A skater changes from \(I_i=3.0\,\mathrm{kg\,m^2}\) to \(I_f=1.5\,\mathrm{kg\,m^2}\) while initially spinning at \(4.0\,\mathrm{rad\,s^{-1}}\). Find \(\omega_f\).

State the condition required for angular momentum conservation about a chosen axis.

If a spinning system's moment of inertia decreases and angular momentum is conserved, what happens to angular speed?

If angular momentum is conserved during an inelastic rotational collision, must kinetic energy also be conserved?

A turntable has \(I=0.80\,\mathrm{kg\,m^2}\) and spins at \(5.0\,\mathrm{rad\,s^{-1}}\). A ring is dropped onto it, increasing the total inertia to \(1.2\,\mathrm{kg\,m^2}\). Find the final angular speed.

A \(0.20\,\mathrm{kg}\) clay ball moving at \(6.0\,\mathrm{m\,s^{-1}}\) sticks to the rim of a stationary disk at radius \(0.40\,\mathrm{m}\). If the final moment of inertia is \(0.18\,\mathrm{kg\,m^2}\), find \(\omega_f\).

A skater doubles their moment of inertia while spinning freely. Find the ratio \(K_f/K_i\).

A disk with \(I_1=0.30\,\mathrm{kg\,m^2}\) spins at \(12\,\mathrm{rad\,s^{-1}}\). A second nonspinning disk with \(I_2=0.20\,\mathrm{kg\,m^2}\) lands on it and sticks. Find the final angular speed.

A particle of mass \(m\) moves past an origin with speed \(v\) and impact parameter \(b\). It is captured by a rotating assembly whose final total moment of inertia about the origin is \(I_f\). Derive the final angular speed.

A person of mass \(m\) stands at radius \(r\) on a frictionless turntable of inertia \(I\) spinning at \(\omega_i\). The person walks to the centre. Derive \(\omega_f\) and explain why the person's walking changes kinetic energy.

A uniform rod of length \(L\) and mass \(M\) is pivoted at its centre and initially at rest. A clay particle of mass \(m\) moving perpendicular to the rod with speed \(v\) sticks to the rod at distance \(a\) from the pivot. Derive the final angular speed and the kinetic energy lost.

Two identical masses \(m\) slide without friction on radial guides and rotate about the centre. Initially each mass is at radius \(r_i\) and the angular speed is \(\omega_i\). An internal string pulls both masses to radius \(r_f\). Derive \(\omega_f\), the change in kinetic energy, and the work that the internal pull must do.