# Torque caused about point of contact by a force passing through CoM of a sphere on a frictionless surface

A solid sphere (or any round object like a cylinder, disc for that matter) is kept on a frictionless surface. A force is applied to the sphere, parallel to the surface, and passing through its center of mass. Weight, Normal force and this applied force all act at CoM and hence would not produce any torque. However, the applied force would produce a torque about the Point of Contact. Normal force and Weight act through the PoC and won't produce torques to counter this one. How come then there is no rotation about PoC and instead, there shall be just translation?

If you apply an unbalanced force to the sphere, then it (and the point of contact) will accelerate. This means that a frame where the bottom of the sphere (the point of contact) is at rest is not an inertial frame.

In this accelerating frame, fictitious forces appear. In particular, $F_{fict} = -ma_{frame}$ can be assumed to act through the center of mass. This force exactly counters the applied force and you get no net torque (or net acceleration) in the frame.

If the applied force is at a different distance from the bottom of the sphere, then the torque about that point will have a different magnitude from that of the fictitious force and they won't cancel out. The rotation about that point will accelerate.

• Sorry, just a small correction related to my comment to Doug's answer. F applied at height R/2 above CoM shall cause pure rolling in case of a disc or a cylinder. It shall be something else for a sphere, since its MI is different. – Prasad Oct 25 '16 at 5:00

Just because there is a force applied at some distance away from some arbitrary point doesn't mean it will torque about that point. Rotation will happen around the PoC if the PoC is fixed.

In this case there is no friction so there is no way rotation about the PoC will occur. The situation you described is analogous to having a sphere in free space since the gravitational and normal forces balance. If you were to push directly on centre in this case, again the sphere would not rotate.

If you were to push off axis from the CoM, this would cause rotation in the case you described and in free space.

• Thanks Doug. Based on the earlier part of your answer I immediately got a doubt that you alluded to in your last line. If the force is applied off center (at height R/2 from the CoM), why does then rotation occur even when friction is absent and PoC isn't fixed? – Prasad Oct 25 '16 at 4:41
• @Prasad if you look at the answer in this post you will see that a force whose line of action is not passing through the centre of mass of a body can be decomposed into a force acting through the centre of mass and hence cause a linear acceleration of the mass and a couple which cause an angular acceleration of the body. physics.stackexchange.com/a/285167/104696 – Farcher Oct 25 '16 at 8:53