# Pendulum Formula Derivation

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#1
I am doing my physics coursework for the "Physics B (Advancing Physics)" course, which involves a written report on how I found the value for the acceleration due to gravity. One of these methods was to use a pendulum. But I can't find how the derivation for the pendulum formula is done. Any help?

Pendulum formula:
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9 years ago
#2
(Original post by Sora)
I am doing my physics coursework for the "Physics B (Advancing Physics)" course, which involves a written report on how I found the value for the acceleration due to gravity. One of these methods was to use a pendulum. But I can't find how the derivation for the pendulum formula is done. Any help?

Pendulum formula:
I don't think there's a closed form expression for the period of a simple pendulum without resorting to elliptic integrals. What you've got there is the approximate period. Have you encountered SHM?
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#3
SHM = Simple harmonic motion? If yes, then no, I have not met this. And all the derivations I have found on the internet involve differential equations, which I also haven't met before. So, I am kind of stuck.
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9 years ago
#4
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9 years ago
#5
The formula is an approximation.
It is derived assuming the amplitude (the angle the pendulum swings through) is less than about 10 degs.
In this case sin θ is very nearly equal to θ
The proof is here

http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c3
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9 years ago
#6
I would say that it would be unreasonable to expect you to derive this at A level. As ben-smith says, it is only an approximate expression in any case. I think that you would be OK just quoting it, so long as you qualify it with the appropriate conditions (most importantly, that it only works with small displacements from the rest position).
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#7
My teacher wrote on my coursework "Where does this come from!" next to the formula when i quoted it... Not sure what to do.
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9 years ago
#8
(Original post by Sora)
My teacher wrote on my coursework "Where does this come from!" next to the formula when i quoted it... Not sure what to do.
Your teacher may just be indicating that you can't just quote the result out of thin air. It may be sufficient to quote it if you back it up with a reference, or indicate that it can be derived with knowledge of simple harmonic motion and differential equations. It does not necessarily mean that you have to do the derivation, especially as it looks like this is way above your current level, just that you cannot quote it from nowhere.

I suggest you talk to your teacher for clarification.
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#9
The coursework is due in for Monday. I got it back with the teachers notes on Friday, and I had the weekend to adjust it. Worried now...
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9 years ago
#10
Well at this stage, you have little choice. I suspect that you would not be able to give a convincing account of where the equation comes from (not a dig, just an observation, it requires parts of both physics and maths that you have yet to meet). You can either scrap the entire section that mentions it, or give a reference as to where it comes from, describe briefly how it can be derived, but leave out the detail. The latter really is fine - I suspect that your teacher is indicating that you cannot just drop in an equation without any mention of where it comes from.
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#11
Would a source that describes it be credible? Or will I have to do a brief description?
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9 years ago
#12
I can't answer that, I've no idea what the requirements of your exam board are.
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#13
I'll do a brief description to be on the safe side.
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9 years ago
#14
there is a derivation... my teacher did it for us.. im doing the same course, so im guessing you need to know it! however as it would happen it would be in my old exercise book which id better hunt out...

*5mins later...*
So i have this written down.. although I'm not exactly sure how some of it follows on.. maybe someone will understand! As far as i know we dont need to be able to differentiate.. so the bit in red we should just know, and dont need to derive.. i'm not even sure if the two bits link tbh! better review it before my retake lol!

F proportional to x
F=-Kx
F=ma

a=dv/dt and v=dx/dt
a=d^2x/dt^2
a=dv/dt

ma=-Kx
a=-Kx/m
d^2x/dt^2 = -Kx/m

x=A cos w t (this and this to the next line is all that I don't understand!)
w=root (k/m)
w=(2 pi) /t = 2 pi f

2 pi F = root (k/m)

f= 1/2pi root (k/m)

f=1/t so t=1/f

t=2pi root (m/k) ------ spring

t=2pi root (l/g) ------ pendulum

x(t) = A sin (2 pi f t + theta)
x(f) = A cos (2 pi f t + theta)
x= A sin (wt + theta)
x= A cos (wt + theta)

f=ma
m (d^2x/dt^2) = -mg sin theta
sin theta is approx theta for small angles (in radians)

d^2x/dt^2 = -g theta

d^2x/dt^2 = -(g/l) ^ x

EDIT:
Ok
so basically its putting the two SHM equations equal and solving:

d^2x/dt^2 = -kx/m = -(2 pi f)^2 * x

then solve!

F= force (N)
x= extension (metres)
k= constant

a= acceleration
m= mass

A= maximum displacement
w= "ro" that curvy w symbol!
f= frequency
t/T = time (secs)

l= length (metres)
g= gravity (9.81)
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#15
That is incredibly confusing... How am i supposed to explain that?
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9 years ago
#16
I've never seen a derivation of the result with the 2pi that doesn't involve differential equations. However, you can argue that the formula has to have sqrt(l/g) because that's the only way to make something that has units of time out of the things that are relevant for the motion of the pendulum.
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1 month ago
#17
Help me in derivation of the above formula
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1 month ago
#18
(Original post by Getobai)
Help me in derivation of the above formula
https://courses.lumenlearning.com/ph...mple-pendulum/
For this kind of thing Google is the first place you should check :-:
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