A uniform metal plate shaped like a triangle $ABC$ has a mass of $540 \mathrm{g}$. The length of the sides $AB,BC$ and $CA$ are $3 \mathrm{cm},5 \mathrm{cm}$ and $4 \mathrm{cm},$ respectively. The plate is pivoted freely about the point $A.$ What mass must be added to a vertex, so that the plate can hang with the long edge horizontal ?

One end of a rod of length $L=1 \mathrm{m}$ is fixed to a point on the circumference of a wheel of radius $R=\frac{1}{\sqrt{3}} \mathrm{m}.$ The other end is sliding freely along a straight channel passing through the center $O$ of the wheel as shown in the figure below. The wheel is rotating with a constant angular velocity $\omega$ (in radian per second) about $O.$

The speed of the sliding end $P$ when $\theta =60°$ is

A solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown in the figure. The cube is free to rotate about a fixed axis $AB.$ A bullet of mass $m(<<M)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height of $\frac{4a}{3}$ from the surface to impart the cube and angular speed $\omega .$ It strikes the face and embeds in the cube. Then $\omega$ is close to (note: the moment of inertia of the cube about an axis perpendicular to the face and passing through the center of mass is $\frac{2M{a}^2}{3}$.)