Let's say that someone is drawing a circle with a compass. As that circle is being drawn a second compass is attached to the first such that the needle leg is attached to the pencil of the first. Only instead of a needle it is a small wheel. As the first compass inscribes its circle the second compass is tracing out the circumference of the first and drawing a circle at twice the rate of rotation of the first. An analogy would be someone at the edge of a merry-go-round that is rotating above a piece of paper. The rider endeavors to trace out a circle on the paper as the merry-go-round rotates at twice the rate of rotation of the merry-go-round. Is there an exponential relation between the relative rates of rotation and the area incribed under the curve created by the second compass? Additionally, if a third compass rode on the path inscribed by the second attempting to sketch a circle at some ratio of the rate of rotation of the first and second could that area be expressed? (Please private message as well as post.)
Thank you,
Duhoc
Thank you,
Duhoc