Difference between the angular momentum for a mass following a trajectory and a mass rotating around a rotational center

Question

I struggle to find a satisfying reason and formulation, why for a point mass,

• that is fixed in a rotational system and the radius is changed (like pulling on a string with a small ball rotating around it), the speed of this point mass would increase due to the conservation of angular momentum.

• that follows a trajectory with decreasing radius (like a Roller Coaster on a Spiral shaped rail), the speed of this point mass would not increase, since conservation of angular momentum is not applicable in this case.

  (I already started doubting, if it really does not increase).

The only reason I found so far is that in the second case, the rail acts as an external force to the mass, while in the first case the radius change is an internal change of the system that comes along with putting energy into the system by working against the centrifugal forces. While I expect this is a somewhat correct answer, I still find this is a little bit hand waving argument.

I expect that I just miss a proper clean definition of external force vs internal force.

I also constructed the following thought experiment for myself, that involves an external rail, but I would expect to react like the first case:

When a point mass rotates on a circular stretchable rail and the radius of that rail would be changed by applying some force against the centrifugal force of the point mass, I'd also expect the velocity of the point mass increases, since some force applied over a distance (energy change) is involved.

But I doubt that the applied force over a distance is the underlying fundamental reason and I expect that there is another more satisfying fundamental difference between the fixed predetermined trajectory and the thought experiment / the first case mentioned in the beginning.

Answer 1

The difference is because you are applying the force to the object in different directions, as it spirals inwards.

For the ball rolling on a frictionless spiral track, the force is always perpendicular to the track, and therefore perpendicular to the direction of motion of the ball. Therefore, the force does no work on the ball.

For the ball on a string that is being pulled inwards, the ball is following the same spiral shaped track, but now the string tension acts towards the center of the spiral, not perpendicular to it. Therefore, there is a component of the tension in the direction of motion of the ball, and the tension in the string does work on the ball as the string is shortened.