1. Title and aim
Calibration of a helical spring and determination of the spring constant. Determine the spring stiffness constant and verify Hooke’s law of linear elasticity by measuring extension as a function of applied tensile load.
2. Abstract
Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of spring stiffness constant of 25.00 N m⁻¹, differing by 0.000% 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
- Helical spring and rigid stand: Provide a linear elastic restoring force with uniform coil pitch and known proportional limit.
- Mass hanger and calibrated slotted weights: Apply reproducible downward gravitational tensile forces in calibrated increments.
- Vertical millimeter mirror scale: Measure pointer displacement accurately while eliminating optical parallax by line-of-sight reflection alignment.
- Lightweight index pointer: Indicate spring elongation directly on the scale without adding significant parasitic tare mass.
- Helical steel spring suspended vertically from a rigid support clamp.
- Mirror millimeter scale used to eliminate optical parallax on pointer alignment.
- Incremental static tensile loading performed strictly within the proportional elastic limit.
4. Theory and graph choice
Explore how elastic bodies deform in response to applied mechanical forces. In this experiment, you will add calibrated loads to a vertically suspended helical spring, measure the resulting elongation against a millimeter scale, and construct a linear regression to determine the spring constant k. The observations follow an ideal elastic spring within its proportional limit.
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
- Ensure the support stand is vertically upright and stable. Suspend the spring and empty mass hanger freely. Note the initial position of the index pointer against the millimeter scale as the zero reference reading.
- Carefully add a 0.50 N load (approx. 51.0 g mass) to the mass hanger. Allow any vertical bouncing or swinging motion to subside completely.
- Position your eye so that the tip of the pointer exactly covers its reflected image in the mirror scale. Read the pointer position to the nearest 0.1 mm.
- Systematically increase the stretching force in steps of 0.50 N up to 2.50 N (or 3.00 N maximum). At each step, record the total applied force and the corresponding pointer position.
- Gradually remove the weights in reverse order (2.50 N down to 0.50 N, and finally zero load). Check that the pointer returns to its initial unloaded position.
- Plot extension δ on the vertical axis against force F on the horizontal axis. Compute the best-fit line δ = mF + b using ordinary least squares. The spring constant is k = 1/m.
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.
| No. | Stretching load F (N) | Spring extension δ (mm) | x (SI) | y (SI) | Fitted y (SI) | Residual (SI) |
|---|---|---|---|---|---|---|
| 1 | 0.50 | 20.0 | 0.50000 | 0.02000 | 0.02000 | -1.3878e-17 |
| 2 | 1.00 | 40.0 | 1.00000 | 0.04000 | 0.04000 | -1.3878e-17 |
| 3 | 1.50 | 60.0 | 1.50000 | 0.06000 | 0.06000 | -1.3878e-17 |
| 4 | 2.00 | 80.0 | 2.00000 | 0.08000 | 0.08000 | -1.3878e-17 |
| 5 | 2.50 | 100.0 | 2.50000 | 0.10000 | 0.10000 | -1.3878e-17 |
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.
Σ(x − x̄)² = 2.50000
Σ(x − x̄)(y − ȳ) = 0.10000
m = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² = 0.04000000
b = ȳ − mx̄ = 1.3878e-17
k = 1/m = 1/(0.04000000) = 25.000000 N m⁻¹
9. Uncertainty and comparison
Percentage difference = |estimate − model value| / |model value| × 100 = 0.0000%. Percentage difference is not experimental uncertainty.
For n = 5 observations, Σresidual² = 9.6296e-34; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.1331e-17. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.8791e-17.
Propagating only the relevant fit coefficient gives a fit-only standard error of 7.0820e-15 N m⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.
Scale alignment, pointer zeroing, weight mass tolerances, and imperfect spring elasticity contribute uncertainty. Real helical springs have an effective dynamic self-mass and initial winding tension. The regression slope error reflects residual scatter but excludes calibration systematic errors. In a physical laboratory, load incrementally and decrementally to verify the absence of elastic hysteresis.
10. Result and discussion
Model comparison value: 25.00 N m⁻¹
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
- Do not load the spring beyond its proportional elastic limit; avoid hanging masses greater than 300 g (approx. 3.0 N).
- Align your eye directly perpendicular to the mirror scale so that the pointer tip perfectly overlaps its reflection to prevent parallax.
- Wait patiently until all oscillatory motion has stopped before reading the scale; do not touch the spring or hanger during measurement.
- Always lower slotted masses gently onto the hanger to avoid dynamic impact forces that could permanently stretch the coils.
The virtual apparatus simulates an ideal helical spring obeying linear Hookean elasticity with uniform wire properties. In physical laboratory experiments, spring self-mass (which contributes an effective dynamic mass), initial coil winding tension, temperature dependence of the steel’s shear modulus, and minor support flexure contribute to real-world deviations. Instrument reading quantization is modeled at 0.1 mm, matching a millimeter mirror scale. The ideal teaching model obeys Hooke’s law δ = F/k across the tested loading range. Real laboratory springs often possess initial coil tension, requiring a small threshold force before extension commences, which manifests as a non-zero horizontal intercept. Fitting extension against force yields slope m = 1/k in SI units (m N⁻¹); the spring stiffness constant is recovered from the inverse gradient k = 1/m in N m⁻¹.
12. Conclusion
The worked example illustrates how to obtain spring stiffness constant from measurements, graph variables and a fitted coefficient. The estimate is 25.00 N m⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.