= Pendulum with a uniform rod and an endpoint mass
{title2=$T=2\pi\sqrt{l(m+M_r/3)/[g(m+M_r/2)]}$}
A uniform rigid rod of length $l$ and <mass> $M_r$, pivoted at one end with a <point mass> $m$ at the other, has pivot <moment of inertia> $I=l^2(m+M_r/3)$. Its gravitational <potential energy> is $-gl(m+M_r/2)\cos\theta$. The planar <Euler-Lagrange equation> is $I\ddot\theta+gl(m+M_r/2)\sin\theta=0$, giving the displayed small-amplitude <oscillation period>. The limit $M_r=0$ gives a <simple pendulum>, whereas $m=0$ gives the uniform-rod <physical pendulum>. Thinness does not specify a rod's mass, so both contributions must be stated or explicitly neglected.
Back to article page
