Three children are riding on the edge of a merry–go–round that is 84 kg, has a 1.40–m radius, and is spinning at 20.0 rpm. The children have masses of 21.0, 24.0, and 29.0 kg. If the child who has a mass of 24.0 kg moves to the center of the merry–go–round, what is the new angular velocity in rpm?
To find the new angular velocity, we need to apply the conservation of angular momentum.
Angular momentum is given by the formula:
L = Iω
Where:
L is the angular momentum,
I is the moment of inertia, and
ω is the angular velocity.
The moment of inertia for a solid disc rotating about its axis is given by the formula:
I = 1/2 * m * r^2
Where:
m is the mass of the object, and
r is the radius of rotation.
Now, let's find the initial angular momentum of the system before the child moves to the center of the merry-go-round.
Initial angular momentum (L_initial) = (moment of inertia of the merry-go-round) * (initial angular velocity)
moment of inertia of the merry-go-round (I_merry-go-round) = 1/2 * m * r^2 (Using the given mass and radius)
L_initial = (1/2 * 84 kg * (1.40 m)^2) * (20.0 rpm * 2π/1 min)
Next, let's find the final angular momentum after the child moves to the center of the merry-go-round.
Final angular momentum (L_final) = (moment of inertia of the merry-go-round with the child at the center) * (final angular velocity)
moment of inertia of the merry-go-round with the child at the center (I_final) = 1/2 * (84 kg - 24 kg) * (1.40 m)^2 (Using the original mass of the merry-go-round minus the mass of the child and the radius)
To find the final angular velocity (ω_final), equate the initial and final angular momentum:
L_initial = L_final
(1/2 * 84 kg * (1.40 m)^2) * (20.0 rpm * 2π/1 min) = (1/2 * (84 kg - 24 kg) * (1.40 m)^2) * ω_final
Solve for ω_final to find the new angular velocity in rpm.