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This is the question:

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And the answer is:

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I understand the $2\sqrt{a^2+x^2}-2l$ part but how does Young modulus $\lambda/2l$ equal the spring constant?

Shouldn't it be $\lambda \times A/2l$ where A is the cross-sectional area?

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For (some) Mathematicians Hooke’s law is the proportionality between the force and the fractional change in the length and they call the constant of proportionality the “modulus os elasticity” and denote it using the symbol $\lambda$. Thus for these Mathematicians $\lambda$, the modulus of elasticity, is not the same a Young’s modulus.
In your example the spring constant, often denoted by symbol $k$, $k=\frac {\lambda}{2l}$ with $2l$ being the original length of the string.

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