Question

A 24-cm-diameter, 2.5 kg solid disk is rotating at 150rpm around an axis that is perpendicular to the plane of the disk and passes through its center. A 24 -cm-diameter, 1.2 kg loop is dropped straight down onto the rotating disk. Friction causes the loop to accelerate until it is riding on the disk. What is the final angular velocity of the combined system? Moment of inertia of a disk about a perpendicular axis passing through its center is given as 1/2mR². Moment of inertia of a loop about a perpendicular axis passing through its center is given as mR².

Answer 1

The final angular velocity of the combined system is approximately 125 rpm. The law of conservation of angular momentum is used to determine this by equating the initial and final angular momenta of the system.

Step-by-step explanation:

To find the final angular velocity of the combined system, we need to apply the law of conservation of angular momentum. The angular momentum of an object is given by the product of its moment of inertia and its angular velocity. Initially, only the disk is rotating, so its angular momentum is given by the product of its moment of inertia (1/2 * 2.5 kg * (0.12 m)²) and its initial angular velocity (150 rpm). When the loop is dropped onto the rotating disk, the total angular momentum of the system remains constant.

After the loop is dropped, it starts rotating with the disk. The moment of inertia of the loop is given by (1.2 kg * (0.12 m)²). By setting the initial and final angular momenta of the system equal, we can solve for the final angular velocity of the combined system.

Final angular momentum = Initial angular momentum

(1/2 * 2.5 kg * (0.12 m)² * 150 rpm) + (1.2 kg * (0.12 m)² * 0 rpm) = (3.7 kg m² * final angular velocity)

Simplifying the equation, we find the final angular velocity of the combined system to be approximately 125 rpm.