Difficult Question in Calculus

[Duhoc]

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

 

[e(ho0n3]

Why are you saying that the rate of rotation is doubled on the second compass? Since the second compass is attached to the first compass, it is rotating at the same rate as the first compass. Maybe I'm not understanding your problem.

Are you trying to model a rotation similar to that of the sun, Earth and moon, i.e. the Earth rotates around the sun and the moon around the earth?

 

[Duhoc]

reply

The second compass is attached to the first but is rotating at double the rate of the first or at any ratio of the first. But I believe that the area inscribed by the second compass is proportional to the relative rate of rotation of the two compasses. Let is envision a baseball pitcher's motion. Some of his motions are synchronous and some simultaneous. But each joint which he uses is rotating at a relative rate to the others. Now, in the problem which I have proposed, if a relationship could be established, it could be integrated to reveal a four dimensional inscription in space time. i.e. a pitcher's motion, a golf swing, walking down the street. So that is the nature of my question.

 

[HallsofIvy]

So do it step by step. If the first compass is drawing a circle of radius r₁, and rotating at $\omega$ radian/sec. Then its position at time t seconds is $(r _1cos(\omega t), r_1 sin(\omega t))$. Now taking that as your starting point, the second compass is tracing a circle of radius r₂ rotating at $2\omega$ radian/sec. It's postion, relative to the first, at time t, is (r_2 cos(2\omega t), r_2 cos(2\omega t))[/itex]. Now add the two vectors to get the position of the second compass relative to the center of the first:
$(r_1cos(\omega t)+ r_2 cos(2\omega t), r_1 sin(\omega t)+ r_2 sin(2\omega t)).$

 

[Duhoc]

That is amazing...

Hallsofivy, but my question is is there an exponential relationship between the rates of rotation and the area inscribed by the second compass under the curve, and can that be integrated to extend this relationship to four dimensional inscriptions in time-space?