Elasticity — Question 1

Question 1

A measurement offset must not be mistaken for material compliance

A wire has unloaded length $L=2\,\mathrm m$ and area $A=5\,\mathrm{mm^2}$. Its extension gauge reads $.30\,\mathrm{mm}$ at $100\,\mathrm N$ and $.70\,\mathrm{mm}$ at $300\,\mathrm N$. In this range the wire is linear elastic, but the gauge has an unknown constant zero offset $b$: $r=FL/(EA)+b$.

1. Infer Young's modulus without assuming the gauge zero is correct.

2. Find $b$ and the true extension at $100\,\mathrm N$.

Question 1 - Solution

A measurement offset must not be mistaken for material compliance

(a) Derivation and calculation

Taking differences eliminates the constant offset: $$\frac{\Delta r}{\Delta F}=\frac{L}{EA} =\frac{(.70-.30)10^{-3}}{300-100}=2.00\times10^{-6}\ \mathrm{m/N}.$$ Thus $$\boxed{E=\frac{2}{(5\times10^{-6})(2\times10^{-6})}=200\ \mathrm{GPa}}.$$

(b) Result and interpretation

The true extension at 100 N is $$\boxed{\delta_{100}=100(2\times10^{-6})=.200\ \mathrm{mm}}.$$ Hence $$\boxed{b=r_{100}-\delta_{100}=.300-.200=.100\ \mathrm{mm}}.$$ At 300 N the model predicts $.600+.100=.700$ mm, recovering the other reading.

Physical check

The modulus depends on the slope of the force–extension relation, so a constant gauge offset does not affect it.