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Dinesh

Dinesh club

Posted: 28 Dec 2021

Filmed: 28 Dec 2021

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Pendulum

www.princeton.edu/~hos/Mahoney/articles/huygens/casestud/casestudfr.html
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 Dinesh
Dinesh club
Galileo's theorem provided Huygens with two tools of analysis. Given a starting point B on the cycloid, draw BF parallel to the horizon, intersecting the standing generating circle at E. First, the time of fall over diameter DA is equal to the time of fall over EA. Second, the time of fall over EA is equal to the time of uniform motion at half the final speed. "We must show that the time over the portion BA of the cycloid is to the said time of uniform motion through EA as the semicircumference of a circle to the diameter." From this point, the proof involved mathematics alone. Taking any point H on the cycloid and drawing IH parallel to FB, Huygens drew the corresponding chord AK and then two parallel lines (both labeled YM) on either side of IH, thereby defining four small segments: MHM tangent to the cycloid at H, PKP on AK, RZR on EA, and SLS tangent to the semicircle AF at L. His demonstration began with the ratio of the time of uniform motion over MHM at the speed acquired by fall over BH (which is equal to the speed acquired in vertical fall from F to I) to the time of uniform motion over RR at half the final speed acquired over EA. The finite mathematics of chords in circles, combined with the special rules regarding infinitesimal tangents and their subtends, showed that ratio to be the same as the ratio of SLS to YIY. Hence, by the reasoning used earlier, the ratio of the time over the whole arc BA to the time over the whole chord EA is equal to the ratio of "all the SLS" to "all the YIY", which is the ratio of the semicircumference to the diameter.

When Huygens began his study of the pendulum, the notion of what we now call "harmonic oscillation" rested on the intuition of the regular motion of a physical device. To show that some other phenomenon displayed that regularity required linking it to the pendulum, as for example Huygens himself once did by harnessing a pendulum to a vibrating string. Huygens' analysis of the pendulum and of motion on the cycloid relocated harmonic oscillation in a mathematical curve and hence redefined the path of reduction. It now sufficed to reduce the mathematics of the phenomenon to that of the cycloid, as Huygens again attempted to do with the vibrating string. At this point, the concept was embodied in a geometrical object and so it remained until 1675. In that year, Huygens' attention was drawn to a thitherto unnoticed mechanical property of the inverted cycloid, namely that the tangential component of the weight of a body placed on the curve is proportional to its arc-distance from the vertex. (35) In other words, the force tending to move the body is proportional to the body's distance from the point of equilibrium. But, he reasoned, the cycloid is a tautochrone; hence, that relation is itself the tautochronic relation. Moreover, the relation holds for springs. Therefore, springs are tautochrones. Indeed, he could think of a host of mechanisms expressing the relation. His notes of the early 1690s are filled with them. All of them must be tautochrones. As one reads through his studies on this subject, it seems clear that from 1659 the physical pendulum that had once embodied his and his predecessors' understanding of what we now call harmonic motion was replaced, not by another physical instance, but first by a mathematical curve and then, from 1675, by a mathematical relation of which the curve was one instance.
3 years ago.