Vector Basis for Circular Motion

Basic Thermodynamics ~ J. Pohl ©

www.THERMOspokenhere.com (53-A470)

Motion of a BODY at constant speed constrained by forces to follow a circular path is kinematically described by the radius of the circle and vectors that depict its position (P(t)), velocity (V(t)) and (A(t)) 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. Position of the BODY in terms of this basis is written as:

P(t)_BODY = r [cosθ(t) I + sinθ(t)J]

(1) Position of a BODY restrained to a
"circular domain" as a function of time.

In what follows we will not carry the notation "BODY." The derivative of position with respect to time yields the velocity of the BODY ( V (t)). Next the derivative of that velocity (which is the second derivative of the position) produces a mathematical expression, the acceleration of the BODY ( A (t)). Each is written in a "magnitude times unit vector" form where the unit vectors, I and J are fixed in direction. The vectors are:

(2)Position.

(3)Velocity or dP(t)/dt.

(4)Acceleration is d(dP(t)/dt)dt or dV(t)/dt.

Problems can be solved with the above set of vectors written just as they are. However a simpler, less writing-intensive vector representation is available. We will use the "truth" of the above set (which you can see) to develop new representations.

First recognize that the above three vectors are written in magnitude-unit vector forms. However, these results can be used to confirm a simpler, 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)

(5) Position: r is constant.)

V(t) = r(dθ/dt)e_θ(t)

(6) dθ/dt is the constant rotation rate.

A(t) = - r (dθ/dt)² e_r(t)

(7) Here the rotation rate (dθ/dt) is squared.

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)

(8) The vectors apply to motion in a constant plane.

The vectors discussed here apply for motion of a body "in a plane" about some point (with constant radius). These represent a number of physical cases. To expend the motion to include variation of the radius (but still in a plane) would be the next step of study.