Timeline for Intuition - why does the period not depend on the amplitude in a pendulum?
Current License: CC BY-SA 3.0
11 events
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| Jul 22, 2017 at 14:58 | comment | added | hmakholm left over Monica | @AndreasBlass: Yes, that was the point I was trying to express by saying not just that the cycloid happens to be approximatable, but that it is smooth enough that it can be approximated by a circle. (But not too smooth; if we had found that the required curve looked like $y=x^4$, a circle probably wouldn't have approximated it well). | |
| Jul 22, 2017 at 14:53 | comment | added | JMac | @AndreasBlass That's a fantastic point, and really helps to highlight how we can treat many systems as linear as long as the range is controlled enough to know it will not deviate too much from that behaviour. | |
| Jul 22, 2017 at 6:06 | comment | added | Andreas Blass | It should perhaps be mentioned that "the cycloid can be approximated near the bottom by a circle" is not some remarkable property of cycloids and circles but rather a property of any sufficiently smooth curves with a local minimum. Near that minimum, the height function and its second derivative can be exactly matched by a circle, and the first derivative vanishes, so you can get approximation accurate up to a 3rd-order error. If the curve is left-right symmetric about the bottom point, then the error shrinks to 4th order. | |
| Jul 20, 2017 at 22:18 | history | edited | hmakholm left over Monica | CC BY-SA 3.0 |
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| Jul 20, 2017 at 22:13 | comment | added | Diracology | That's a great answer. | |
| Jul 20, 2017 at 21:30 | comment | added | JMac | @NoamChai I was literally about to finish adding that to my answer anyways, for a bit more completeness. | |
| Jul 20, 2017 at 21:28 | comment | added | Noam Chai | @jmac i think it's a great thing to add. And a credit for henning would be appreciated... | |
| Jul 20, 2017 at 21:21 | comment | added | JMac | @NoamChai I thought about editing in the Tautochrone curve into my answer as well. I think one of the important things to note about this is that near the end of the curve, the path looks approximately circular. That's one good semi-intuitive way to see how for small angles this applies to a pendulum. | |
| Jul 20, 2017 at 20:20 | history | edited | hmakholm left over Monica | CC BY-SA 3.0 |
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| Jul 20, 2017 at 20:14 | comment | added | Noam Chai | I like your answer, I know about cycloid but I haven't thought about it before. Even if it's not a simple answer , it's pretty neat, thank's :) | |
| Jul 20, 2017 at 19:31 | history | answered | hmakholm left over Monica | CC BY-SA 3.0 |