Re: Re: Re: Solve or Reduce?

• To: mathgroup at smc.vnet.net
• Subject: [mg64446] Re: [mg64412] Re: [mg64398] Re: Solve or Reduce?
• From: Math Novice <math_novice_2 at yahoo.com>
• Date: Fri, 17 Feb 2006 04:12:04 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```I tried to graph the angle of rotation, b, of the smaller circle as a function of the angle of rotation, a, of the larger circle. Consider that the points on the circumferences of the two circles are connected by a link of length 7 that can pivot at the two points on the circumferences and as the point (13,0) on the larger circle sweeps out an angle in the counterclockwise direction (starting from a=0 degrees) the point (13,7) on the smaller circle (that corresponds to b=0) will be forced to sweep out an angle in the counterclockwise direction.

The relationship between the two points on the circumferences of the circles is given by

(13 Cos[a]-(5+8 Cos[b]))^2+(13 Sin[a]-(7+8 Sin[b]))^2==49

and this is the equation that I used to try to solve for b as a function of a. I didn't impose any restrictions on the solution in Mathematica only because I don't know how but I am initially interested only in finding b as a function of a where both a and b are in the first quadrant. In the solutions I have tried, b increases as a increases up to ~ a= 32 degrees and then b starts to decrease when my measurements on the printouts of the graph of the two circles doesn't indicate any reason why b wouldn't still increase with a at that point. Any help you can give me would be appreciated.

Daniel Lichtblau <danl at wolfram.com> wrote:

Geometrically, think of taking a circle of radius 7 centered at any
given point in the first quadrant lying on your larger circle. It will
intersect your smaller circle twice. If you restrict to solutions this
is where you begin to have two such solutions; prior to that one
intersection was in the fourth quadrant.

Without having an idea of what exactly you did to get solutions, or what restrictions you are using, it is not possible to say much more.

Daniel Lichtblau
Wolfram Research
Math Novice wrote:
I am trying to find the angle b corresponding to the points on the circumference of the circle (5+8 Cos[b], 7+8 Sin[b]) that are a distance of 7 units from a point on the circumference of the circle (13 cos[a], 13 Sin[a]) for angles a in the first quadrant. (5+8 Cos[b], 7+8 Sin[b]) is a circle of radius 8 and center (5,7) and (13 cos[a], 13Sin[a]) is a circle of radius 13 and center (0,0).

(13,0) on the circle (13 cos[a], 13 Sin[a]) and (13,7) on (5+8 Cos[b], 7+8 Sin[b]) are the first set of points that are 7 units apart when the angle a (of the larger circle ) is equal to 0. As I increase the value of a and use my compass (set at 7 units) to measure on the printout of the diagram of the two circles it seems that there should always be an angle for b that corresponds to a and b should always increase as a increases but something happens at about 32 degrees. For some calculations b starts to decrease as a increases past 32 degrees or with some other calculations b becomes negative as a increases past 32 degrees. Any idea of what I'm going wrong?

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