A 1.80 m radius playground merry-go-round has a mass of 120 kg and is rotating with an angular velocity of 0.300 rev/s. What is its angular velocity after a 22.0 kg child gets onto it by grabbing its outer edge? The child is initially at rest.
To determine the angular velocity of the playground merry-go-round after the child gets onto it, we can use the principle of conservation of angular momentum.
The angular momentum of the merry-go-round before the child gets on is given by:
L1 = I1 * ω1
Where:
L1 is the initial angular momentum
I1 is the moment of inertia of the merry-go-round
ω1 is the initial angular velocity of the merry-go-round
The moment of inertia for a solid cylinder rotating about its axis is given by:
I = (1/2) * m * r^2
Where:
m is the mass of the object (in this case, the merry-go-round)
r is the radius of the merry-go-round
The angular momentum of the child after getting on the merry-go-round is given by:
L2 = I2 * ω2
Where:
L2 is the final angular momentum
I2 is the moment of inertia of the merry-go-round with the child on it
ω2 is the final angular velocity of the merry-go-round
Since angular momentum is conserved, we have:
L1 = L2
Substituting the values into the equation:
I1 * ω1 = I2 * ω2
The initial angular velocity, ω1, is given as 0.300 rev/s. We need to convert it to radians per second by multiplying it by 2π:
ω1 = 0.300 rev/s * 2π rad/rev = 0.300 * 2π rad/s
The moment of inertia of the merry-go-round, I1, can be calculated using the formula:
I1 = (1/2) * m * r^2 = (1/2) * 120 kg * (1.80 m)^2
Similarly, the moment of inertia of the merry-go-round with the child on it, I2, can be calculated using the same formula, but with the total mass (m1 + m2) and the same radius:
I2 = (1/2) * (120 kg + 22 kg) * (1.80 m)^2
Now we can solve the equation for ω2:
I1 * ω1 = I2 * ω2
Plug in the known values and solve for ω2:
(1/2) * 120 kg * (1.80 m)^2 * (0.300 * 2π rad/s) = (1/2) * (120 kg + 22 kg) * (1.80 m)^2 * ω2
Simplifying the equation will give the angular velocity, ω2, of the merry-go-round after the child gets onto it.