The angular velocity of the merry-go-round after a 24 kg child gets onto it can be calculated as follows:
Step 1: Calculate the moment of inertia of the system before the child gets on. The moment of inertia of the merry-go-round with no children on it is given by:I = (1/2)mr²I = (1/2)(145 kg)(1.7 m)²I = 216.13 kg m²
Step 2: Calculate the moment of inertia of the system after the child gets on.The moment of inertia of the system with the child on the edge is given by :I = I₁ + I₂ where I₁ is the moment of inertia of the merry-go-round with no children on it, and I₂ is the moment of inertia of the child. I₁ = 216.13 kg m²I₂ = mr² where m is the mass of the child and r is the distance of the child from the center of the merry-go-round. In this case, m = 24 kg and r = 1.7 m.I₂ = (24 kg)(1.7 m)²I₂ = 69.84 kg m²I = I₁ + I₂I = 216.13 kg m² + 69.84 kg m²I = 285.97 kg m²
Step 3: Apply the law of conservation of angular momentum.L1 = L2where L1 is the angular momentum of the system before the child gets on, and L2 is the angular momentum of the system after the child gets on.L1 = I₁ω₁where ω₁ is the initial angular velocity of the merry-go-round.L2 = Iωf where ωf is the final angular velocity of the system.ω₁ = 2πfiω₁ = 2π(0.42 rev/s)ω₁ = 2.64 rad/sL1 = I₁ω₁L1 = (216.13 kg m²)(2.64 rad/s)L1 = 570.68 kg m²/sL2 = I₂ωfL2 = (69.84 kg m² + 216.13 kg m²)ωfL2 = 285.97 kg m²ωfωf = L2/Iωf = (570.68 kg m²/s)/(285.97 kg m²)ωf = 1.996 rad/s
Therefore, the magnitude of the angular velocity of the merry-go-round after a 24 kg child gets onto it is 1.996 rad/s.
Angular Velocity : Precise speed, time rate at which an article turns, or spins, about a hub, or at which the rakish relocation between two bodies changes. The angle between two lines on one body and the other in the figure depicts this displacement.