Physics

The key insight is that angular velocity is a continuous physical quantity—it describes how fast the disk is spinning at a given moment, and the disk's rotation can't suddenly "jump" to a different speed. Even though the angular acceleration $\alpha$ changes discontinuously (from one constant value to another), the angular velocity $\omega$ itself must remain the same on both sides of that transition instant. Think of it like a car smoothly changing from one constant acceleration to another: the speedometer reading doesn't flicker; only the rate of change of speed shifts.

The key insight is that angular velocity is a continuous physical quantity—it describes how fast the disk is spinning at a given moment, and the disk's rotation can't suddenly "jump" to a different speed. Even though the angular acceleration $\alpha$ changes discontinuously (from one constant value to another), the angular velocity $\omega$ itself must remain the same on both sides of that transition instant. Think of it like a car smoothly changing from one constant acceleration to another: the speedometer reading doesn't flicker; only the rate of change of speed shifts.