Radial Acceleration

If the angular velocity ω is constant, if we have a particle traveling at constant speed in a circle, then α = dω/dt = 0 and there is no tangential acceleration a_t . However, we have known from almost the beginning of the course that a particle traveling at constant speed v in a circle of radius r has an acceleration directed toward the center of the circle, of magnitude v² /r , as shown in Figure (1). We will now call this center directed acceleration the radial acceleration a_r

Figure 1: Particle moving at a constant speed in a circle of radius r accelerates toward the center of the circle with an acceleration of magnitude a_r = v²/r.

If a particle is traveling in a circular orbit, but its speed v_t is not constant, then it has both a radial acceleration a _r=v_t²/ r , and a tangential acceleration a_t = rα . The radial acceleration is always directed toward the center of the circle and always has a magnitude v² /r . The tangential acceleration, if it exists, is tangential to the circle, pointing forward (counterclockwise) if α is positive and backward if α is negative. These accelerations are shown in Figure (2).

Figure 2: Motion with radial and tangential acceleration.