I think this has lost direction a bit.

If we look for the simplest analysis of the simplest system first we have the best chance of understanding what's going on. Good science tends to be reductionist.

Let's, initially, ignore friction or the changing diameter of the reel holding the string. The only force is along the string. If we can find a good reason for 1. a speed increase and ,2. an increase in tension, then we have made a valid start.

I think that was what the original question was really about.

We know that, as the radius of rotation of a rotating object decreases, its angular velocity goes up - this can be explained in terms of conservation of angular momentum. This principle yields Keppler's laws for planetry orbits and relies on a simple, central, attractive force. It explains skaters spinning fast as they bring their legs closer inwards and many other phenomena.

The simple sums show that, for the same basic reason, the tension increases in our string.

Just looking at the 'before and after' conditions and using angular momentum gives the right answer.

If you don't like my momentum ideas for this particular system then you could replace the central pivot with a balanced system of two equal masses, rotating round their common centre of mass, like a bola, used for hunting. The string could be wound in by each mass, at the same rate, to keep things balanced and we would have a completely isolated system, eliminating any possible apparent violation of conservation. The same result would be produced and could be a bit more acceptable?

Putting energy in or taking it out should still not violate the momentum conservation. Friction will, eventually, slow things down, in this case, but is it relevant to the basic system?

Of course, this is not the same situation as a satellite in orbit, where the central force follows the inverse square law. The loss in energy caused by friction reduces the orbital height and, as the satellite falls further and further into the potential well, there is more and more (gravitational potential) energy available; the satellite speeds up. Angular momentum is transferred to the atmosphere (and speeds up the world a bit), so the model, in that case relies on the satellite losing some momentum - due to a non-radial force. Without the friction, the satellite would just follow a Newtonian orbit, not spiraling in at all.

If the arrangement was like the diagram in Daveshorts' diagram and if the spring was 'open' and the coils never touched each other, following Hooke's Law, there would be no loss of energy and the mass would follow some sort of oscillating path around the centre, with varying radius - either spiraling in and out or like flower petals /a fried egg, depending on the rate of rotation and the stiffness of the spring. (I tried, briefly, to draw it but failed).

These discussions certainly make you think, don't they?

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