A playground carousel (“merry-go-round”) is free to rotate frictionlessly in the horizontal plane (and air resistance is negligible). Without riders, the carousel has a moment of inertia of 152 kg·m2. But there is a single rider, initially standing 1.85 m from the axis of rotation, as the carousel turns at an angular speed of 0.640 rad/s. Then the person moves to another location, 0.75 m from the axis, and the angular speed is then 0.973 rad/s. Find the person’s mass.
To solve this problem, we can use the principles of conservation of angular momentum. Angular momentum is given by the equation:
L = I * ω
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
In the initial state, the person is standing 1.85 m from the axis of rotation, and the angular speed is 0.640 rad/s. So, we can write the equation for the initial angular momentum:
L_initial = I * ω_initial
In the final state, the person moves to a location 0.75 m from the axis of rotation, and the angular speed is 0.973 rad/s:
L_final = I * ω_final
Since angular momentum is conserved, we can equate the initial and final angular momentum:
L_initial = L_final
I * ω_initial = I * ω_final
Simplifying the equation, we can cancel out the moment of inertia:
ω_initial = ω_final
Now, we can solve for the person's mass. The formula for moment of inertia of a point mass is given by:
I = m * r^2
where I is the moment of inertia, m is the mass, and r is the distance from the axis of rotation.
In the initial state, the person is standing 1.85 m from the axis of rotation:
I_initial = m * r_initial^2
In the final state, the person moves to a location 0.75 m from the axis of rotation:
I_final = m * r_final^2
Since the moment of inertia is the same in both states, we can set I_initial equal to I_final:
m * r_initial^2 = m * r_final^2
Now we can solve for the person's mass:
m = (r_final^2 * I) / r_initial^2
Substituting the given values, we have:
m = (0.75^2 * 152 kg·m^2) / 1.85^2
m = 90.953 kg
Therefore, the person's mass is approximately 90.953 kg.