What is formula for radial acceleration and tangential acceleration in circular motion?

Radial (centripetal) acceleration points toward the center of a circular path and is calculated as a_r = v²/r or a_r = ω²r, where v is tangential speed, r is radius, and ω is angular velocity. This acceleration is necessary to continuously change the velocity vector's direction, keeping the object moving in a circle rather than a straight line. A car navigating a 50-meter radius curve at 20 m/s experiences a_r = 400/50 = 8 m/s² toward the curve's center—this is the acceleration the road friction must provide to prevent skidding outward.

Tangential acceleration a_t = dv/dt or a_t = αr (where α is angular acceleration) describes changes in the object's speed along the circular path. When a Ferris wheel is starting up, it has both centripetal acceleration (keeping riders in circular motion) and tangential acceleration (increasing rotational speed). If the Ferris wheel's 15-meter radius reaches 2 rad/s angular velocity while accelerating at 0.1 rad/s², riders experience a_t = 0.1(15) = 1.5 m/s² tangentially and a_c = (2)²(15) = 60 m/s² radially—combined magnitude of √(2.25 + 3600) = 60.02 m/s².

Total acceleration in non-uniform circular motion combines both components vectorially: a_total = √(a_r² + a_t²). For a car accelerating through a turn, if it's speeding up at 3 m/s² (tangential) while the turn demands 7 m/s² centripetal acceleration, the total is √(9 + 49) = 7.62 m/s² pointing diagonally—not purely toward center nor purely forward. This distinction matters in racing, where drivers balance throttle (tangential acceleration) against turn radius limits (centripetal acceleration) to maximize speed without losing traction. Phone accelerometers measure both components to track motion in three dimensions, enabling step counting, orientation detection, and crash detection features.