A particle with mass m is moving with constant speed v along a circular orbit (radius r).
The centripetal force F=m\frac{v^2}{r} is provided by gravitation force from another mass M=F/g.
A string is connected from mass m to the origin then connected to mass M.
Because the force is always in the \hat{r} direction, so the angular momentum \vec{L}=m\,\vec{r}\times \vec{v} is conserved. i.e. L=mr^2\omega is a constant.

For particle with mass m:

m \frac{d^2r}{dt^2}=m\frac{dv}{dt}= m \frac{v^2}{r}-Mg=\frac{L^2}{mr^3}- Mg

The following is a simulation of the above model.

You can change the mass M or the radius r with sliders.
The mass M also changed to keep the mass m in circular motion when you change r.
However, if you change mass M , the equilibrium condition will be broken.

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