Young’s modulus measures how stiff a material is in tension or compression. In a wire-extension experiment,
a known load is hung from a wire, the extension is measured, and Young’s modulus is calculated from stress
and strain.
1. Stress in the wire
Tensile stress is force divided by cross-sectional area:
\[
\sigma=\frac{F}{A}.
\]
For a circular wire of diameter \(d\), the area is
\[
A=\frac{\pi d^2}{4}.
\]
2. Strain in the wire
Tensile strain is extension divided by original length:
\[
\varepsilon=\frac{\Delta L}{L}.
\]
Strain is dimensionless, because it is a ratio of two lengths.
3. Young’s modulus
Young’s modulus is stress divided by strain:
\[
E=\frac{\sigma}{\varepsilon}.
\]
Substituting \(\sigma=F/A\) and \(\varepsilon=\Delta L/L\):
\[
E=\frac{F/A}{\Delta L/L}.
\]
\[
\boxed{
E=\frac{FL}{A\Delta L}
}.
\]
4. Using micrometer readings
If the extension is found from two micrometer readings, then
\[
\Delta L=R_{\mathrm{final}}-R_{\mathrm{initial}}.
\]
The reading difference must be converted to metres before using the formula for \(E\).
5. Self-weight correction
A vertical wire also has its own weight. If the measured extension is from a stress-free natural length, the
tension is not the same at every point in the wire. The average tension due to the wire’s own weight is
\(W/2\).
The wire weight is
\[
W=\rho A L g.
\]
A corrected effective force can be written as
\[
F_{\mathrm{eff}}=F+\frac{W}{2}.
\]
Then
\[
\boxed{
E=\frac{F_{\mathrm{eff}}L}{A\Delta L}
}.
\]
In many school or university labs, the extension is measured incrementally after the wire is already hanging in
place. In that case, the self-weight contribution is already present before the added load, so it usually cancels
from the added-extension calculation.
6. Safety-load check
The wire should remain below an allowable stress:
\[
\sigma_{\mathrm{allow}}=\frac{\sigma_y}{n},
\]
where \(\sigma_y\) is the yield or limiting stress and \(n\) is the required safety factor.
The experiment is within the selected safety limit if
\[
\sigma=\frac{F_{\mathrm{eff}}}{A}\le \sigma_{\mathrm{allow}}.
\]
The maximum safe applied load is approximately
\[
\boxed{
F_{\mathrm{safe,max}}
=
\sigma_{\mathrm{allow}}A-\frac{W}{2}
}.
\]
7. Graph interpretation
A load-extension graph is linear for a wire that obeys Hooke’s law:
\[
F=k\Delta L.
\]
The slope is the wire stiffness:
\[
k=\frac{F}{\Delta L}.
\]
For a wire,
\[
k=\frac{EA}{L}.
\]
Therefore,
\[
E=\frac{kL}{A}.
\]
A stress-strain graph has slope
\[
\boxed{
E=\frac{\Delta\sigma}{\Delta\varepsilon}
}.
\]
8. Common sources of error
| Source of error |
Effect |
Good practice |
| Diameter measurement error |
Large error in area because \(A\propto d^2\) |
Measure diameter at several positions and average |
| Parallax in micrometer or scale |
Wrong extension reading |
Read at eye level and use zero correction |
| Wire not straight or slipping clamp |
Extension too large |
Pre-load lightly and check clamps |
| Load too high |
Plastic deformation or breakage |
Stay below safe working stress |
| Temperature change |
Thermal expansion changes length |
Keep temperature stable |
9. Worked example
Suppose a wire has
\[
L=3.0\ \mathrm{m},
\qquad
F=80\ \mathrm{N},
\qquad
\Delta L=2.1\ \mathrm{mm}.
\]
Convert extension:
\[
\Delta L=2.1\ \mathrm{mm}=2.1\times10^{-3}\ \mathrm{m}.
\]
If the diameter is \(d=1.43\ \mathrm{mm}\), then
\[
A=\frac{\pi d^2}{4}
=
\frac{\pi(1.43\times10^{-3})^2}{4}.
\]
\[
A\approx1.61\times10^{-6}\ \mathrm{m^2}.
\]
Young’s modulus is
\[
E=\frac{FL}{A\Delta L}.
\]
\[
E\approx
\frac{(80)(3.0)}{(1.61\times10^{-6})(2.1\times10^{-3})}.
\]
\[
E\approx7.1\times10^{10}\ \mathrm{Pa}
=
71\ \mathrm{GPa}.
\]
This is in the range expected for an aluminium-like material.
Key idea: Young’s modulus is the slope of stress versus strain. In a wire lab, accurate diameter and extension
measurements are essential.
