One Circle Revolving Around AnotherDate: 05/26/99 at 09:16:45 From: Mike Aumueller Subject: Circle circumference/revolution question If one circle has a circumference of 7.8 cm and a second circle has a circumference of 2.6 cm, how many revolutions will the smaller circle make while rotating around the perimeter of the larger circle? I have tried for hours to create a formula that will answer this question, but I have had no success. Help. I have tried the following: C1/C2 = Revolution1/Revolution2 where I have measured the revolution of circle 2 around circle 1. But I would like to figure out the answer without actually measuring. Thanks. Date: 05/26/99 at 12:18:47 From: Doctor Rick Subject: Re: Circle circumference/revolution question Hi, Mike. This is pretty hard to picture, because it involves a sort of "motion-picture geometry." I hope I can help a little. Let the center of the big circle (radius R1) be A and let the center of the small circle (radius R2) be B. Label one point on the circumference of the small circle as point C. Start with AB at angle 0 (say, the x-axis), and with point C at the point of tangency of the two circles. As AB rotates, the small circle rolls along the big circle. When the center of the small circle has moved to B', sweeping out the angle alpha = BAB', the point of tangency has moved along a certain arc of the big circle. Because the circle rolls without slipping, point C will have moved to point C' sweeping out an equal arc of the small circle. Draw this out and think about it. There is enough information here for you to figure out how the angle beta = ABC' is related to the angle alpha. But then you must be careful, because angle beta is NOT what the question is asking about! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 05/26/99 at 13:22:30 From: Mike Aumueller Subject: RE: Circle circumferece/revolution question Thank you very much for your assistance and have a great day. Date: 06/22/99 at 15:00:59 From: Mike Aumueller Subject: Going in circles Dear Sir, I need to know the answer and proof to the question I have presented. A mechanical engineering professor states that the answer is 4 revolutions; however many other people insist that the answer is 3. Thanks. Date: 06/22/99 at 21:38:34 From: Doctor Rick Subject: Re: Going in circles Hi again, Mike. I gave you the outline of the proof last time. The key to the difference between the answers of 3 revolutions and 4 revolutions is in my last statement: "But then you must be careful, because angle beta is NOT what the question is asking about!" Let's draw a picture. ******* ** ** * * * * C' * B * ******* * | * ** | ** * | * * | * ** | ** * | * ******* * B' * * S|C * * /| * * | ** / | * * | * | ** * | / D******* * | / * S' * | / * * | / * * | / * * |/ * * A * * * * * * * * * * * * * * * * * * * ****** Let the center of the big circle (radius R1) be A and let the center of the small circle (radius R2) be B. Label one point on the circumference of the small circle as point C. Start with AB at angle 0 (say, the y-axis), and with point C at the point of tangency of the two circles. As AB rotates, the small circle rolls along the big circle without slipping. This means that, if the circle has rolled so that the point of tangency is at D, then the arc length CD on the big circle is equal to the arc length DC' on the small circle. Since the radius of the small circle is 1/3 the radius of the large circle, angle DB'C' = arc DC' / R2 (radians) = arc CD / (R1/3) = 3 arc CD / R1 = 3 angle CAD But we are not interested in angle DC'B'. The angle of rotation of the small circle is the angle S'B'C', where B'S' is parallel to BS, so that we are reckoning the rotation relative to a fixed direction. Now angle S'B'C' = angle S'B'D + angle DB'C' = angle CAD + 3 angle CAD = 4 angle CAD Thus when the small circle has gone 360 degrees, that is, angle CAD = 1 revolution, angle S'B'C' will have swept out 4 revolutions. See if this satisfies both factions. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 06/23/99 at 09:40:27 From: Mike Aumueller Subject: RE: Going in circles Thank you very much for your help. The only other question I have is: Why is it that when you physically simulate the rotation of the two circles the result is 3 rotations, not 4? Date: 06/23/99 at 10:07:36 From: Doctor Rick Subject: RE: Going in circles Hi, Mike. Are you absolutely SURE you are measuring the rotations relative to a fixed direction (like north) and not relative to the line from the center of the big circle to the small circle? And are you holding the big circle fixed and revolving the small circle around it, rather than turning the big circle like a gear and keeping the small circle in one place? If you answer yes to both questions, then you should count 4 rotations per revolution. You should find that, when the small circle has gone from north to east of the big circle, the point that was touching the big circle at first has come back around to the bottom (south) of the small circle. That's one full rotation in 1/4 revolution. I remember in fifth grade being asked by my teacher what was the period of the moon's rotation. I said one month, and my teacher said no, it does not rotate at all. I didn't talk back to my teacher, but the fact is, I was thinking of the sidereal period - counting rotations relative to the stars - while he was thinking of the geocentric period, I guess you'd call it, counting rotations relative to a line from the earth to the moon. It's the same problem of being off by one rotation per revolution because the reference direction is itself rotating. It can be easy to miss! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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