displacing the centre of mass.

or

1. By work done by force in displacing its point of application

Now I understand that both of my methods 1 and 2 are equally valid for the circular motion case

But what I do not understand is that why isn't the work done by both force and torque (about point of rotation) added in the circular motion case? Why are these two works synonymous here, but not in method 1?

Is method 1 (as described above) applicable only only when torque is calculated about Centre of Mass and no other point?

In the circular motion case we only have a particle, so the centre of mass itself is not defined. Since force is tangential, $$dW=F ds$$, so that $$W=F s$$ considering constant force. If you apply apply torque method you get $$W=\tau \theta =F r\theta =F s$$. Thus they are the same result. You can apply method (1) about any point, but be careful to consider the work of a pseudo torque due to the pseudo force that may arise if your chosen point is accelerated. You do not have to concern yourself with that if you choose COM as point of analysis because any pseudo force would pass through COM itself, hence creating no torque.