A solid cylinder of mass $$2 \mathrm{~kg}$$ and radius $$0.2 \mathrm{~m}$$ is rotating about its own axis without friction with angular velocity $$5 \mathrm{~rad} \mathrm{s}^{-1}$$. A particle of mass $$1 \mathrm{~kg}$$ moving with a velocity of $$5 \mathrm{~ms}^{-1}$$ strikes the cylinder and sticks to it as shown in figure.

The angular velocity of the system after the particle sticks to it will be

Joule second is the unit of

A body of mass $$5 \mathrm{~kg}$$ at rest is rotated for $$25 \mathrm{~s}$$ with a constant moment of force $$10 \mathrm{~Nm}$$. Find the work done if the moment of inertia of the body is $$5 \mathrm{~kg} \mathrm{~m}^2$$.

A thin circular ring of mass ,$$M$$ and radius $$R$$ rotates about an axis through its centre and perpendicular to its plane, with a constant angular velocity $$\omega$$. Four small spheres each of mass $$m$$ (negligible radius) are kept gently to the opposite ends of two mutually perpendicular diameters of the ring. The new angular velocity of the ring will be