Circular-Motion Projection Model for Simple Harmonic Motion

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How does uniform circular motion model simple harmonic motion?

Project the position of a point moving uniformly around a circle onto a diameter; the projection oscillates back and forth.

What does the circle's radius represent in the projection model?

The amplitude: the projected point moves between $x=+A$ and $x=-A$.

What is the position equation for projected motion with phase shift $\phi$?

$x(t)=A\cos(\omega t+\phi)$; the phase shift sets the starting point in the oscillation.

How are angular frequency, frequency, and period related?

$\omega=2\pi f$ and $T=1/f=2\pi/\omega$; one revolution corresponds to one oscillation.

How are projected velocity and acceleration expressed in simple harmonic motion?

$v(t)=-A\omega\sin(\omega t+\phi)$ and $a(t)=-\omega^2x(t)$.

Quiz questions

1. How can uniform circular motion produce simple harmonic motion?Answer: By projecting the moving point's position onto a diameter.
2. Which equation gives the position of a projected oscillator with amplitude $A$, angular frequency $\omega$, and phase $\phi$?Answer: $x(t)=A\cos(\omega t+\phi)$
3. What relation connects angular frequency $\omega$ to oscillation frequency $f$?Answer: $\omega=2\pi f$
4. Where does the projected oscillator reach its greatest speed?Answer: As it passes through the equilibrium position.
5. What does the relation $a=-\omega^2x$ indicate about acceleration in simple harmonic motion?Answer: Acceleration points toward the equilibrium position.