Angular Speed

Angular velocity ( ) refers to how fast an object is rotating, and it is related to the angular displacement over time. The relationship between angular velocity and angular acceleration is given by: Where: - is the initial angular velocity, - is the angular acceleration, - is the time. - (A) is incorrect because angular velocity is not necessarily constant. It can change depending on the angular acceleration. - (B) is incorrect because angular velocity is not proportional to the moment of inertia; it is related to angular acceleration, not the moment of inertia. - (D) is incorrect because angular velocity depends on the axis of rotation in cases of non-uniform rotation (for example, in non-central forces). Thus, the correct answer is that angular velocity depends on angular acceleration.

When a rotating disc shrinks in radius, its moment of inertia decreases. According to the **conservation of angular momentum**, which states that the angular momentum of a system is conserved if no external torque acts on it, we have: Where: - is the moment of inertia, - is the angular velocity. Since no external torque is applied, angular momentum is conserved: Where: - and are the moment of inertia and angular velocity initially, - and are the moment of inertia and angular velocity after the radius shrinks. Since the moment of inertia for a disc is given by: If the radius decreases, the moment of inertia decreases. To conserve angular momentum, the angular velocity must increase. Thus, the correct answer is: