I Force on Body Attached to Spring at Displacement x - A.P. French

AI Thread Summary
The discussion focuses on understanding the forces acting on a body attached to a spring at small displacements, specifically the derivation of the restoring force equation F = -kx. It highlights that the first equation is a Taylor series expansion, which allows for approximating the restoring force in various systems, not limited to Hooke's law. The conversation also clarifies that while the restoring force is typically proportional to displacement, it can take different forms, such as in the case of a pendulum where the force is proportional to the sine of the angle. By applying a Taylor series, one can simplify the analysis for small displacements, justifying the approximation of the sine function. Overall, the thread emphasizes the mathematical foundation behind modeling forces in oscillatory systems.
Slimy0233
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Source: A.P. French's Vibrations and Waves

I do not recognize the first equation, can someone explain how it came to be? The reasoning behind it.

How can force on a body attached to a spring at small displacement x be represented as

1685899566771.png


? I know recognize F = - kx (restoring force)

I realize that the mass is at equilibrium and not rest, thus there were/are multiple forces acting on the spring, thus, I guess my question simplifies, what is the nature of the forces

1685899663451.png


if -kx is restoring force, what are the rest of the forces, can someone please state an example for better understanding?

edit: Good God, creating a post is no joke :')
edit 2: The math which was visible at first is not visible now.
 
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The first equation is just a Taylor series. Whatever form the restoring force takes (not just Hooke's law), you can Taylor expand it into a series, a polynomial of some high (possibly infinite) order. But if we agree to restrict ourselves to the region where ##x## is small then ##x^2##, ##x^3##, etcetera, must be really small and we can neglect all terms except the ##x## one. Then the force reduces to the linear restoring force you are familiar with.

(That's a paraphrase of the paragraph between the two marked equations, by the way.)
 
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Ibix said:
The first equation is just a Taylor series. Whatever form the restoring force takes (not just Hooke's law), you can Taylor expand it into a series, a polynomial of some high (possibly infinite) order.
I am sorry, can you please explain this more. Especially the "whatever form the restoring force takes" part.
 
Well, a restoring force is just how strongly a system resists deformation or displacement. It doesn't have to be directly proportional to the displacement. An obvious example is a pendulum, where the restoring force is proportional to the sine of the displacement angle, ##\theta##. But you can expand that sine as a Taylor series, and as long as you keep the angle small then you can neglect the ##\theta^3## and higher terms (the even power terms are zero in this case). This is the formal justification for writing "##\sin\theta\approx\theta## for small ##\theta##". Once you have done that, you have justified modelling a small-amplitude pendulum as a simple harmonic oscillator.

(And, although French doesn't mention it above, you can find out how big ##\theta## has to be for the ##\theta^3## term to matter, and hence how small a "small" amplitude actually must be.)
 
So, in the case of the pendulum, the first equation you have marked would be $$\begin{eqnarray*}
F(\theta)&=&-mg\sin\theta\\
&=&-mg\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\ldots\right)
\end{eqnarray*}$$As long as ##\theta## is small this is approximately ##F(\theta)\approx-mg\theta##. This is the right hand side of the second equation you have marked, which would therefore be$$ml\frac{d^2\theta}{dt^2}=-mg\theta$$
 
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@Ibix These are one of the best answers I have ever received. Thank you very much!!

Beautifully explained!
 
You're very welcome. I don't think the explanation is quite as unparalleled as you say, but I'm glad it helped you.
 

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