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CTK Exchange
Cliff P
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Aug-24-07, 04:08 PM (EST) |
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"Semicubical parabola, tautochrone"
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We read on the web that around 1687, Dutch mathematician and physicist Christiaan Huygens (1629 – 1695), showed that the semicubical parabola is a curve along which a particle may descend under the force of gravity so that it moves equal vertical distances in equal times. I assume that these properties are different from the famous tautochrone or brachistochrone curve properties? Thanks.
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mr_homm
Member since May-22-05
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Aug-25-07, 00:33 AM (EST) |
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1. "RE: Semicubical parabola, tautochrone"
In response to message #0
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Yes, this is a different property. In this curve, the particle has zero vertical acceleration, provided that it is started on the curve at the correct speed. In contrast, but brachistochrone is the curve connecting two given fixed points, which allows the particle to reach the lower point in the shortest possible time, if it is released from rest at the upper point. Similarly, the tautochrone is a curve for which a particle started from rest at any point on the curve will reach the bottom point in an equal time. The brachistochrone and tautochrone are both actually the same curve, the cycloid, but the starting conditions are different. A cycloid is a brachistochrone if the upper point is on the cusp of the cycloid and the lower point is anywhere elso on it. It is a tautochrone if the lower point is at the bottom of the cycloid and the upper point is anywhere else on it. Hope this helps! --Stuart Anderson |
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Sandro
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Aug-31-07, 11:13 AM (EST) |
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2. "RE: Semicubical parabola, tautochrone"
In response to message #1
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What does "zero vertical acceleration" mean? Does the particle feel the gravitational force or it does not? I can't write the equations to demonstrate this property of semicubical parabola (isochronous curve). Can someone help me???
Thanks
Sandro
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alexb
Charter Member
2075 posts |
Sep-01-07, 09:21 AM (EST) |
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3. "RE: Semicubical parabola, tautochrone"
In response to message #2
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Even without the equations, it is not difficult to describe how such a motion is possible. For one, the assertion relates to the curve with a vertical cusp: y3 = x2. Also, the initial speed as assumed non-zero. Now, say, if the particle moves rightwards, the gravitational force acts to slow it down but the farther it moves from the origin, the less the effect of the gravitational force becomes. For this particular curve it is claimed that in the same amount of time, although the horizontal distance covered will grow with the distance from the origin, the vertical distance will remain the same. |
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