Final Answer:
The final angular speed Wf is C) Equal to Wi.
Step-by-step explanation:
When the boys move towards the center of the turntable, they reduce their moment of inertia, causing an increase in angular speed to conserve angular momentum.
According to the law of conservation of angular momentum, initial angular momentum (Iωi) equals final angular momentum (Iωf), where I is the moment of inertia and ω is the angular speed. Since the initial angular momentum is Ii * ωi (where Ii is the initial moment of inertia and ωi is the initial angular speed) and the final angular momentum is If * ωf (where If is the final moment of inertia and ωf is the final angular speed), the equation can be expressed as Ii * ωi = If * ωf.
When the boys move inward on the turntable, reducing their distance from the axis of rotation, the moment of inertia decreases. Assuming no external torque, the product of moment of inertia and angular speed remains constant. Therefore, as the boys move closer to the center, the moment of inertia decreases, causing the angular speed (ωf) to increase to maintain the conservation of angular momentum.
This increase in angular speed compensates for the decreased moment of inertia, resulting in the final angular speed (Wf) being equal to the initial angular speed (Wi). This explanation is based on the principle of conservation of angular momentum, where any changes in the distribution of mass affect the angular speed to maintain the total angular momentum constant. Mathematically, this can be shown as Ii * ωi = If * ωf, leading to Wf = Wi.(C)