Problem 54710. Compute the period of a pendulum started from a finite initial angle

Created by ChrisR

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<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 94.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 47.05px; transform-origin: 407px 47.05px; vertical-align: baseline; "><div style="block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/49830">Cody Problem 49830</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.45px 8px; transform-origin: 61.45px 8px; unicode-bidi: normal; "> asks for the period T<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 240.792px 8px; transform-origin: 240.792px 8px; unicode-bidi: normal; "> of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle <img src="data:image/png;base64,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" alt="theta0" style="width: 15px; height: 20px;" width="15" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.883px 8px; transform-origin: 306.883px 8px; unicode-bidi: normal; "> that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). </div><div style="block-size: 42.1px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.05px; text-align: left; transform-origin: 384px 21.05px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.683px 8px; transform-origin: 168.683px 8px; unicode-bidi: normal; ">Write a function that takes the initial angle and returns <img src="data:image/png;base64,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" alt="T\sqrt{g/L}" style="width: 52.5px; height: 20.5px;" width="52.5" height="20.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; ">, where L<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 107.358px 8px; transform-origin: 107.358px 8px; unicode-bidi: normal; "> is the length of the pendulum and g<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; "> is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce <img src="data:image/png;base64,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" alt="2pi" style="width: 19.5px; height: 18px;" width="19.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; ">.</div></div></div>

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Last Solution submitted on Jul 07, 2022

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Dyuman Joshi on 20 Oct 2022

It seems integral() is not able to resolve upto the required precision.
I will have to think of another method or function.

ChrisR on 28 Oct 2022

Hint: 45982

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Solution 8489925

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Problem 54710. Compute the period of a pendulum started from a finite initial angle

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Problem 54710. Compute the period of a pendulum started from a finite initial angle

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