A playground merry-go-round of radius 2.00 m has a moment of inertia I = 255 kg·m2 and is rotating about a frictionless vertical axle. As a child of mass 25.0 kg stands at a distance of 1.00 m from the axle, the system (merry-go-round and child) rotates at the rate of 14.0 rev/min. The child then proceeds to walk toward the edge of the merry-go-round. What is the angular speed of the system when the child reaches the edge?


rad/s

To find the angular speed of the system when the child reaches the edge, we need to apply the principle of conservation of angular momentum.

Initially, the angular momentum of the system is given by:

L1 = I1 * ω1

Where:
L1 is the initial angular momentum of the system
I1 is the moment of inertia of the merry-go-round
ω1 is the initial angular speed of the system

When the child walks towards the edge of the merry-go-round, the moment of inertia of the system changes. The new moment of inertia, I2, is given by:

I2 = I1 + m * r^2

Where:
m is the mass of the child
r is the distance of the child from the axis of rotation (which is the radius of the merry-go-round)

The final angular momentum of the system, L2, can be calculated using the principle of conservation of angular momentum:

L2 = I2 * ω2

Where:
L2 is the final angular momentum of the system
ω2 is the final angular speed of the system (which is what we need to find)

Since angular momentum is conserved, we have:

L1 = L2

Substituting the expressions for L1, L2, I1, and I2, we get:

I1 * ω1 = (I1 + m * r^2) * ω2

Rearranging this equation to solve for ω2, we have:

ω2 = (I1 * ω1) / (I1 + m * r^2)

Now we can plug in the given values:

I1 = 255 kg·m^2
ω1 = 14.0 rev/min = (14.0 rev/min) * (2π rad/rev) * (1 min/60 s) ≈ 4.647 rad/s
m = 25.0 kg
r = 1.00 m

Substituting these values into the equation, we get:

ω2 = (255 kg·m^2 * 4.647 rad/s) / (255 kg·m^2 + 25.0 kg * (1.00 m)^2)

Evaluating this expression, we find:

ω2 ≈ 4.317 rad/s

Therefore, the angular speed of the system when the child reaches the edge is approximately 4.317 rad/s.

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