Question #43824

A physical pendulum is constructed using a rigid, massless rod of length $L$ pivoted freely at its upper end. Two small, massless containers are attached to the rod at distances $L/2$ and $L$ from the pivot. Initially ($t=0$), the container at $L/2$ holds a mass $m_0$ of a radioactive isotope A, and the container at $L$ is empty. Isotope A decays into isotope B with a decay constant $\lambda$. By a continuous internal mechanism, isotope B is instantly transferred to the container at $L$. Isotope B then decays into a stable gas C with a decay constant $2\lambda$, which instantly escapes the system completely. Assuming the atomic masses of A, B, and C are equal, find the angular frequency $\omega$ of small oscillations of the pendulum at the exact instant the mass of isotope B is maximized. (Let $g$ be the acceleration due to gravity)