A thin but rigid semicircular wire frame of radius $$r$$ is hinged at $$O$$ and can rotate in its own vertical plane. A smooth peg $$P$$ starts from $$O$$ and moves horizontally with constant speed $$2 v_0$$ lifting the frame upward as shown in figure. Find the angular velocity $$\omega$$ of the frame when its diameter makes an angle of $$60^{\circ}$$ with vertical.

Four holes of radius 5 cm are cut from a thin square plate of 20 cm and mass 1 kg. The moment of inertia of the remaining portion about Z-axis is

A solid sphere of 80 kg and radius 15 m moving in a space becomes a circular disc of radius 20 m in 1 h. The rate of change of moment of Inertia in this process is .......

The angular momentum of a body placed at origin of mass 1 kg and having position vector $$r = 3t\widehat i + 4\widehat j$$ is