How does uniform circular motion model simple harmonic motion?

The projection of a point moving uniformly around a circle onto a fixed axis moves back and forth in SHM. The circular point travels around the circle, while its projection oscillates along a line.

What do the radius $X$ and angular velocity $\omega$ represent in the projection model?

$X$ is the amplitude of the projected oscillation, and $\omega$ sets how quickly the circular point and its projection move through their cycles.

What are the position and angular-frequency relations for the projection?

For a projection starting at the positive endpoint, $x(t)=X\cos(\omega t)$. The angular velocity and period are related by $\omega=2\pi/T$.

How does the projection’s speed vary during an oscillation?

Its speed is greatest at the center, where $x=0$, and zero at the endpoints, where $x=\pm X$. Its magnitude is $|v|=v_{\max}\sqrt{1-x^2/X^2}$.

What is the period of an ideal mass–spring oscillator?

For a mass $m$ attached to a spring with spring constant $k$, the period is $T=2\pi\sqrt{m/k}$.

General physicsLesson 6 of 9 in the course Oscillatory Motion and Waves

Circular-Motion Projection: Position, Speed, and Mass–Spring Period

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Uniform circular motion provides a useful geometric model for understanding simple harmonic motion (SHM). Imagine a ball rotating at a steady rate on a vertical turntable. Its shadow, projected onto a fixed horizontal axis, moves back and forth in simple harmonic motion.

Consider a point $P$ traveling around a circle of radius $X$ at constant angular velocity $\omega$ . Its projection onto a fixed axis moves between $-X$ and $+X$ .

All flashcards

How does uniform circular motion model simple harmonic motion?: The projection of a point moving uniformly around a circle onto a fixed axis moves back and forth in SHM. The circular point travels around the circle, while its projection oscillates along a line.
What do the radius $X$ and angular velocity $\omega$ represent in the projection model?: $X$ is the amplitude of the projected oscillation, and $\omega$ sets how quickly the circular point and its projection move through their cycles.
What are the position and angular-frequency relations for the projection?: For a projection starting at the positive endpoint, $x(t)=X\cos(\omega t)$ . The angular velocity and period are related by $\omega=2\pi/T$ .
How does the projection’s speed vary during an oscillation?: Its speed is greatest at the center, where $x=0$ , and zero at the endpoints, where $x=\pm X$ . Its magnitude is $|v|=v_{\max}\sqrt{1-x^2/X^2}$ .
What is the period of an ideal mass–spring oscillator?: For a mass $m$ attached to a spring with spring constant $k$ , the period is $T=2\pi\sqrt{m/k}$ .

Quiz questions

In the circular-motion model of simple harmonic motion, what does the circle’s radius represent?Answer: The amplitude of the oscillation.
A point completes one revolution around its circle in time $T$ . How many complete oscillations does its projection make in that time?Answer: One complete oscillation.
Which expression gives the projection’s position when it starts at the positive endpoint at $t=0$ ?Answer: $x(t)=X\cos(\omega t)$
At what position does the projection have its greatest speed?Answer: At equilibrium, where $x=0$ .
What is the period of an ideal mass–spring oscillator with mass $m$ and spring constant $k$ ?Answer: $T=2\pi\sqrt{m/k}$

Based on College Physics 2e by Paul Peter Urone, Roger Hinrichs (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.

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