Motion of a particle in a circular path. Suppose that the path of the particle P is a circle in the plane, with centre at the origin O and radius r0. Let i and j be unit vectors in the directions of the positive x- and y-axes. Let r, v and a be the position vector, velocity and acceleration of P. If P has polar coordinates (r0, θ), then
r = r0 (i cos θ + j sin θ),
v = ṙ = r0(−θ̇i sin θ + θ̇j cos θ),
a = r̈ = r0(−θ̈i sin θ − θ̇2i cos θ + θ̈j cos θ − θ̇2j sin θ).
Let er=i cos θ+j sin θ and eθ=−i sin θ+j cos θ, so that er is a unit vector along OP in the direction of increasing r, and eθ is a unit vector perpendicular to this in the direction of increasing θ. Then the equations above become
r=r0er, v=ṙ=r0θ̇eθ, a=r̈=−r0θ̇2er+r0θ̈eθ.
If the particle, of mass m, is acted on by a force F, where F=F1er+F2eθ, then the equation of motion mr̈=F gives−mr0θ̇2=F1 and mr0θ̈=F2. If the transverse component F2 of the force is zero, then θ̇=constant and the particle has constant speed.
See also angular velocity and angular acceleration.