SHO & Phasors
Phasor + real oscillation. Builds on Simple Harmonic Motion.
Imagine a ball moving in a circle at constant speed. Now imagine shining a flashlight from the side so the ball’s shadow falls on a wall. You also have a mass bouncing on a spring next to the wall.
How does the shadow of the circling ball compare to the mass on the spring?
The shadow matches perfectly! Simple harmonic motion IS the projection of uniform circular motion onto one axis. A rotating arrow (phasor) of length A spinning at angular frequency ω projects to x(t) = A cos(ωt + φ) — exactly the motion of a mass on a spring. This is not an approximation; it’s exact.
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Watch the rotating phasor and its projection onto the x-axis. Compare with the mass-spring system below. They move in perfect lockstep!
x(t) = A cos(ωt + φ)
SHM is circular motion viewed from the side. The phasor spins at constant ω, but its projection accelerates and decelerates — just like a mass on a spring.
A Ferris wheel rider’s shadow on the ground traces out simple harmonic motion. The shadow speeds up through the middle and slows at the edges.
Electrical engineers use phasors to analyze AC circuits. Voltages and currents are rotating arrows, and impedance is just arrow arithmetic. It turns calculus into geometry.
In audio engineering, every sound is a sum of phasors at different frequencies. Adding phasors with different phases explains constructive and destructive interference.
The ocean tides are a sum of many phasors — from the Moon, Sun, and Earth’s rotation. Tidal prediction is literally adding rotating arrows of different frequencies.
“A phasor turns the hardest part of oscillation physics — keeping track of phase — into something you can see: a spinning arrow. When in doubt, draw the phasor.”
Phasor Oscillator Lab
Build oscillations from phasors. Watch a rotating arrow trace out sine waves in real time. Add multiple phasors to see how they combine — constructive interference, destructive interference, and everything in between.
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Start with a single phasor and verify that its projection gives a perfect cosine. Now add a second phasor with the same frequency but different phase. Watch how the sum phasor changes length as the phase difference varies. At 0° difference you get maximum amplitude; at 180° you get cancellation. Try two phasors with slightly different frequencies — you’ll see beats emerge!
The phasor representation is not just a visualization trick — it’s the foundation of Fourier analysis, AC circuit theory, and quantum mechanics. Every oscillation, no matter how complex, can be decomposed into a sum of spinning arrows.