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I'm trying to calculate the separation between coils on a massive spring with no weights attached under different conditions.

I will present my solution to the problem in the case of a uniform gravitational field, which gives answers which I know by experiment to be false. I would be great full if some one will point out a mistake in my reasoning.

Let $\rho$ be the linear density of the spring in it's unextended state, and $k$ its spring constant. Let $f(x)$ be the vertical component of position in the extended state of the point which was originally had vertical component of position $=x$. I'm not sure how to better phrase this, see diagram below:

setup

Let us consider find the resultant force acting on the piece (in the blue box) of the spring between $x$ and $x+h$ equating it to zero as the position we are considering is equilibrium. 3 Forces are acting:

  • (Up) The tension from the part of the spring above it: $k*(f(x)-x)$ by Hooke's law

  • (Down) Gravity: $\rho hg$

  • (Down) Tension from the part of the spring below it $k*(f(x+h)-x-h)$. By analogy of the first force acting on the bottom part of the spring, and then using newton's 3rd law to conclude that the force must be equal and opposite.

Now the equation we get is:

$$k*(f(x)-x)=\rho hg+k*(f(x+h)-x-h)$$

Rearranging:

$$0=\frac{\rho g}{k}+\frac{f(x+h)-f(x)}{h}-\frac{h+x-x}{h}$$

And taking the limit as $h \rightarrow 0: $ $$0=\frac{\rho g}{k}+f'(x)-1$$

Rearranging and integrating:

$$f(x)=x*(1-\frac{\rho g}{k})+c$$

$c=0$ as $f(0)=0$ (the top of the spring is attached).

This means that the spring extends linearly, and the separation between successive coils are the equal. Yet intuitively the parts of the spring which are higher support more weight and therefore should be extended more! In the image below we can clearly see that the calculation is wrong and the intuition is correct. But where is the error in the calculation?

experiment

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The mistake in your reasoning is that you are forgetting the effect of the mass above the box on the tension. Your "Hooke's law" expression does not apply the way you think it does - the tension in the spring decreases linearly with distance along the spring (in the unextended state) because of the mass of the spring.

This modifies your equations - there is an additional term $\rho g x$ that must appear.

That term will result in a quadratic expression - just as you would expect.

See if you can get it from here...

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  • $\begingroup$ Surly the mass above the blue box has no effect on the extension of the spring in the blue box. If we were to cut the spring at the top of the blue box and rigidly attach it somewhere, I have a hard time believing it would "know" about it and the extension on the part in the blue box would change. Is there a book that you might recommend to help me understand this more? $\endgroup$ – Michal Oct 12 '15 at 22:25
  • $\begingroup$ The expression appears on the left hand side of your equation... the force pulling on the top of your blue box is not simply $k(f(x)-x)$, but $k(f(x)-x)-\rho g x$. $\endgroup$ – Floris Oct 12 '15 at 22:30
  • $\begingroup$ would the force pulling the bottom of the blue box also be changed to $k(f(x+h)-x-h)-\rho g (x+h)$? In the case $f(x)$ also seams to come out linear... I've also realised another mistake I made: the spring constant $k$ is dependant on the length on the spring $\endgroup$ – Michal Oct 12 '15 at 23:06

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