Physics

Physical constants from gradients

Calculate a physical constant from a fitted gradient.

Your measurements

Number formats & units

Use decimal dots in datasets. Single values also accept decimal commas and scientific notation such as 1.2e-3 or 1.2 × 10⁻³.

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Result & working

Example

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Formula & example

The relationship

T^2=\frac{4\pi^2}{g}L\implies g=\frac{4\pi^2}{m}

\text{extension}=\frac{F}{k}\implies k=\frac1m

R=\frac{\rho}{A}L\implies\rho=mA

\ln\!\left(\frac{y}{y_{\mathrm{ref}}}\right)=\text{constant}-\frac t\tau\implies\tau=-\frac1m

t_{1/2}=-\frac{\ln2}{m}

\text{relative output uncertainty}\approx\frac{\Delta m}{|m|}\quad\text{for an inverse}

What the variables mean

Symbol	Meaning
$m$	Gradient of exactly the selected vertical-against-horizontal relationship.
$\Delta m$	Absolute uncertainty from an appropriate graph method.
$A$	Cross-sectional area, with uncertainty already propagated from diameter if needed.

Worked example

Plotted relationship: T² against L → gravitational acceleration
Positive gradient m: 4.03
Absolute gradient uncertainty: 0.05

g = (9.8 ± 0.1) m/s²

Follow the example steps

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.

Common mistakes

Rearrange the fitted relationship before finding the constant.

Incorrect approach

T²/L = 4 s²/m: g = 4

Correct approach

g = 4π²/4 ≈ 9.87 m s⁻²

More checks before you submit

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.

Method notes

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.

Physical constants from gradients

Your measurements

Result & working

Download results

Formula & example

The relationship

Worked example

Common mistakes

✕ Incorrect approach

✓ Correct approach

Linearization & log plots

Error bars & acceptable lines

Uncertainty propagation

Incorrect approach

Correct approach