A merry-go-round (radius \(R\), rotational inertia \(I_{\mathrm{i}}\) ) spins with negligible friction. Its initial angular velocity is \(\omega_{i}\) A child (mass \(m\) ) on the merry-go-round moves from the center out to the rim. (a) Calculate the angular velocity after the child moves out to the rim. (b) Calculate the rotational kinetic energy and angular momentum of the system (merry-go-round + child) before and after.

Question: A merry-go-round with moment of inertia \(I_i\) is spinning with an initial angular velocity \(\omega_i\). A child initially standing at the center moves out to the rim of the merry-go-round, a distance \(R\) away from the center. Calculate the new angular velocity, rotational kinetic energy, and angular momentum before and after the child moves to the rim, given the values for \(R\), \(I_{\mathrm{i}}\), \(m\), and \(\omega_{i}\). Answer: First, calculate the final angular velocity using the conservation of angular momentum equation: \(\omega_{f} = \frac{I_{\mathrm{i}}\omega_{i}}{I_{\mathrm{i}} + mR^2}\) Next, calculate the rotational kinetic energy before and after the child moves: Before: \(K_{\text{initial}} = \frac{1}{2} I_{\mathrm{i}}\omega_{i}^2\) After: \(K_{\text{final}} = \frac{1}{2} I_{\mathrm{i}}\omega_{f}^2 + \frac{1}{2} mR^2\omega_{f}^2\) Finally, calculate the angular momentum before and after: Initial angular momentum: \(L_{\text{initial}} = I_{\mathrm{i}}\omega_{i}\) Final angular momentum: \(L_{\text{final}} = I_{\mathrm{i}}\omega_{\text{f}} + mR^2\omega_{\text{f}}\) Plug in the given values for \(R\), \(I_{\mathrm{i}}\), \(m\), and \(\omega_{i}\) to find the final angular velocity, rotational kinetic energy, and angular momentum before and after the child moves to the rim.