Figure: How uniform circular motion relates to sinusoidal motion and the cosine and sine functions
Caption: An diagram illustrating how uniform circular motion relates to sinusoidal motion.
The red arrow rotates with uniform circular motion starting from 0°. The angular velocity ω is uniform
The arrow's vertical component traces out the sine function sin(ωt).
The arrow's horizontal component (not shown) traces out the cosine function cos(ωt).
Both motions are considered sinusoidal motion.
The two trigonometric functions are identical except that the cosine function has a phase shift of -90° from the sine function.
Obviously, uniform circular motion seen edge-on projects into sinusoidal motion.
A classic example of this effect is the Galilean moons of Jupiter. Their orbits are nearly circular with nearly uniform circular motion and they are seen approximately edge-on from the Earth. Hence on the sky, they oscillate approximately sinusoidally with amplitudes approximately equal to their mean orbital radii.
The mathematical relationship between uniform circular motion and sinusoidal motion has technological implications. The riprocating engine operating at constant rate converts the approximate sinusoidal motion (the linear motion of a piston in an engine cylinder) into the approximate uniform circular motion of a wheel of some kind.
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