Final answer:
To find the new angular velocity of the merry-go-round after the child moves to the center, the conservation of angular momentum principle is used, resulting in a calculation based on changes in the system's moment of inertia.
Step-by-step explanation:
To solve for the new angular velocity of the merry-go-round when the child of mass 34.8 kg moves to the center, we need to apply the principle of conservation of angular momentum, as there are no external torques. The system's initial angular momentum must equal the final angular momentum. The initial angular momentum can be found by considering the moment of inertia of the merry-go-round and the children, where each child is treated as a point mass at the edge (radius 1.60 m).
The initial moment of inertia (Iinitial) is the sum of the merry-go-round's moment of inertia and the moments of inertia of each child: Iinitial = Imr + Σ Ichild. The merry-go-round's moment of inertia is Imr = MR2, where M is the mass of the merry-go-round and R is its radius. The moment of inertia for a child located at the edge is Ichild = mchildR2, where mchild is the mass of the child.
After one child moves to the center, their moment of inertia becomes zero as their radius is now zero. The new moment of inertia (Inew) is just the moment of inertia of the merry-go-round plus the moments of inertia of the remaining two children.
The angular momentum must be conserved, so Iinitialωinitial = Inewωnew, which allows us to solve for the new angular velocity ωnew. Since the initial angular velocity (ωinitial) is given as 21.3 rpm, we can substitute and solve for ωnew.