K&K Question 3.22 – Mass, String, and Ring

1. The problem statement, all variables and given/known data

A mass m whirls around on a string which passes through a ring, as shown. Neglect gravity. Initially the mass is distance r0 from the center and is revolving at angular velocity ω0. The string is pulled with constant velocity V starting at t = 0 so that the radial distance to the mass decreases. Draw a force diagram and obtain a differential equation for ω. This equation is quite simple and can be solved either by inspection or by formal integration.

2. Relevant equations

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[tex]r=Vdt[/tex]
[tex]\frac{V^2}{r}=Vd\omega[/tex]
[tex]T=\frac{mV^2}{r}=mr\omega^2=mV^2dtd\omega[/tex]
[tex]N=m\ddot{y}[/tex]

3. The attempt at a solution

I have absolutely no idea on how to start this problem. I cannot obtain a valid differential equation from the tension, and I do not know how to relate the normal force to everything. Even if I write:
[tex]\frac{dv}{dt}=V^2dtd\omega[/tex]
It is still impossible to get a valid differential equation. I don’t even think the relation [itex]\theta=s/r[/itex] would help.

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