Is the spring constant k changed when you divide a spring into parts?

I've always been taught that the spring constant $k$ is a constant — that is, for a given spring, $k$ will always be the same, regardless of what you do to the spring.

My friend's physics professor gave a practice problem in which a spring of length $L$ was cut into four parts of length $L/4$. He claimed that the spring constant in each of the new springs cut from the old spring ($k_\text{new}$) was therefore equal to $k_\text{orig}/4$.

Is this true? Every person I've asked seems to think that this is false, and that $k$ will be the same even if you cut the spring into parts. Is there a good explanation of whether $k$ will be the same after cutting the spring or not? It seems like if it's an inherent property of the spring it shouldn't change, so if it does, why?

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– Qmechanic♦ Nov 16 '11 at 21:09

4 Answers

Well, the sentence

It seems like if it's an inherent property of the spring it shouldn't change, so if it does, why?

clearly isn't a valid argument to calculate the $k$ of the smaller springs. They're different springs than their large parent so they may have different values of an "inherent property": if a pizza is divided to 4 smaller pieces, the inherent property "mass" of the smaller pizzas is also different than the mass of the large one. ;-)

You may have meant that it is an "intensive" property (like a density or temperature) which wouldn't change after the cutting of a big spring, but you have offered no evidence that it's "intensive" in this sense. No surprise, this statement is incorrect as I'm going to show.

One may calculate the right answer in many ways. For example, we may consider the energy of the spring. It is equal to $k_{\rm big}x_{\rm big}^2/2$ where $x_{\rm big}$ is the deviation (distance) from the equilibrium position. We may also imagine that the big spring is a collection of 4 equal smaller strings attached to each other.

In this picture, each of the 4 springs has the deviation $x_{\rm small} = x_{\rm big}/4$ and the energy of each spring is $$E_{\rm small} = \frac{1}{2} k_{\rm small} x_{\rm small}^2 = \frac{1}{2} k_{\rm small} \frac{x_{\rm big}^2}{16}$$ Because we have 4 such small springs, the total energy is $$E_{\rm 4 \,small} = \frac{1}{2} k_{\rm small} \frac{x_{\rm big}^2}{4}$$ That must be equal to the potential energy of the single big spring because it's the same object $$= E_{\rm big} = \frac{1}{2} k_{\rm big} x_{\rm big}^2$$ which implies, after you divide the same factors on both sides, $$k_{\rm big} = \frac{k_{\rm small}}{4}$$ So the spring constant of the smaller springs is actually 4 times larger than the spring constant of the big spring.

You could get the same result via forces, too. The large spring has some forces $F=k_{\rm big}x_{\rm big}$ on both ends. When you divide it to four small springs, there are still the same forces $\pm F$ on each boundary of the smaller strings. They must be equal to $F=k_{\rm small} x_{\rm small}$ because the same formula holds for the smaller springs as well. Because $x_{\rm small} = x_{\rm big}/4$, you see that $k_{\rm small} = 4k_{\rm big}$. It's harder to change the length of the shorter spring because it's short to start with, so you need a 4 times larger force which is why the spring constant of the small spring is 4 times higher.

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For a given spring, $k$ is a constant, As long as you're talking about an ideal spring. In other words, the definition of the ideal spring is that it applies the force proportional to its deformation length (at both endings of course).

I'm afraid both you and your professor are wrong. The correct formula should be:

$k_{new} = k_{orig}*4$

To show that let's do the following gedankenexperiment. Suppose you have your original spring in tension. It's deformed length $L$, and applies the appropriate force $F$.

Now imagine that your spring is actually 4 consequently connected springs of length $L/4$. Each spring is at rest, this means that for every spring the forces applied to both endings are equal. Since all the springs are connected and apply forces on each other - this means that all the forces applied to all the spring endings are the same. And obviously they equal to $F$.

OTOH each spring is deformed by only $L/4$. Hence - their "constants" are 4 times higher

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In other way $k \times l = \text{constant}$ so $K$ varies $1/l$.

So $K\times l=K' \times l/4$ and $K=K'/4$ so $K'=4K$ $K'$ will be $4k$ ($k$=original spring constant)

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To supplement the answer by Luboš Motl, I will come to this problem from a Material Science point of view.

What you mean by the inherent property of the string is not the spring constant, in fact, it is Young's modulus $E$, which is defined to be the same for some material (and not only for a string):

$$E = \frac{\text{tensile stress}}{\text{tensile strain}} = \frac{\sigma}{\varepsilon} = \frac{\text{force per area}}{\text{extension per length}} = \frac{F / A}{x / l} = \frac{F l }{x A}$$

Now use this definition to construct the Hooke's Law:

$$F = \frac{EA}{l} x = k x$$

where

$$k = \frac{EA}{l}$$

Now if you think, when cutting a string in half, you are keeping $E$ and $A$ constant (the same material and the same cross-sectional area), but you change $l$ by a factor of $4$. This makes the new $k$ to be 4 times smaller.

Also, $E$ is not always constant for some type of material, but greatly depends on the micro-structure of it, such as tiny imperfections in the crystalline structure, which influence $E$ in different ways.

Note, that a rigorous proof, would involve the geometry of the spring as well, but it would only influence the effective size of the $EA$ product - the scaling with length would remain the same.

Some rambling on $E$

I thought I might talk about why $E$ is always constant for some type of material. We can imagine a slab to be constructed by lots of tiny springs, which obey the Hooke's law. (This is possible because of one approximation, that the atoms in the material do not move far from their equilibrium position.)

Because of the energy conservation we already know (the answer by Luboš Motl), that if we connect several springs, then we will change the effective spring constant:

$$k_{new} = n * k$$ where n is the number of the springs.

This means, that the percentage part of the extension is more sensible to measure, i.e.: $$F = const * x / l = const * \varepsilon$$ using the previous definition.

The same if we connect several springs in parallel, the effective $k$ will scale with the number of springs we connect and the number of the strings will be proportional to the cross sectional area of the spring. $$F = const*A*x$$ Where we can denote the unknown constant as the Young's modulus.

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