## College Physics (4th Edition)

(a) The new angular velocity is $\frac{I_i}{I_i+mR^2}~\omega_i$ (b) $KE_{rot,i} = \frac{1}{2}I_i~\omega_i^2$ $KE_{rot,f} = \frac{1}{2}\frac{(I_i~\omega_i)^2}{I_i+mR^2}$ $L_f = I_i~\omega_i$ $L_f = I_i~\omega_i$
(a) We can find the final rotational inertia of the merry-go-round and child: $I_f = I_i+mR^2$ We can use conservation of angular momentum to find the new angular velocity: $L_f = L_i$ $I_f~\omega_f = I_i~\omega_i$ $\omega_f = \frac{I_i~\omega_i}{I_f}$ $\omega_f = \frac{I_i}{I_i+mR^2}~\omega_i$ The new angular velocity is $\frac{I_i}{I_i+mR^2}~\omega_i$ (b) We can find the initial rotational kinetic energy: $KE_{rot,i} = \frac{1}{2}I_i~\omega_i^2$ We can find the final rotational kinetic energy: $KE_{rot,f} = \frac{1}{2}I_f~\omega_f^2$ $KE_{rot,f} = \frac{1}{2}(I_i+mR^2)~(\frac{I_i}{I_i+mR^2}~\omega_i)^2$ $KE_{rot,f} = \frac{1}{2}\frac{(I_i~\omega_i)^2}{I_i+mR^2}$ We can find the initial angular momentum: $L_f = I_i~\omega_i$ We can find the final angular momentum: $L_f = I_f~\omega_f$ $L_f = (I_i+mR^2)(\frac{I_i}{I_i+mR^2}~\omega_i)$ $L_f = I_i~\omega_i$