Final answer:
The new angular velocity of the merry-go-round can be calculated using the conservation of angular momentum. By considering the initial angular momentum of the merry-go-round and the child after they grab the outer edge, we can determine the final angular velocity. The new angular velocity is approximately 0.414 rad/s.
Step-by-step explanation:
To calculate the new angular velocity of the merry-go-round, we can use the conservation of angular momentum. The initial angular momentum of the merry-go-round is equal to the sum of the angular momentum of the original system and the child after they grab the outer edge. The initial angular momentum of the merry-go-round is given by Li = Imerry-go-round ● ωi, where Imerry-go-round is the moment of inertia of the merry-go-round and ωi is the initial angular velocity. The angular momentum of the original system is zero since the children are initially at rest. The angular momentum of the child after they grab the outer edge is equal to child ● child ● ω, where child is the mass of the child, the child is the distance of the child from the axis of rotation, and ω is the angular velocity.
Applying the principle of conservation of angular momentum, we have:
Li = (Imerry-go-round + child ● child) ● ωf
Solving for ωf, we get:
ωf = Li / (Imerry-go-round + child ● child)
Substituting the given values, we have:
ωf = (1000.0 kg.m² ● 6.0 rev/min) / (1000.0 kg.m² + 22.0 kg ● 0.455 m)
Converting rev/min to rad/s, we get:
ωf = (1000.0 kg.m² ● (6.0 rev/min ● 2π rad/rev) / (60 s/min)) / (1000.0 kg.m² + 22.0 kg ● 0.455 m)
Simplifying the expression, we find that the new angular velocity of the merry-go-round is approximately 0.414 rad/s.