= Hoop with a suspended counterweight
{title2=$(M+m)R\ddot\theta=g(m-M\sin\theta)$}
A point mass $M$ attached to a massless rotating hoop of radius $R$ is connected by a taut string wrapping around the hoop to a vertically suspended mass $m$. Measure $\theta$ from the downward radius toward the string's tangency side. The wrapped length decreases by $R\theta$, so the suspended mass descends by the same amount. Therefore $T=(M+m)R^2\dot\theta^2/2$ and $V=-MgR\cos\theta-mgR\theta$ up to a constant. The <Euler-Lagrange equation> gives the displayed equation. For $0<m<M$, the stable equilibrium has $\sin\theta_0=m/M$, and the small-oscillation <angular frequency> is $\omega=\sqrt{g/R}[(M-m)/(M+m)]^{1/4}$. When $M\gg m$, the system approaches a <simple pendulum> of length $R$.
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