Young’s modulus lab report

1. Title and aim

Young’s modulus of a wire. Determine the Young modulus of a wire from its extension under a series of increasing tensile forces, and interpret the gradient of an extension–force graph.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Young’s modulus of 200.018 GPa, differing by 0.009% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Wire and rigid support: Provide a straight specimen with a defined 2.000 m gauge length and a secure upper clamp.
• Load system or calibrated force gauge: Apply additional tensile force from 2 to 20 N in controlled increments.
• Micrometer screw gauge: Measure the small diameter at several positions and orientations before calculating cross-sectional area.
• Extension indicator and length scale: Record the initial gauge length and changes in length relative to a fixed reference.
• Initial measured wire length L₀ = 2.000 m.
• Circular wire diameter d = 0.500 mm; area A = πd²/4.
• Extension increments are rounded to 0.001 mm; force values are exact in this model.

4. Theory and graph choice

A metal wire may look rigid while extending by a fraction of a millimetre. This experiment makes that small deformation measurable. Measure the wire geometry, increase the force in steps, record extension relative to a reference, and use the complete set of readings to estimate stiffness of the material. The virtual specimen is 2.000 m long and 0.500 mm in diameter; its model Young modulus is 200 GPa.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units m N⁻¹.

5. Method

1. Check the clamp, identify the fixed gauge endpoints, and record the 2.000 m gauge length. Note that the animation exaggerates extension.
2. Check micrometer zero, measure at several positions in two orientations, and record a representative mean diameter. Use 0.500 mm for the virtual specimen.
3. Straighten the wire with the setup tension and establish the extension reference. Treat displayed force as added force relative to that reference.
4. Select an added force of 2 N, allow settling, and record extension. Repeat at 4, 6, 8 and 10 N or continue through the allowed 20 N range.
5. Revisit one load and then remove the added load. In a real lab, record unloading readings and inspect for slip, drift or residual extension.
6. Plot extension against added force, obtain the gradient, convert it to m/N, calculate A = πd²/4 and then Y = L/(As). Report Y in GPa with the model assumptions.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0090%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 3.0000e-13; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 2.5000e-08. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.8723e-07.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.09819 GPa. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Young’s modulus is proportional to L₀ and inversely proportional to the extension/force slope and diameter squared. For independent inputs, (uY/Y)² ≈ (uL/L)² + (2ud/d)² + (um/m)². A shared diameter error affects all readings and is not represented by scatter around the fitted line.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• In a physical lab secure clamps and keep feet clear of suspended loads; use a load catcher and follow the apparatus load limit.
• Add loads gently, keep the wire straight, and wait until oscillation stops before taking a reading.
• Use consistent micrometer contact and check for zero error, diameter variation and parallax where applicable.
• Avoid temperature drift and do not extend a real specimen beyond the established proportional range.

This is an ideal, uniform circular wire with constant modulus and small deformation. Added force is measured from a reference state; wire weight, support motion, creep, temperature change, calibration errors and plasticity are not modeled. The extension drawing is exaggerated for visibility. Perfect simulated readings demonstrate the equation and do not establish the uncertainty achievable with physical equipment. Fit scatter alone is not a complete uncertainty budget. The unknown model modulus is 200 GPa. A light initial preload defines the reference length; F and δ are additional increments relative to that reference. All loads stay in the linear elastic regime. The diagram exaggerates extension. Creep, temperature drift, backlash and plastic deformation are omitted.

12. Conclusion

The worked example illustrates how to obtain Young’s modulus from measurements, graph variables and a fitted coefficient. The estimate is 200.018 GPa. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.