What is combined translation and rotation motion?

In the absence of external torque for a body revolving about any axis, the quantity that remains constant is

$A$ solid sphere with uniform density and radius $R$ is rotating initially with constant angular velocity $\omega_1$ about its diameter. After some time,it starts losing mass at a uniform rate,with no change in its shape. The angular velocity of the sphere when its radius becomes $R/2$ is $x\omega_1$. The value of $x$ is . . . . . . .

$A$ mass $m$ moves in a circle on a smooth horizontal plane with velocity $v_0$ at a radius $R_0$. The mass is attached to a string which passes through a smooth hole in the plane as shown. The tension in the string is increased gradually and finally $m$ moves in a circle of radius $\frac{R_0}{2}$. The final value of the kinetic energy is

$A$ horizontal platform is rotating with uniform angular velocity around the vertical axis passing through its centre. At some instant of time, a viscous fluid of mass $m$ is dropped at the centre and is allowed to spread out and finally fall off. The angular velocity during this period

$A$ disc of moment of inertia $I_1$ is rotating in a horizontal plane about an axis passing through its center and perpendicular to its plane with a constant angular speed $\omega_1$. Another disc of moment of inertia $I_2$ having zero angular speed is placed coaxially on the rotating disc. Now,both the discs are rotating with a constant angular speed $\omega_2$. The energy lost by the initial rotating disc is