= Spring-loaded centrifugal governor
{title2=$\omega^2=[g+2kL(1-\cos\theta)/m]/(L\cos\theta)$}
Consider two equal balls of <mass> $m$, four light rods each of length $L$, a fixed upper pivot and a light sliding lower collar. Each upper and lower rod makes angle $\theta$ with the spindle. A <spring> of <spring constant> $k$ is uncompressed at $\theta=0$ and is compressed by collar motion. Its compression is $\delta=2L(1-\cos\theta)$. Collar balance gives $2T_l\cos\theta=k\delta$, while vertical ball balance gives $(T_u-T_l)\cos\theta=mg$. Radial <Newton's second law> gives $(T_u+T_l)\sin\theta=m\omega^2L\sin\theta$. Eliminating the rod <tensions> proves the displayed relation for nonzero $\theta$.
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