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 For: θ = wt

        w = 2π/T

Where: T is the period for circular motion, which is the time it takes to complete one full cycle of motion. 

So we can express the exponential function as a function of time(t) with a radius R on the real axis:

F(t)ᵣₒₜ = Re^(iwt)

As the value of t increases, this function traces a circle of radius R in the complex number plane. Which describes a rotating vector of magnitude R, that points away from the center, and rotates counterclockwise as t increases.

On the real axis the function equals R which has units of length. The exponential function[e^(iwt)] is dimensionless so the function has units of length.

When we take the derivative of this function with respect to time we get units for speed. So the derivative of the function gives us a velocity vector which represents circular motion.

F(t)ᵣₒₜ = Re^(iwt)

F'(t)ᵥₑₗ = iwRe^(iwt)

Now this new function has units of R/sec which is dist/sec or speed.

We notice this new exponential function is multiplied by the operator i, so this new velocity vector will be perpendicular to the original rotational vector on the real axis.

By taking the tail end of the velocity vector and connecting it perpendicular to the front tip of the rotational vector at coordinates (0,R); we notice that as the rotational vector traces a circle, the velocity vector forms a tangent line at every point on the circle.

*This suggests that linear motion is just the limiting case of circular motion. Occurring over a tiny piece of the circle. But as the circle radius increases, linear motion approximations occur over larger pieces of the circle.

This is why locally, every curve looks linear, and globally, linear motion can be viewed as circular motion with infinite radius.




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