Suppose that the particle P is moving in the plane, in a circle with centre at the origin O and radius r0. Let (r0, θ) be the polar coordinates of P. At an elementary level, the angular acceleration may be defined to be , where a dot denotes differentiation with respect to time.
At a more advanced level, the angular acceleration α of the particle P is the vector defined by α = , where ω is the angular velocity. Let i and j be unit vectors in the directions of the positive x- and y-axes and let k = i × j. Then, in the case above of a particle moving along a circular path, ω = k and α = k. If r, v and a are the position vector, velocity and acceleration of P, then
where er = i cosθ + + j sinθ and eθ=−i sinθ + j cosθ (see circular motion). Using the fact that v = ω × r, it follows that the acceleration a = α × r + ω × (ω × r).