# Acceleration of Center of Mass in Rotational Motion

I have a question regarding the acceleration of the center mass during rotational motion.

From my understanding, Fnet = m*a(center of mass).

Also, Torque = angular acceleration * moment of inertia.

Would this mean that the same force can do a different amount of work on an object depending on where it is applied?

For example, consider a rod floating in space. If it is pushed by force F at its center of mass, there will be no torque. However, if it is pushed near one of its ends, there will be the same acceleration of the center of mass as before, plus some torque.

• Also torque in 3D is $\vec{\tau} = I \vec{\alpha} + \vec{\omega} \times I \vec{\omega}$ not just $\tau=I \alpha$ as stated. – ja72 Nov 9 '14 at 18:16
• Related answer physics.stackexchange.com/a/80449/392 – ja72 Nov 9 '14 at 18:22

## 1 Answer

Yes, you are right! Only when a force is applied purely through the center of mass it results in the body gaining a linear action with no rotational components.When any force is applied at a distance from the center of mass, it results in the body gaining the linear acceleration mentioned above plus an angular acceleration which depends on the moment arm (perpendicular distance to the point of application). In all, yes, the work done by a force does depend on its point of application. (and of course the time interval through which it acts.)