Question

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.

Short answer

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.

Step-by-step solution

Identify the initial angular momentum

Initially, the system consists of the merry-go-round and the child at the center. The total initial angular momentum (L_initial) is given by the sum of the merry-go-round's angular momentum and the child's angular momentum. Since the child is initially at the center, its angular momentum is 0. So the total initial angular momentum is just the merry-go-round's angular momentum: \(L_{\text{initial}} = I_{\mathrm{i}}\omega_{i}\)

Identify the final angular momentum

After the child moves to the rim, two components contribute to the total angular momentum: the merry-go-round and the child. The child is now a distance \(R\) away from the axis of rotation, and the merry-go-round's moment of inertia is still \(I_{\mathrm{i}}\). Let's use \(\omega_f\) to denote the final angular velocity. The total final angular momentum (L_final) is given by: \(L_{\text{final}} = I_{\mathrm{i}}\omega_{\text{f}} + mR^2\omega_{\text{f}}\)

Applying the conservation of angular momentum

We know that the total angular momentum is conserved, which means that \(L_{\text{initial}} = L_{\text{final}}\). We can use this to find the final angular velocity: \(I_{\mathrm{i}}\omega_{i} = I_{\mathrm{i}}\omega_{\text{f}} + mR^2\omega_{\text{f}}\) Now, we can solve for \(\omega_{\text{f}}\): \(\omega_{f} = \frac{I_{\mathrm{i}}\omega_{i}}{I_{\mathrm{i}} + mR^2}\)

Calculate the rotational kinetic energy before and after

Before the child moves, the rotational kinetic energy (K_initial) is just the energy of the spinning merry-go-round: \(K_{\text{initial}} = \frac{1}{2} I_{\mathrm{i}}\omega_{i}^2\) After the child moves, the total rotational kinetic energy (K_final) is the sum of the kinetic energy of the merry-go-round and the child: \(K_{\text{final}} = \frac{1}{2} I_{\mathrm{i}}\omega_{f}^2 + \frac{1}{2} mR^2\omega_{f}^2\)

Calculate the angular momentum before and after

We have already calculated the initial and final angular momentum in steps 1 and 2. To summarize: Initial angular momentum: \(L_{\text{initial}} = I_{\mathrm{i}}\omega_{i}\) Final angular momentum: \(L_{\text{final}} = I_{\mathrm{i}}\omega_{\text{f}} + mR^2\omega_{\text{f}}\) Now you can plug in the given values for \(R\), \(I_{\mathrm{i}}\), \(m\), and \(\omega_{i}\) to calculate the final angular velocity, rotational kinetic energy, and angular momentum before and after the child moves to the rim.