A playground merry–go–round has a mass of 118 kg and a radius of 1.80 m and it is rotating with an angular velocity of 0.470 rev/s. What is its angular velocity after a 20.0–kg child gets onto it by grabbing its outer edge? The child is initially at rest.
I omega = constant
before
I suppose we could model the device as a flat disk
I = .5 M r^2 = .5(118)(1.8)^2
I omega = I (.47*2 pi)
now after
Inew = I + 20(1.8)^2
so
I omega = Inew Omeganew
.5(118)(1.8)^2(.47*2 pi)=Inew* Omeganew
that will give omega new in radians/sec
divide by 2 pi to get revs/s
To find the final angular velocity of the merry-go-round with the child on it, we can apply the principle of conservation of angular momentum. Angular momentum is conserved when no external torque acts on an object.
Angular momentum is given by the formula:
L = I * ω
Where:
L = Angular momentum
I = Moment of inertia
ω = Angular velocity
In this case, the initial angular momentum of the merry-go-round is equal to the final angular momentum of the merry-go-round plus the angular momentum of the child.
Initial angular momentum = Final angular momentum + Angular momentum of the child
The moment of inertia of the merry-go-round can be calculated using the formula:
I = m * r^2
Where:
m = Mass of the merry-go-round
r = Radius of the merry-go-round
The initial angular momentum of the merry-go-round is:
L_initial = I_merry-go-round * ω_initial
The moment of inertia of the merry-go-round with the child can be calculated by adding the moment of inertia of the merry-go-round and the child:
I_final = I_merry-go-round + I_child
The final angular momentum of the merry-go-round with the child is:
L_final = I_final * ω_final
Since angular momentum is conserved, we can set the initial angular momentum equal to the final angular momentum:
L_initial = L_final
Substituting the formulas for angular momentum and moment of inertia, we get:
I_merry-go-round * ω_initial = (I_merry-go-round + I_child) * ω_final
Solve for ω_final to find the final angular velocity of the merry-go-round with the child.