Tangential and Centripetal Acceleration on a Rotating Body

Picture a point on the rim of a wheel that is spinning up. It is already curving around the center, so it needs acceleration pointing inward just to keep turning — that's centripetal acceleration, a_c = r·omega^2. But the wheel is also gaining speed, so the point speeds up along its circular path — that's tangential acceleration, a_t = r·alpha. These are two genuinely different reasons to accelerate, not two names for the same thing, and they point in different directions: tangential along the direction of motion, centripetal straight toward the center.

Because a_t and a_c are perpendicular, you cannot just add their sizes. The total acceleration is the vector sum, found with the Pythagorean theorem: a = √(a_t² + a_c²). The angle between the total vector and the radius tells you how much of the acceleration is 'spin-up' versus how much is 'stay on the circle.' At the instant alpha = 0, the point still accelerates centripetally — constant speed around a circle is still accelerating. At the instant omega = 0, only the tangential piece exists.

This splitting is what makes Newton's second law for rotation work cleanly: torque causes alpha, and alpha directly sets a_t, while a_c is just bookkeeping to keep the point on its circular path. The next step in the unit, moments of inertia, cares only about the tangential response to torque — centripetal acceleration never does work and never appears in rotational dynamics equations, since it's always perpendicular to the motion.