How can I accelerate the linear speed of a spinning rock on a string when the rope can exert no net torque on the rock?

Question

If I take a rock attached to a mass-less rope and start spinning it over my head (or on a friction-less surface, I don't think it matters) I can accelerate the linear speed of the rock (I can spin it faster).

That means there is some form of angular acceleration, which implies a torque acting on the rock. But because there only force acting on the rock is through the tension in the rope, which is always perpendicular to it's direction of travel, where does the torque come from?

My best guess is that in a perfect world you wouldn't actually be able to accelerate the rock, but because you can sort of move your hand a little outside of the centre of the rock's orbit that you can pull it along a little bit. If this is the case, is it possible to accelerate a rock on a string using a stationary machine that spins the rope around, always pulling from the centre of the orbit?

Answer 1

Your guess is correct: you accelerate the rock by moving your hand a little so that the direction of the rope is slightly along the direction of motion of the rock (not quite perpendicular to it). Tugging the rope then accelerates the rock.

A machine that only exerted forces exactly perpendicular to the motion would not cause the speed to either increase or decrease. A good example of this occurs in the motion of a charged particle through a magnetic field.

You can also consider the case of a machine that sits exactly still, with the rock going around in a circle to begin with, and then the machine starts to wind the rope in. Now the rock follows a spiral, and therefore not quite a circular path. While the rope is winding in it will exert a force slightly along the direction of motion of the rock, causing the rock's speed to increase, in such a way that its angular momentum about the centre stays constant.