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Basic assumptions and consequences
We have a ball of mass m and radius rₘ spinning with speed v at a distance r from a pivot point. The tether is pulled in at constant speed u from the initial distance r₁ to the final one r₂. Correspondingly the speed reaches the new, to be determined level v₂. The following assumptions are applied:

- The pull on the theter results in a Coriolis force acting on the ball: F = m u v/r.
- A friction force acts at a distance rₚ from the pivot. This is modelled with a friction coefficient µ and considering only the centripetal force: F = µ m v²/r. Its effect on the ball is scaled by rₚ/r because of the reduced lever.
- Drag force in the common form F = 1/2 C ρ A v² acts on the ball too. Here C is the drag coefficient, ρ is air density, A = πrₘ² is the ball's cross surface.
- The wobbling of central pivot is modelled with a mass mₐ attached to the fixed centre by a spring of constant k.

This final assumption results in a variable displacement of the mass mₐ by an amount rₐ given by the centripetal force: rₐ = F/k = m v²/kr. The effect of this displacement on the ball can be modelled considering the loss of angular momentum in favour of the mass mₐ. It can be easily shown that the resulting Coriolis force on the ball gets modified according to (see below for the detailed calculation):

F = -2 m mₐ rₐ²/(mr² + mₐrₐ²) uv/r

We can factor in all these forces in a discrete-time simulation to obtain the time-dependece of the resulting speed v by means of numerically integrating the resulting differential equation for which no trivial analytic solution is available (as shown further below):

dv/dt = F/m = (1 - 2 mₐ rₐ²/(mr² + mₐrₐ²)) uv/r - µ rₚ v²/r² - (CρA/2m) v²

We can thus in particular obtain the final value v₂ as shown in the sheet "Results".
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Model for non-fixed pivot
The calculation of the modified Coriolis force reads as follows. We can determine the dragging effect of the mass mₐ on the ball in absence on other forces and transfer the result to the general case. This allows us a computation based on instantaneous conservation of angular momentum between the ball of mass m at distance r and the pivotal mass mₐ at distance rₐ:

dL/dt = 0 d(Iω)/dt = d(m r² + mₐ rₐ²)(v/r)/dt = 0
d(m r v + mₐ rₐ² v/r)/dt = 0
m dr/dt v + m r dv/dt + mₐ rₐ² dv/dt 1/r - mₐ rₐ² v/r² dr/dt = 0

where we neglected the time dependence of rₐ = mv²r/k i.e. drₐ/dt ~ 0. We can notice that dr/dt = -u by definition and express the result in terms of the acceleration dv/dt:

-m u v + m r dv/dt + mₐ rₐ² dv/dt 1/r + mₐ rₐ² v/r² u = 0
(m r + mₐ rₐ²/r) dv/dt = - mₐ uv rₐ²/r² + m u v
(m r² + mₐ rₐ²) dv/dt = - mₐ u v rₐ²/r + m u v r
dv/dt = (-mₐ rₐ² + m r²) / (m r² + mₐ rₐ²) uv/r = uv/r - 2 mₐ rₐ² / (m r² + mₐ rₐ²) uv/r

The first term is the Coriolis force acting on the ball and the second term is the "drag" induced by the presence of the mass mₐ at distance rₐ. We can thus finally write down the corresponding "effective" force caused by it:

F = -2 mₐ rₐ² / (m r² + mₐ rₐ²) uv/r

It is worth to notice that we implcitely approximated the ball with a point-mass here which means the result only holds for r >> rₘ.
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