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

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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