Continuing from the following scenario from my previous question Centripetal force of a rotating rigid body? :
Consider someone pushing a roundabout in a playground. Initially the
roundabout is stationary, but when it is pushed, it rotates with
increasing rotational speed.The force of the push is balanced by the reaction force exerted by the
support at the centre of the roundabout. The forces are equal in
magnitude and opposite in direction, so the roundabout is in
translational equilibrium. But they have different lines of action, so
there is a resultant torque, causing the playground to rotate and have
angular momentum.
Okay, suppose the roundabout's rotational speed now stabilises (ie. the resultant torque becomes zero and the roundabout is in rotational equilibrium). I infer that this happens only when the pushing force is removed (otherwise there would be a resultant torque as described in the yellow box). But if so, what (force) is keeping the roundabout rotating at its constant rotational speed (assuming no friction)? Isn't circular/rotational motion a forced motion??
Best Answer
Imagine two masses connected by a spring, like this:
If the entire thing spins at a constant angular velocity, the masses are moving in circles. Then $m_1$ and $m_2$ must have forces on them, since circular motion is a form of accelerated motion.
These forces are supplied by the spring. The spring will stretch somewhat, exerting forces pulling the masses back towards their equilibrium position. The magnitude of these forces is the usual centripetal force $m \omega^2 r$.
Now suppose you have a disk rotating like this:
The situation is very similar. If you take a point on this disk (besides the center), it's going in a circle. Therefore, this point has a centripetal acceleration, and must feel a force. Where is this force coming from?
The answer is that the force is coming from the disk itself. Imagine the disk as being a mesh of many points, all connected by springs. When you start the disk spinning, all the springs will stretch a little bit, exerting forces on the masses they're connected to. The net force of the springs on any given mass will be the centripetal force.
The way we usually express this is to say the entire disk is under tension (negative pressure). The tension decreases from a maximum at the center to zero at the edge. The gradient of the tension gives the force per unit volume, and that force is $\omega^2 r \rho \mathrm{d}V$ on a volume element $\mathrm{d}V$.