 Bead on a parabolic wire

For a smooth stationary wire $z=x^2/(4a)$ and release from rest at height $h$, the normal reaction does no work. The energy equation is $[1+x^2/(4a^2)]{\dot{x}}^2/2+g{x}^2/(4a)=gh$. Integrating the travel time between turning points gives the displayed period. Its small-amplitude expansion is $T=2\pi \sqrt{2a/g}[1+h/(4a)+O((h/a{)}^2)]$. The mass cancels because both gravitational and inertial forces are proportional to it.