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 $\theta$ with the vertical is

Seven identical circular planar disks, each of mass $M$ and radius $R$ are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point $P$ is 

From a uniform circular disc of radius $R$ and mass $9 \mathrm{M}$, a small disc of radius $\frac{R}{3}$ is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is:

An equilateral triangle $ABC$ is cut from a thin solid sheet of wood. (see figure) $D, E$ and $F$ are the mid-points of its sides as shown and $G$ is the centre of the triangle. The moment of inertia of the triangle about an axis passing through $G$ and perpendicular to the plane of the triangle is $I_0.$ If the smaller triangle $DEF$ is removed from $ABC,$ the moment of inertia of the remaining figure about the same axis is $I.$ Then

An L-shaped object made of thin rods of uniform mass density is suspended with a string as shown in figure. If $AB=BC$ and the angle made by $AB$ with downward vertical is $\theta$, then,

A rod of length $50 \mathrm{cm}$ is pivoted at one end. It is raised such that if makes an angle of $30°$ from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $\mathrm{rad} {\mathrm{s}}^{-1}$) will be $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis (see figure). When released from the initial horizontal position, its instantaneous angular acceleration will be: