Question

What kind of motion is caused by nonzero torques about two parallel axes?

Answer 1

Let's take a particular example of a disk rotating around an axis through its centre:

If the component of the torque parallel to the axis is TaTa then the disk will start to rotate according to the angular version of Newton's second law:

Ta=Iω˙(1)(1)Ta=Iω˙

where ω˙ω˙ is the angular acceleration and II is the moment of inertia. The obvious way to make this rotate is to apply a tangential force:

In this case the magnitude of the torque is just T=rFT=rF and the direction of the torque vector is parallel to the axis so Ta=T=rFTa=T=rF and equation (1) tells us that the disk will start rotating.

Now suppose we change the direction of the force so it points directly towards the axis:

Obviously this isn't going to make the disk rotate, and equally obviously it's because the force and radial vectors are in a line so T=r×F=0T=r×F=0 i.e. the torque is zero. That means TaTa is zero and equation (1) tells us the disk won't rotate.

In this case the torque is non-zero, but the torque vector T=r×FT=r×F points out of the screen towards us i.e. the torque vector is at 90º to the axis. That means the component of the torque parallel to the axis, TaTa, is zero and that's why the disk won't rotate.