A point mass $m$ is glued inside a massless hollow rod of length $L$ at an unknown location. The rod lies on a frictionless horizontal table. In the first experiment, the rod is pivoted at end A. An ideal spring of constant $k$ is attached to end B such that its axis is perpendicular to the rod, with the other end of the spring fixed to the table. The period of small oscillations about the equilibrium position is $T$. In the second experiment, the rod is pivoted at end B, and the same spring is attached to end A in the same perpendicular manner. The period of small oscillations is $2T$. How far is the mass from the center of the rod? (Answer in terms of $L$)