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When I was in the tea room eating my lunch, I noticed a piece of tissue
paper with crossword data scrawled across it. Being the mischifous soul
I am, I added the text "T = 2 pi sqrt(L/G)".
Apparently my handwriting is very unique. Either that or everybody knows
that only I would be stupid enough to write such a thing. Either way, a
guy just came other to thank me for my help with the crossword, and to
ask what the hell the formula is for.
3 points to the first person who can tell me what this formula describes.
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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> 3 points to the first person who can tell me what this formula
> describes.
Period for a ideal pendulum of length L in a homogeneous graviational
field of a acceleration G in the limit of small elongations (sin(x) appr.
x).
I don't mind the points, I'll dontate them to charity...
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Michael Zier wrote:
> Period for a ideal pendulum of length L in a homogeneous graviational
> field of a acceleration G in the limit of small elongations (sin(x) appr.
> x).
>
> I don't mind the points, I'll dontate them to charity...
You can have a few more points if you explain the "in the limit of small
elongations" part...
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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>
> You can have a few more points if you explain the "in the limit of small
> elongations" part...
That makes me suspicious, where do you get the points from, so you can
shell them out like no good? ;)
Try to set up the differential equations for the pendulum, and you'll run
into non-linear parts somewhere. So in order to keep the solution simple,
usually one assumes elongations small compared to the pendulum length (so
that you can approximate "sin(x)" by a much more linear "x").
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Michael Zier <mic### [at] mirizi de> wrote:
> So in order to keep the solution simple,
> usually one assumes elongations small compared to the pendulum length (so
> that you can approximate "sin(x)" by a much more linear "x").
Or in other words, sin(x) is very close to x for the range -y <= x <= y,
where y is a small positive value.
--
- Warp
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> You can have a few more points if you explain the "in the limit of small
> elongations" part...
You can write an exact differential equation for the motion of a pendulum,
but it will have a sin or cos in it because the string/rod is at an angle to
gravity. Usually you replace sin(x) with just x to make it easy to solve
and get a simple solution. x (in radians!) is actually a pretty good
approximation to sin(x) for angles up to 10 or 20 degrees - try it.
Obviously the bigger swings your pendulum makes, the more inaccurate the
"simple" equation will be.
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scott wrote:
> You can write an exact differential equation for the motion of a
> pendulum, but it will have a sin or cos in it because the string/rod is
> at an angle to gravity. Usually you replace sin(x) with just x to make
> it easy to solve and get a simple solution. x (in radians!) is actually
> a pretty good approximation to sin(x) for angles up to 10 or 20 degrees
> - try it. Obviously the bigger swings your pendulum makes, the more
> inaccurate the "simple" equation will be.
Damn. And I thought it was an "exact" equation. Oh well...
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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>
> Damn. And I thought it was an "exact" equation. Oh well...
http://en.wikipedia.org/wiki/Image:Pendulum_period.svg
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Michael Zier wrote:
> http://en.wikipedia.org/wiki/Image:Pendulum_period.svg
"The graph as well as other important points are simply wrong: The time
at 90° amplitude is 18% larger than T0 and not 7.3% as shown in the
plot. Then, the author plots the integrand instead of the integral.
Furthermore, he calculates sin(th0)/2 instead of sin(th0/2). Means, this
page is seriously flawed!"
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Am Tue, 26 Aug 2008 14:40:18 +0100 schrieb Invisible:
> Michael Zier wrote:
>
>> http://en.wikipedia.org/wiki/Image:Pendulum_period.svg
>
> "The graph as well as other important points are simply wrong: The time
> at 90° amplitude is 18% larger than T0 and not 7.3% as shown in the
> plot. Then, the author plots the integrand instead of the integral.
> Furthermore, he calculates sin(th0)/2 instead of sin(th0/2). Means, this
> page is seriously flawed!"
"Author
Alessio Damato; thanks to John wayman, he let me notice a mistake in the
code."
If you look at the plot, you'll notice that at 90° it shows a deviation
of 18% indeed. So the file might actually be the corrected one.
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