[Kryptid]

For angular momentum to be conserved the tangential velocity must remain constant.

Not if the radius is decreasing (as in your spiral examples). The tangential velocity must increase in order to compensate for the decrease in radius.

Angular momentum is conserved because it is based on Noether's theorem: https://en.wikipedia.org/wiki/Noether%27s_theorem

[David Cooper (OP)]

This is another instance of Dark Motion. Angular Momentum is not conserved.
...
As several posts point out, there is a tangential force in addition to the centripetal force when a mass is moved on a spiral path. That force is reacted by change in the magnitude of the velocity and hence a change in momentum.

I think you'll find that you've introduced an extra factor which introduces external torque. If a string is being pulled in to make the ball follow a spiral path, there is no external torque: pulling the string adds to the speed of the ball and force has to be put in, but that force can be stored internally, though even if it's fed in from outside by such means as shining light on a solar panel, that doesn't involve external torque.

In the case of a ball following a spiral path due to the string wrapping itself round a pole, the force is being applied from off centre and will apply a deceleration force to the ball which must be equal to the acceleration being applied by pulling it inwards, so the ball does not go round the pole at a higher speed: we have conservation of momentum with no energy being expended to pull the ball in, but the angular momentum goes down due to the braking force of the pull coming from off centre, and that off centre pull has to be resisted by the pole not rotating, so the pole is applying torque to something external.

If the pole isn't attached to anything that you consider to be external because the system is floating in space, then that off-centre pull on the rod to make it rotate will rotate the object the pole is anchored in, so there will be conservation of angular momentum when you include the change in rotation to that part of the apparatus holding the rod. Alternatively, you can call that part external to the rod, string and ball system, and then you have to class it as external torque.

[Momentus]

I think you'll find that you've introduced an extra factor which introduces external torque

The description of Fig 2 is quite explicit. There is no external force. No external force present and none required for the masses to be constrained to a spiral path. This is an interpretation of your original post.

string wrapping itself round a pole

It is not necessary to introduce an additional complication.

[David Cooper (OP)]

I think you'll find that you've introduced an extra factor which introduces external torque

The description of Fig 2 is quite explicit. There is no external force. No external force present and none required for the masses to be constrained to a spiral path. This is an interpretation of your original post.

My attempt to work out what point you were trying to make may well have failed. At the top you said "Angular Momentum is not conserved", so that influenced the way I interpreted the rest. If you have something spiraling in by shortening the string and you lack air resistance, then angular momentum will be conserved, but you're claiming it won't be, so I don't know where you're getting that claim from.

string wrapping itself round a pole

It is not necessary to introduce an additional complication.[/quote]

I thought that perhaps it was something you were doing as part of your way of backing your claim, and it's certainly worth commenting on because it superficially looks like the same kind of spiraling action and it maintains the momentum of the ball rather than increasing it.

I have no idea what point your post was actually trying to get across. If you don't want people to have to guess, you need to spell things out more clearly.

[Momentus]

Post in haste, eat humble pie at leisure.

I was wrong to say that the tangential speed of the spiralling gyro must remain constant to fulfil the conditions for the conservation of angular momentum.

Assuming that they have the same mass and the same angular momentum a smaller, faster flywheel has a greater energy potential than a slower, larger flywheel.

My Fig 2 and the force graphic fig 3, shows how that energy is transferred.

The example discussed in this post does not violate the conservation of angular momentum. I was wrong to say so

The title of your post is most apt.