Tangential and Centripetal Acceleration

Any point moving in a circle has centripetal acceleration, pointing inward toward the axis, just from the fact that its direction keeps changing. That piece is a_c = omega² r, and it exists even at constant spin speed. But if the wheel is also speeding up or slowing down, the point's speed along the circle is changing too, which needs a second acceleration: tangential acceleration, a_t = alpha r, pointing along the direction of motion, tangent to the circle.

These two pieces point in different directions: centripetal is always radial (toward the center), tangential is always perpendicular to that (along the rim). Because they are perpendicular, you combine them like any two perpendicular vectors, with the Pythagorean theorem: a_total = √(a_c² + a_t²). The direction of the total acceleration arrow tilts away from straight-inward, leaning toward whichever way the point is speeding up or slowing down.

This topic rebuilds the same idea from straight-line motion where acceleration could point anywhere relative to velocity, now applied to circles. It connects forward to torque and Newton's second law for rotation, since alpha comes from net torque divided by rotational inertia, the same alpha that sets the tangential piece here. Once a wheel reaches constant angular velocity, alpha drops to zero, the tangential piece vanishes, and the total acceleration becomes purely centripetal again, pointing straight at the axis.