= Solution
Each freely moving particle has position, relative to its own initial radial and tangential unit <vectors>,
$$
x_i(t)=R,e_{r,i}+vt,e_{\theta,i}.
$$
Thus all particles lie on a circle of radius
$$
\boxed{R(t)=\sqrt{R^2+v^2t^2}}.
$$
Their <angular momenta> are constant and sum to
$$
\boxed{L=mRv,\hat z}.
$$
Since $I(t)=mR(t)^2$, the angular <velocity> is
$$
\dot\theta=\frac{L}{I(t)}
=\frac{Rv}{R^2+v^2t^2}.
$$
With $\theta(0)=0$,
$$
\boxed{\theta(t)=\arctan\frac{vt}{R}}.
$$
This is the geometry of a <freely expanding rotating particle ring>.
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