Young’s modulus theory

The same hanging force acts through every section of an ideal straight wire in static equilibrium when its own weight is neglected. Dividing this force by the original cross-sectional area gives tensile stress. A thin wire therefore experiences greater stress than a thick wire under the same force. Stress tells us how intensely the material is loaded, rather than describing the load alone. Use force in newtons and area in square metres to obtain pascals.

Extension is the increase in gauge length, measured from a chosen reference. Strain divides that increase by the original gauge length, so two specimens of different lengths can be compared fairly. An extension of 0.50 mm is a much smaller fractional change for a two-metre wire than for a short wire. Convert both lengths to matching units before taking their ratio. Strain has no unit.

Young modulus relates tensile stress to strain within the straight portion of the material response. Larger modulus means smaller strain for a given stress. This model keeps every selected load within a linear elastic range, so removing the added load restores the reference length. In a physical specimen, elastic recovery and strict proportionality must be checked; an elastic response need not remain perfectly linear at every stress.

Substituting stress and strain into the modulus relation gives extension proportional to added force. Put force F on the horizontal axis and extension δ on the vertical axis. The gradient s then has units metres per newton and equals L divided by AY. Calculate modulus using the inverse of this gradient. Reversing the graph axes reverses the gradient, so always label axes before choosing a formula.

The micrometer reading must be corrected for any verified zero error. Take readings at several positions and rotate the instrument around the wire to detect nonuniformity or an oval cross section. Use gentle, repeatable contact rather than squeezing the wire. Diameter enters the area as a square; consequently a small fractional diameter error has approximately twice that fractional effect on area and inferred modulus. Repeated measurements reveal variation but cannot remove a shared calibration error.

A small initial tension can straighten a real wire. Define zero extension after this setup condition and express subsequent forces as additional forces relative to it. Wait for movement to settle before reading. An unloaded repeat checks whether the apparatus slips or the specimen retains deformation. A constant indicator offset can appear as a graph intercept; it should be investigated rather than automatically forcing a fit through zero.

For this specimen the predicted gradient is about 0.05093 mm/N, corresponding to 0.1019 mm extension at 2 N and 1.0186 mm at 20 N. Convert a graph gradient expressed in mm/N to m/N before calculating pascals. Examine deviations from the fitted line, not only the final modulus. Scatter in a fitted gradient captures only one part of uncertainty; length, diameter, force calibration, indicator resolution and temperature can also contribute. An ideal data set may have zero scatter while real measurement uncertainty remains finite.