Final answer:
The final rotation rate of the disk can be determined using the law of conservation of angular momentum. The initial angular momentum is calculated using the moment of inertia of the disk and its initial angular velocity. The final rotation rate is found by substituting the values and solving for the angular velocity.
Step-by-step explanation:
The final rotation rate of the disk can be determined using the law of conservation of angular momentum. According to this law, the initial angular momentum of the system is equal to the final angular momentum. The initial angular momentum can be calculated by multiplying the moment of inertia of the disk by its initial angular velocity.
Angular momentum = Moment of inertia * Angular velocity
The moment of inertia of a solid disk is given by the formula I = (1/2) * m * r^2, where m is the mass of the disk and r is its radius. Substituting the given values, we find that the initial angular momentum is equal to 0.5 kg * (0.6 m)^2 * (2π rad/s).
When the small mass slides gradually to the center of the disk, the moment of inertia decreases. The moment of inertia of a disk with a small mass at the edge is greater than the moment of inertia of a disk with the small mass at the center.
Since the initial angular momentum is conserved, the final rotation rate of the disk will increase. The final angular momentum can be calculated by multiplying the moment of inertia of the disk with the small mass at the center by the final angular velocity.
Angular momentum = Moment of inertia * Angular velocity
Using the new moment of inertia, which is equal to the moment of inertia of the disk minus the moment of inertia of the small mass at the center (I_new = I_disk - I_small_mass), we can solve for the final angular velocity. Substitute the known values to find the final rotation rate of the disk.
Final angular momentum = (0.5 kg * (0.6 m)^2 * (2π rad/s)) = (I_disk - I_small_mass) * Angular velocity_new
Solving for Angular velocity_new, we find that the final rotation rate of the disk is (0.5 kg * (0.6 m)^2 * (2π rad/s)) / (I_disk - I_small_mass).