Talk:Pendulum (mechanics)

Mathematics?

Why isn't this article called Pendulum (physics)? The content reminds me a lot more of a (theoretical) physics lecture than anything I had in mathematics. bamse (talk) 20:38, 7 December 2016 (UTC)

The article at Pendulum is less technical and may be closer to what the average reader wants to know. The theoretical background is here at Pendulum (mathematics). So-called exact solutions can be more easily discussed in this article, where you predict the motion without needing the small-angle approximation. You can imagine that other titles are possible. But it would be strange to have Pendulum (physics) when we already have Pendulum, so it's not obvious how to improve the title. See also Pendulum (disambiguation). EdJohnston (talk) 21:44, 7 December 2016 (UTC)

But it would be strange to have Pendulum (physics) when we already have Pendulum I would not have a problem with it. I guess it depends on what you call "mathematics". bamse (talk) 22:13, 7 December 2016 (UTC)

I prefer Pendulum (mathematics). The Pendulum article contains a lot of the physics of pendulums, just not discussed with equations. I think the current title gives general readers a clearer idea of what this article is about than Pendulum (physics) would. --Chetvorno^TALK 23:00, 7 December 2016 (UTC)

I think the current title gives general readers a clearer idea of what this article is about .... You mean scares them away ;-)? I don't feel strongly about the title and don't want to start a long discussion. It is just that I (as a physicist) stumbled upon it expecting some kind of mathematical concept called "pendulum". If you call everything with equations in it "mathematics" fine. At least we have Power (physics) and Work (physics).bamse (talk) 10:22, 8 December 2016 (UTC)

This unbalanced article focuses almost entirely on solving the large angle motion of the ideal frictionless gravity pendulum, a mathematical model not applicable to real world pendula. The is within the mathematical field of dynamical systems. I agree there is a lot of interesting physics connected with pendulums, but this article covers almost none of it. It probably should. For example, if this article included such topics as the motion of real pendulums with damping, Q factor and critical damping, driven pendulums, sources of error, effect of thermal expansion of the rod, temperature compensated pendulums such as the gridiron pandulum and mercury pendulum, calculating the period of arbitrarily shaped pendulums (compound pendulums), the center of oscillation and radius of gyration, the Kater pendulum, the effects of the escapement on accuracy (the Airy condition for isochronism), effects on period due to air pressure (Boyle correction) and viscosity, or coupled pendulums, then I would support your proposal to change the name to Pendulum (physics). As it is.... --Chetvorno^TALK 10:30, 9 December 2016 (UTC)

Well, I guess we both agree that there is some overlap between mathematics and physics. Where to draw the line is a personal thing. Some people might argue that only pure mathematics is mathematics and as soon as concepts like time (as in dynamical systems) appear, that would be physics (or engineering or whatever). On the other hand there is also applied mathematics, etc. Regarding your first argument, isn't basically all of physics a mathematical model not applicable to real world [problems]. (I mean all of the models used in physics have limitations and give only approximate results.) ? bamse (talk) 13:21, 9 December 2016 (UTC)

Could somebody add closed form expression for the period T

There exists a closed form expression for the period T

${\begin{alignedat}{2}T&=2\pi {\sqrt {\frac {\ell }{g}}}\left(1+\left({\frac {1}{2}}\right)^{2}\sin ^{2}{\frac {\theta _{0}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}\sin ^{4}{\frac {\theta _{0}}{2}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right)^{2}\sin ^{6}{\frac {\theta _{0}}{2}}+\cdots \right)\\&=2\pi {\sqrt {\frac {\ell }{g}}}\cdot \sum _{n=0}^{\infty }\left(\left({\frac {(2n)!}{(2^{n}\cdot n!)^{2}}}\right)^{2}\cdot \sin ^{2n}{\frac {\theta _{0}}{2}}\right)\\&=2\pi {\sqrt {\frac {l}{g}}}{\frac {1}{\sqrt {\cos ^{2}\left({\frac {\theta _{0}}{2}}\right)}}}.\end{alignedat}}$

Looks interesting. Do you have a source? --Chetvorno^TALK 02:01, 23 June 2017 (UTC)

I may have just requested something similar in a "clarify" tag at the bottom. Something about the final equation above gives me trouble. I am SO not a mathematician, but wouldn't the square root of the cosine squared just be the cosine?

Riventree (talk) 11:48, 23 August 2018 (UTC)

LOL I did't notice that! The whole expression is just equal to the small angle period divided by cos θ/2, which is horse manure. Something went wrong in the last transition. Seriously, as far as I know, there is no "closed form" solution for the large angle period. All the formulas involve evaluating an infinite series, or a recursive function, like the arithmetic-geometric mean M. These are all approximate methods, since the evaluation has to stop at some point. --Chetvorno^TALK 03:33, 24 August 2018 (UTC)

Simplified "Energy Derivation"

The potential and kinetic energy of the pendulum remain constant:

$mgl(1-\cos \theta )+{\dfrac {mv^{2}}{2}}=k$

Differentiating,

${\dfrac {mgl\sin \theta d\theta }{dt}}+{\dfrac {2mvdv}{2dt}}=0$

$g\sin \theta +a=0$

Shubjt (talk) 18:44, 24 January 2022 (UTC)

Initial velocity

In the "Examples" section there is "The animations below depict the motion of a simple (frictionless) pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity."

That doesn't make sense. Initial angle $\neq$ initial velocity. Initial velocity is zero. A1E6 (talk) 21:08, 29 March 2023 (UTC)

It is saying that the motion shown in the drawings could have been started by either of two initial conditions: pulling the pendulum to the side and releasing it (initial displacement) or giving it a push (initial velocity). But I agree the last phrase is a confusing and unnecessary addition. My advice is be bold and delete it. --Chetvorno^TALK 00:06, 30 March 2023 (UTC)

Oh I see. One could argue that even though initial angle and initial velocity are not the same, increasing the initial angle does indeed correspond to increasing initial velocity. Also the pendulum has to have some initial velocity (or energy) in order to perform a full swing, as shown in the last two animations. For this reason I'll rather keep the article as-is. A1E6 (talk) 11:22, 30 March 2023 (UTC)

Now this might be controversial but I would suggest removing the last two animations and rewriting the paragraph so that it mentions increasing initial angle only. Because in the rest of the article we assume that initial velocity of the pendulum is zero (see also this https://en.wikipedia.org/wiki/Pendulum_(mechanics)#/media/File:Oscillating_pendulum.gif – at the extremities, the velocity vector is zero). What do you think? A1E6 (talk) 13:07, 30 March 2023 (UTC)

Incomplete elliptic integral incorrect?

IMO, the expression F(pi/2,k) after Eq. 3. and other occurences of F should be F(1,k). 178.197.198.157 (talk) 05:14, 15 June 2023 (UTC)