please check the attached file below
Writing Prompts
In lecture, we showed that the parametric equations describing the path of a point on a circle of radius r as it rolls around the inside of a circle of radius R are
We’ve seen that if R = 1 and r = 1/4, the path traced out is an asteroid, which can be described by the equation . Here, we will find a more general form. Let R/r = 4 (or, rearranged r = R/4). Eliminate the parameter theta in the parametric equations to derive the general equation. The identities and will be helpful.
You may conjecture that if R/r = 5, then maybe the path traced out has the form . Give a simple argument about why this cannot be the correct equation. (What I mean by “simple argument” is that it does not take lots of computations, just some simple observations about algebra and geometry. Finding the argument itself may not necessarily be easy).
Can you follow similar steps to eliminate the parameter if R/r = 3 or R/r = 5 to get the true equation? If not, what gets in the way?
What about if R/r = 2?
Adapt the steps we took in lecture to find the parametric equations related to the circle rolling inside to produce the parametric equations of rolling the smaller circle around the outside of the larger circle.
Side note (pun intended):
After examining these shapes, we should be convinced that the quantity R/r defines how these shapes look. Use ::this desmos graph:: to examine what happens as you change the ratio b. See what happens when you pick nice numbers like 4 or 5 or 4/3 or 5/4, or not nice numbers like 101/99 or pi.
A musical pitch is created when the air vibrates at a particular frequency. Musical pitches played together sound good (consonant) or bad (dissonant). Open up two windows of this ::pure tone generator:: and experiment with what happens when you play two notes that have nice ratios and not nice ratios. For instance, try playing one note at 100 Hz, and the other at 400 Hz, or one note at 300 Hz and the other at 400 Hz. On the other hand, try playing one note at 299 Hz and the other at 305 Hz. (Be sure your sound isn’t up too high).
How do the sounds of the combinations of notes relate to the pictures of these families curves?