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My book has this true/false question $$\pmb {Question}$$

The stretching of a coil is determined by its sheer modulus.

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True

But I am not able to comprehend as how does stretching (i.e., involving Tension and hence Young's modulus) can involve shear modulus? The derivation of the mathematical expression(if not explain explanation) would do a good job.

Thanks!

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I think the question could be clearer (that wording leaves a lot of wiggle room); but I believe I understand what it is trying to say.

I find it's easy to picture the coil when we think of a helical spring. As you stretch the spring, the windings of the coil get further and further apart.

If you look at it from the perspective of the wire though, the wire isn't being stretched or bent at all (in an ideal coil). It's actually just twisting the wire as the spring stretches. Because of the helical coil, when the wire is twisted the right direction, it brings all the coils further apart, which stretches the whole length of the coil.

Because it is pure twisting on the wire, it is determined by the shear modulus instead of Young's modulus.

So the answer to your question is because the material of the spring isn't stretching, it's twisting, which ends up stretching the overall coil length.

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  • $\begingroup$ I would argue that it is both stretching and twisting. $\endgroup$
    – Bob D
    Commented Dec 3, 2019 at 13:54
  • $\begingroup$ @BobD for an ideal coil, it’s typically pure twisting without stretching. That’s all that’s required for a spring to work. The distance between ends of the coil stretches, but the actual material of the wire is just twisting. $\endgroup$
    – JMac
    Commented Dec 3, 2019 at 14:16
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The shear modulus $G$ and Young's modulus $E$ for a homogeneous and isotropic material are related by the equation:

$$G=\frac{E}{2(1+ν)}$$

where $ν$ is Poisson's ratio, or -(lateral strain)/(longitudinal strain).

Poisson's ratio takes into account that when a material stretches longitudinally it also contracts laterally and when it is compressed longitudinally it also expands laterally.

Hope this helps.

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  • $\begingroup$ I don’t think this really addresses the question of why they would say the stretching of a coil depends on shear modulus. $\endgroup$
    – JMac
    Commented Dec 3, 2019 at 14:25

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