Vectors will set you FREE! Here we change the straight, fixed in space basis, I, J K into a set, two of which move in a plane.

Vectors for Circular Motion Motion of a mass at constant speed constrained by forces to follow a circular path is kinematically described by the circle radius, and system vectors - position, velocity and acceleration.

The graphic shows the geometry with positive change of angle, θ(t) and d(θ(t))/dt being counter-clockwise. We use the I, J vector basis in our development. Position, in terms of the I - J basis is:

P(t) = r [cos( θ(t) )I + sin( θ(t) )J]

Two derivatives are taken to yield successively the velocity vector, V(t), and A(t), the acceleration vector. Each is written in a "magnitude times unit vector" form where the unit vectors, I and J are fixed in direction.

The above is correct in terms of the constant vector basis: I and J. Problems can be solved with this set - just as they are. However, these results can be used to confirm a simpler, less writing-intensive way of representing P(t), V(t) and A(t).

Compare the directions (unit vector parts) of the vectors, V and A with that of the initial unit vector, the direction of P. Immediately we see the direction of A is opposite the direction of P. Using a simple sketch, we see the direction of velocity, V is perpendicular to the direction of P, and directed toward increasing angle, θ. Thus we define e_r(t), and e_θ(t).

The graphic representation of the unit vectors with analytic statements of position, velocity and acceleration vectors are shown.

P(t) = r e_r(t)

V(t) = r(dθ/dt)e_θ(t)

A(t) = - r (dθ/dt)² e_r(t)

Extension of this idea to apply to non-circular motion, that is for a radius that varies in time, r = r(t), follows easily since the radius is a scalar. The new vector basis described here is:

e_r(t),   e_θ(t)  and   K, (when needed).