Finite-radius spherical pendulum bob ($T=2\pi \sqrt{(L^2+2a^2/5)/(gL)}$)

A homogeneous spherical bob rigidly fixed to a massless support has centre-of-mass distance $L$ from its pivot and intrinsic moment of inertia $2M{a}^2/5$. The parallel axis theorem and linear gravitational restoring torque give the title period for small oscillations. Relative to a point mass at the same centre, the leading fractional period increase is $a^2/(5L^2)$. If $L$ is instead measured to the sphere's surface, the centre distance must first be changed to $L+a$.