= Gravitational constant measurement sensitivity
{title2=$G_{\mathrm{inferred}}=Fr^2/(Mm)$}
A laboratory determination of the <gravitational constant> must connect a sensor's reading to a gravitational signal from independently characterized <masses>. For an ideal two-<point mass> <force> measurement, invert the model:
$$
G_{\mathrm{inferred}}=\frac{Fr^2}{Mm},\qquad \frac{\delta G}{G}\simeq\frac{\delta F}{F}+2\frac{\delta r}{r}-\frac{\delta M}{M}-\frac{\delta m}{m}.
$$
These are signed small changes obtained by taking the <logarithmic derivative>, not an instruction to add <standard uncertainties> with signs. For independent small input uncertainties, linear <propagation of uncertainty> instead gives
$$
\left(\frac{u(G)}G\right)^2\simeq\left(\frac{u(F)}F\right)^2+4\left(\frac{u(r)}r\right)^2+\left(\frac{u(M)}M\right)^2+\left(\frac{u(m)}m\right)^2.
$$
Correlated inputs require their <covariances> too. In particular, common geometric calibration errors must not be treated as independent simply because several distances were recorded.
For a concrete model check, hold the actual <force> and <masses> fixed but overestimate a true $r=0.100\,\mathrm m$ by $1\,\mathrm{\mu m}$. The exact inferred ratio is
$$
\frac{G_{\mathrm{inferred}}}{G_{\mathrm{true}}}=(1+10^{-5})^2=1.0000200001.
$$
The resulting bias is $20.0001$ parts per million, already comparable with the relative <standard uncertainty> of the recommended <gravitational constant>. A sensor can resolve very small changes while its inferred answer remains limited by source geometry. The example concerns one idealized separation; it is not an uncertainty estimate for any actual experiment.
\Image[/gravitational-constant-scale-and-sensitivity.png]
{title=Weak laboratory attraction and the geometric sensitivity of an inferred gravitational constant}
{description=Left: ideal <force> between two stipulated $1\,\mathrm{kg}$ <point masses> using the 2022 recommended <gravitational constant>. Right: exact model bias when only the separation entered in the inference changes. These curves are calculations, not measurement data.}
Making the source <mass> larger or bringing it closer strengthens the signal, but also makes detailed <mass density> and geometry important. Nearby apparatus supplies its own <Newtonian gravitational field>; electromagnetic shielding does not remove that attraction. Sensor offsets, changing <temperature>, vibration and source motion can imitate or alter a signal. The <Cavendish experiment> illustrates calibration of a <torsion constant>; a different sensor requires its own calibrated response and a full source model.
Repeating readings can reduce independent <measurement noise>, but a fixed scale error survives averaging. A useful experiment deliberately changes the source configuration, checks whether the signal follows the predicted change, and characterizes what moves with the source. A <systematic error> that changes with that configuration also survives background subtraction. Comparing methods with different sensors and different dominant errors is therefore valuable.
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