Question

I've been debating with a classmate about whether weight cause a torque on a sphere of uniform density on a friction-less incline. My perspective is that since the force of gravity acts from the center of mass, if we're considering a sphere then weight will be unable to produce a torque since it will act at the pivot point (the center of the sphere). My classmate contends that the pivot point is actually at the point of contact between the sphere and the surface. So, if this is true the parallel component of the weight would act on a lever arm extending from the point of contact to the center of mass, and therefore cause torque. Can you settle this for me?

Answer 1

Without any doubt the sphere slides without rotating when placed on a friction-less inclined surface.

Explanation:

When calculating torque we first need to be careful while choosing the axis about which we wish to compute the torque. The axis should pass through a non accelerating point, the only exception is the COM i.e., the axis passing through COM is valid even if COM is accelerating. In the case of the sphere on a friction-less inclined plane every point of the object is an accelerating point. So, only the axis passing through the COM is a valid axis about which we need to compute the torque. Since the force due to gravity passes through COM, it produces a net zero torque.

If we choose any other axis like the one passing through the pivot point, we need to consider the pseudo force before calculating the torque. We know that the pseudo force is also equal to mgsin(theta). Hence the net torque about the axis passing through the pivot is zero.