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 $\frac{2M{a}^2}{3}$.)

A uniform rectangular plate of mass $m$ which is free to rotate about the smooth vertical hinge passing through the centre and perpendicular to the plate is lying on a smooth horizontal surface. A particle of mass $m$ moving with speed $u$ collides with the plate and sticks to it as shown in the figure. The angular velocity of the plate after the collision will be

A hollow tilted cylindrical vessel of negligible mass rests on a horizontal plane as known. The diameter of the base is a and the side of the cylinder makes an angle $\theta$ with the horizontal. Water is then slowly poured into the cylinder. The cylinder topples over when the water reaches a certain height $h,$ given by-