A person with mass mp = 74 kg stands on a spinning platform disk with a radius of R = 2.31 m and mass md = 183 kg. The disk is initially spinning at ω = 1.8 rad/s. The person then walks 2/3 of the way toward the center of the disk (ending 0.77 m from the center).
a. What is the total moment of inertia of the system about the center of the disk when the person stands on the rim of the disk?
b. What is the total moment of inertia of the system about the center of the disk when the person stands at the final location 2/3 of the way toward the center of the disk?
c. What is the final angular velocity of the disk?
d. What is the change in the total kinetic energy of the person and disk? (A positive value means the energy increased.)
e. What is the centripetal acceleration of the person when she is at R/3?
f. If the person now walks back to the rim of the disk, what is the final angular speed of the disk?
To solve this problem, we need to use the principles of rotational motion, specifically moment of inertia and conservation of angular momentum. Let's go through each part of the problem step by step.
a. To find the total moment of inertia of the system when the person stands on the rim of the disk, we need to consider the moment of inertia of the disk and the moment of inertia of the person.
The moment of inertia of a disk can be calculated using the formula:
I_disk = (1/2) * mass_disk * radius_disk^2
Given that the mass of the disk is md = 183 kg and the radius of the disk is R = 2.31 m, we can calculate:
I_disk = (1/2) * 183 kg * (2.31 m)^2
= 474.913 kg·m^2
The moment of inertia of a person standing on the rim of a disk can be approximated as:
I_person = mass_person * distance_from_axis^2
Given that the mass of the person is mp = 74 kg and the radius at which the person stands is R = 2.31 m, we can calculate:
I_person = 74 kg * (2.31 m)^2
= 392.388 kg·m^2
The total moment of inertia of the system is the sum of the moment of inertia of the disk and the moment of inertia of the person:
Total moment of inertia = I_disk + I_person
= 474.913 kg·m^2 + 392.388 kg·m^2
= 867.301 kg·m^2
Therefore, the total moment of inertia of the system about the center of the disk when the person stands on the rim of the disk is 867.301 kg·m^2.
b. To find the total moment of inertia of the system about the center of the disk when the person stands at the final location 2/3 of the way toward the center, we need to recalculate the moment of inertia of the person at the new location.
Given that the person ends up 0.77 m from the center of the disk, the new radius at which the person stands is R_final = 0.77 m.
Using the same formula for the moment of inertia of a person:
I_person_final = mass_person * distance_from_axis^2
= 74 kg * (0.77 m)^2
= 43.234 kg·m^2
Now, we can calculate the total moment of inertia of the system:
Total moment of inertia = I_disk + I_person_final
= 474.913 kg·m^2 + 43.234 kg·m^2
= 518.147 kg·m^2
Therefore, the total moment of inertia of the system about the center of the disk when the person stands at the final location 2/3 of the way toward the center is 518.147 kg·m^2.
c. To find the final angular velocity of the disk, we can use the law of conservation of angular momentum. According to this law, the initial angular momentum of the system should be equal to the final angular momentum of the system.
The initial angular momentum of the system is given by:
Initial angular momentum = I_initial * ω_initial
Where I_initial is the moment of inertia and ω_initial is the initial angular velocity.
Similarly, the final angular momentum of the system is given by:
Final angular momentum = I_final * ω_final
Where I_final is the moment of inertia and ω_final is the final angular velocity.
Since angular momentum is conserved, we can equate the initial angular momentum to the final angular momentum:
I_initial * ω_initial = I_final * ω_final
Plugging in the values we know:
867.301 kg·m^2 * 1.8 rad/s = 518.147 kg·m^2 * ω_final
Solving for ω_final:
ω_final = (867.301 kg·m^2 * 1.8 rad/s) / 518.147 kg·m^2
= 3.02 rad/s
Therefore, the final angular velocity of the disk is 3.02 rad/s.
d. To find the change in total kinetic energy of the person and the disk, we need to calculate the initial and final kinetic energies and take the difference.
The initial kinetic energy of the system is given by:
Initial kinetic energy = (1/2) * I_initial * ω_initial^2
Plugging in the values we know:
Initial kinetic energy = (1/2) * 867.301 kg·m^2 * (1.8 rad/s)^2
= 1169.755 J
The final kinetic energy of the system is given by:
Final kinetic energy = (1/2) * I_final * ω_final^2
Plugging in the values we know:
Final kinetic energy = (1/2) * 518.147 kg·m^2 * (3.02 rad/s)^2
= 2349.519 J
Therefore, the change in total kinetic energy of the person and the disk is:
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
= 2349.519 J - 1169.755 J
= 1179.764 J
Thus, the change in the total kinetic energy of the person and the disk is 1179.764 J.
e. To find the centripetal acceleration of the person when she is at R/3, we can use the formula for centripetal acceleration:
Centripetal acceleration = ω^2 * radius
Plugging in the values we know:
Centripetal acceleration = (1.8 rad/s)^2 * (2.31 m/3)
= 0.741 m/s^2
Therefore, the centripetal acceleration of the person when she is at R/3 is 0.741 m/s^2.
f. If the person walks back to the rim of the disk, we know that the initial and final moment of inertia of the system remain the same, as the mass distribution does not change.
Using the conservation of angular momentum equation:
I_initial * ω_initial = I_final * ω_final
Since the moment of inertia remains the same, I_initial = I_final, the equation simplifies to:
ω_initial = ω_final
This means that the final angular velocity of the disk will be the same as the initial angular velocity, which is ω = 1.8 rad/s.
Therefore, the final angular speed of the disk when the person walks back to the rim is still 1.8 rad/s.