A light rigid rod is connected rigidly with two identical particles each of mass m as shown. The free end of the rod is smoothly pivoted at $O$. The rod is released from rest from its horizontal position at $t=0$. Find:

(i) angular acceleration of the rod, at $t=0$.

(ii) reaction offered by the pivot, at $t=0.$

Two blocks each of mass m are hanging from a stepped pulley by an inextensible light string which passes over a stepped pulley. The mass of the pulley is $m$ and its radius of gyration is $k$. If the string does not slide on the pulley, find the accelerations of the blocks.

$ABC$ is a triangular plate of uniform thickness. The sides are in the ratio shown in the figure. $I_{AB}, I_{\mathit{B}\mathit{C}}, I_{\mathit{C}\mathit{A}}$ are the moments of inertia of the plate about $AB, BC,CA$ respectively. Which one of the following relations is correct?

A sphere is moving towards the positive $x$-axis with a velocity $v_c$ and rotates clockwise with angular speed $\omega$ as shown in figure, such that $v_c>\omega R$. The instantaneous axis of rotation will be

In a rectangle $ABCD\mathit{,}$ $\mathrm{AB}=21 \mathrm{cm}$ and $\mathrm{BC}=1 \mathrm{cm}.$ Axes $x{x}^{\text{'}}$ and $yy\text{'}$ passes through centre of the rectangle. The moment of inertia is least about:

About which axis moment of inertia in the given triangular lamina is maximum?