Physics
Physical Constant from Graph Gradient Calculator
Match the plotted axes to the model before substituting a gradient. All numerical inputs here use SI units.
Your measurements
Start with the example, or enter your own values. Use a dot for decimals and e for scientific notation.
Calculated locally. No account or uploads.
Result & working
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Step-by-step working
The formula
T² = (4π²/g)L ⇒ g = 4π²/m
Extension = F/k ⇒ k = 1/m
R = (ρ/A)L ⇒ ρ = mA
ln(y/y_ref) = constant − t/τ ⇒ τ = −1/m; t_half = −ln 2/m
For an inverse: relative output uncertainty ≈ Δm/|m|
| Symbol | Meaning |
|---|---|
| m | Gradient of exactly the selected vertical-against-horizontal relationship. |
| Δm | Absolute uncertainty from an appropriate graph method. |
| A | Cross-sectional area, with uncertainty already propagated from diameter if needed. |
When to use this calculator
Derive the straight-line form first and compare with y = mx + c. Swapping the axes usually inverts the gradient relationship. The result includes propagated uncertainty and matched-place rounding.
For logarithmic decay, subtract a justified background before taking logs. The half-life here is a general mathematical decay scale, suitable for analysing a supplied dataset. A non-zero physical baseline can make the logarithmic plot curved.
Constants π and ln 2 are treated as exact. Products add relative uncertainties using the school-lab first-order convention.
Worked example
The calculator opens with these example values. All steps below are available even with JavaScript disabled.
- g = 4π²/m = 39.47842/(4.03) = 9.796133 m/s².
- Relative uncertainty ≈ Δm/|m| = 0.05/|4.03| = 0.01240695.
- Absolute uncertainty = 0.1215401 m/s². Rounded result: g = (9.8 ± 0.1) m/s².
g = (9.8 ± 0.1) m/s²
Common mistakes
- The gradient of T² against L is 4π²/g, not g.
- Keep the slope sign in an exponential model.
- Do not use a fit standard error as a maximum/minimum gradient half-range without explanation.