A rod $PQ$ of mass $M$ and length $L$ is hinged at end $P$. The rod is kept horizontal by a massless string tied to point $Q$ as shown in figure. When the string is cut, the initial angular acceleration of the rod is,

Three identical spherical shells, each of mass $m$ and radius $r$, are placed as shown in figure. Consider an axis $X{X}^{\text{'}}$ which is touching two shells and passing through the diameter of third shell. The moment of inertia of the system consisting of these three spherical shell about $XX\text{'}$ axis is,

Two point masses $m_1$ and $m_2$ are placed at the opposite ends of a rigid rod of length $L$ and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point $P$ on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ${\omega }_0$ is minimum is given by,