If the system is released, then the acceleration of the centre of mass of the system:

A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion is the $x-y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated about $O$ and $P$ are denoted by ${\overrightarrow{L}}_{\mathrm{O}}$ and ${\overrightarrow{L}}_{\mathrm{P}}$, respectively, then

A uniform rod $OB$ of length $1 \mathrm{m}\text{,}$ cross-sectional area $0.012 {\mathrm{m}}^2$ and relative density $2.0$ is free to rotate about $\mathrm{O}$ in vertical plane. The rod is held with a horizontal string $AB$ which can withstand a maximum tension of $45 \mathrm{N}.$ The rod and string system is kept in water as shown in figure. The maximum value of angle $\alpha$ which the rod can make with vertical without breaking the string is

Four thin rods of same mass $M$ and same length $l$ form a square as shown in the figure. Moment of inertia of this system about an axis through centre $O$ and perpendicular to its plane is

A slender uniform rod of mass $M$ and length $l$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $\mathrm{\theta }$ with the vertical is.