Pendulum Equation

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Anonymous

Pendulum Equation 2015-11-09 23:45:37 Post No. 7649536

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Pendulum Equation 2015-11-09 23:45:37 Post No. 7649536

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How and why does the pendulum equation work?

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>>7649536

did you try looking up the explanation?

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>>7649536

Small angle approximation and dimensional analysis.

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Tâ2sqrt(L) because Ï/sqrt(g)â1

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>>7649536

Because gravity works, potencial energy is converted to cinetic energy and because the distance of the movement is a circle.

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>>7649536

it doesn't though

you don't need an especially big pendulum to see it breaking down

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>>7649702

Because of friction or is there another reason?

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>>7649753

don't know what that guy was talking about but the equation in OP is an approximation and falls apart if your starting angle is bigger

the closed form for the correct equation includes annoying integrals or an infinite power series so the approximation is often used even if it's fairly inaccurate

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>>7649775

>hurr it's an approximation

Like every physics equation in existence.

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>>7649786

ya but equations like [math] F = mv^{2}[/math] and [math] F = G\frac{mM}{r^{2}}[/math] are reasonably accurate in real world situations but the pendulum formula is such a rough approximation it's unusable for all but the smallest of angles, most of which are selected precisely because otherwise the equation is just flat out wrong

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>>7649801

obviously the first equation was supposed to be [math]E_{k} = \frac{1}{2}mv^{2}[/math]

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>>7649891

>just a fun little equation

more like shitty useless equation

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>>7649902

Stop dissing my equation mother fucker.

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>>7649536

That equation is a shitty approximation for the period of a pendulum.

The force of gravity acting on the swinging pendulum is [math]F_g = m*g*Sin(\theta)[/math]. However, to derive your equation [math]Sin(\theta)[/math] is approximated to be equal to [math]\theta[/math]. This approximation is only appropriate for very small values of [math]\theta[/math].

>http://www.pha.jhu.edu/~broholm/l25/node2.html

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>>7649983

It's not that shitty of an approximation, though. Empirically it works for angles up to like 20 degrees.

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>>7649693

Br?

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>>7649536

I can't be bothered to write up the derivation, but basically you find that the force driving the pendulum is equal to [math]m g \sin \theta[/math]. If you approximate [math]\sin \theta = \theta[/math], you get a second-order differential equation in [math]\theta[/math]. Solve that and you get a solution with [math]\sin[/math] and/or [math]\cos[/math]. By looking at the terms inside the [math]\sin[/math] or [math]\cos[/math], you can find the period [math]T[/math].

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>>7649993

It's the shittiest

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>>7650034

It's techinically accurate to a 2nd order taylor series.

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>>7650040

Practically, theoretically and now technically shit as well. Good job

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>>7650055

You're delusional if you think its shitty https://en.m.wikipedia.org/wiki/Kater%27s_pendulum

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>>7649993

Hm

Looking at it again, the difference between Sin(x) and x at 22.5 degrees is about 0.01. Didn't realize it at first but it might be a better approximation than I originally thought.

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fckn pendulums, how do they werk?

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The Lagrangian for a mathematical pendulum is

[math]\frac{1}{2} m L^2 \dot \varphi^2 + m g L \cos(\varphi)[/math].

Plug it in the E-L equation to get

[math]m L^2 \ddot \varphi + m g L \sin(\varphi) = 0[/math].

Approximate [math]\sin(\varphi) \approx \varphi[/math]

[math] \ddot \varphi + \frac{g}{L} \varphi = 0[/math]

which is the differential equation of a harmonic oscillator.

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>>7650319

>approximate

why would you do that.

there are solutions without this

https://en.wikipedia.org/wiki/Pendulum_(mathematics)#Arbitrary-amplitude_period

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>>7650374

he's a pop scientist, that' why

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