Two coaxial discs,having moments of inertia $I_1$ and $\frac{I_1}{2}$,are rotating with angular velocities $\omega_1$ and $\frac{\omega_1}{2}$ respectively,about their common axis. They are brought into contact with each other and thereafter they rotate with a common angular velocity. If $E_f$ and $E_i$ are the final and initial total energies,then $(E_f - E_i)$ is

$A$ disc of mass $1 \, kg$ and radius $0.1 \, m$ is rotating with angular velocity $20 \, rad/s$. What is the angular velocity (in $rad/s$) if a mass of $0.5 \, kg$ is placed on the circumference of the disc?

$A$ rod of length $L$ and mass $M$ is free to move on a frictionless surface. $A$ ball of mass $m$ moving with speed $v$ strikes the rod as shown in the figure. What should be the mass of the ball so that it remains at rest after the collision?

$A$ horizontal disc of moment of inertia $4.25 \,kg \cdot m^2$ with respect to its axis of symmetry is spinning counter-clockwise at $15 \,rps$ about its axis,as viewed from above. $A$ second disc of moment of inertia $1.80 \,kg \cdot m^2$ with respect to its axis of symmetry is spinning clockwise at $25 \,rps$ as viewed from above about the same axis and is dropped on top of the first disc. The two discs stick together and rotate as one about their axis of symmetry. The new angular velocity of the system as viewed from above is close to

$A$ 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

$A$ circular disc has a moment of inertia $I_2$ about an axis passing through its center and perpendicular to its plane. Another disc with a moment of inertia $I_1$ is placed on it. If the system rotates about the same axis with an initial angular velocity $\omega$,what will be the final angular velocity of the combined system?