Circles and Sines

If you have studied the behavior of an oscillator, either a mass bouncing up and down on a spring, or a pendulum swinging back and fourth, you have encountered a puzzle. The formula for the period of the motion has a factor of 2π out in front. What is that doing there? The number π shows up when we are dealing with circles, but a mass bouncing on a spring is not going in a circle! So why the 2π?

Well, there actually is a circle involved. You know it from your math class as "the unit circle." And, as you may remember from that math class, sines, and cosines, and circles are somehow all tangled up together in something called "trignometry."

This app is not a substitue for paying attention in math class, but it does let you visualize the secret relationship between circles and sines.

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When I do my spring-oscillator lab, I have the students measure the period vs. mass and period vs. 1/k and analyze the fitting constants. When analyzing period vs. mass, they are able find that this constant is C/√k, where C has no units and, if they were careful, has a value in the neighborhood of 6.3.

A similar thing happens with the period vs. mass data, with the unitless number again (hopefully) close to 6.3. When I ask them what this number is, they have no idea. I don't want to just tell them, I want them to see the connection! When I first did this lab, I could not find a website with a demo like the one you are seeing here. So now there is one and I know where to find it.

If students leave the simulation running, it will stop after the ball makes 20 orbits.

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Circles and Sines v1.00

© Geoff Nunes 2022

Questions, brickbats, and requests for enhacements can be sent to doc at this website (noragulfa.com). Do not expect a rapid response.

Change Log

v1.03

Minor Aesthetic Change: Angle display behaves better (one could argue...).

v1.02

Change: Ball now orbits counterclockwise.

Minor Enhancement: Additional graphics to highlight the relation to trigonometry.

v1.01

Minor Enhancement: Simulation can now be re-started when it stops after 20 orbits.

v1.00

No changes yet...