Question #84309

A simple pendulum consists of a small bob of mass $m$ and positive charge $+q$ suspended by a massless, inextensible string of length $L$. The pendulum is attached to the upper plate of a horizontal parallel-plate capacitor with plate separation $d$. The capacitor is connected to an ideal battery of constant voltage $V$, such that the upper plate is maintained at a positive potential relative to the lower plate. The entire setup is located in a mine at a depth $h = R/2$ below the Earth's surface, where $R$ is the radius of the Earth and $g$ is the acceleration due to gravity at the surface. It is given that $qV = \frac{1}{2} mgd$. Initially, the space between the plates is vacuum. A solid, uncharged dielectric slab of dielectric constant $K = 3$ and thickness $d/2$ is then slowly inserted and placed on the lower plate. Assuming the pendulum bob always oscillates in the vacuum region ($L < d/2$) and edge effects are negligible, what is the new time period of small oscillations of the pendulum?