Enter measurement values
Use standard uncertainties. All values should use SI units.
Example measurement data
The default values model an Earth–Moon force estimate. Replace them with your own experiment values.
| Quantity | Example value | Standard uncertainty | Unit |
|---|---|---|---|
| G | 6.67430 × 10−11 | 1.5 × 10−15 | m³ kg⁻¹ s⁻² |
| m1 | 5.972 × 1024 | 2.0 × 1020 | kg |
| m2 | 7.342 × 1022 | 1.0 × 1018 | kg |
| r | 3.844 × 108 | 1.0 × 105 | m |
Formula used
The model begins with Newton’s universal gravitation equation.
F = Gm1m2 / r2
∂F/∂G = m1m2 / r2
∂F/∂m1 = Gm2 / r2
∂F/∂m2 = Gm1 / r2
∂F/∂r = −2Gm1m2 / r3
For independent inputs, add the squared derivative-weighted uncertainties. For correlated inputs, add the covariance terms using the selected correlation coefficients.
How to use this calculator
- Enter the two measured masses and centre-to-centre separation.
- Enter standard uncertainty for each measured quantity.
- Keep the constant values, or replace them with your selected reference values.
- Use a coverage factor that matches your reporting requirement.
- Enable correlations only when a shared source affects two inputs.
- Calculate, review the variance shares, then export the results.
Understanding uncertainty in universal gravitation
Why measurement uncertainty matters
Universal gravitation connects two masses through their separation. The calculated force may look precise, yet every measured input contains uncertainty. Masses have calibration limits. Distance readings have resolution limits. The gravitational constant also has an accepted uncertainty. Propagating these values shows how trustworthy the calculated force really is. This process is essential when comparing laboratory results, simulation outputs, and theoretical predictions.
The force model
The force magnitude follows the stated governing force equation. Force rises directly with either mass. It falls rapidly as separation increases. This squared distance term often dominates the final uncertainty. A small distance error can therefore matter more than a similar percentage error in mass. Use consistent SI units. Enter kilograms for both masses, metres for separation, and newtons for the resulting force.
First-order propagation
This calculator applies first-order uncertainty propagation. It differentiates the force equation with respect to each input. Each derivative measures local sensitivity. The derivative is then multiplied by the standard uncertainty of that input. The squared terms are added to create the variance. Taking the square root produces the combined standard uncertainty. This method works best when uncertainties are small, measurements are approximately independent, and the force model behaves smoothly near the selected values.
Relative uncertainty insight
Relative uncertainty makes different units easier to compare. For independent inputs, the gravitational-force relative uncertainty is approximately the square root of the summed squared relative uncertainties. The separation contribution is doubled before squaring because distance has an exponent of negative two. This provides a fast quality check. Reducing distance uncertainty usually provides a strong improvement. However, use the full derivative result whenever an uncertainty is large or a correlation exists.
Correlation and covariance
Measurements can share systematic effects. A common scale may affect both mass readings. A shared alignment method may connect distance and mass estimates. These relationships are represented by correlation coefficients. Positive correlation can increase total uncertainty. Negative correlation can reduce it. The covariance terms must be used carefully. Impossible correlation combinations can produce an invalid matrix. Review the measurement process before entering nonzero coefficients. Set every coefficient to zero when the quantities are reasonably independent.
Reporting a result
Report the force with its combined standard uncertainty. Add expanded uncertainty when a coverage factor is required. A factor near two is often used for an approximate 95 percent interval, subject to the assumptions behind that interpretation. Round uncertainty to sensible significant figures. Round the force to the same decimal place. Record units, measurement conditions, instrument resolution, and the chosen coverage factor. Clear reporting lets another person reproduce, assess, and improve the experiment. Before relying on a result, inspect units and input magnitudes. A centimetre entered as a metre changes the force drastically. Repeat measurements where possible. Averaging random variation can reduce standard uncertainty, but it cannot automatically remove systematic bias. Independent checks remain valuable during every experimental stage. Document environmental conditions because temperature, vibration, and alignment can significantly alter instruments, spacing, and repeatability during otherwise similar trials.
Frequently asked questions
1. Which equation calculates the force?
The calculator uses F = Gm1m2 / r2. G is the gravitational constant, m1 and m2 are masses, and r is centre-to-centre separation.
2. Why does separation affect uncertainty strongly?
Separation is squared in the denominator. Its derivative contains a factor of negative two. Therefore, a relative distance uncertainty is weighted about twice before it enters the independent first-order uncertainty sum.
3. What is a standard uncertainty?
A standard uncertainty is an uncertainty expressed like one standard deviation. It can come from repeated readings, calibration data, instrument resolution, or a justified measurement model.
4. Why is the distance derivative negative?
Force decreases when separation increases. The negative derivative describes that direction. Its squared contribution to variance is positive, so the combined uncertainty remains nonnegative for a valid uncertainty model.
5. Should I include uncertainty in G?
Include it when your reporting model treats G as an uncertain input. For many classroom calculations, its contribution is very small compared with measurement uncertainty in masses or separation.
6. Can I use grams or centimetres?
Yes, but convert every input and uncertainty consistently before calculation. The displayed force is in newtons only when masses use kilograms, separation uses metres, and G uses compatible SI units.
7. What does the coverage factor do?
The coverage factor multiplies the combined standard uncertainty to give expanded uncertainty. A value near two is often used for an approximate 95 percent interval, provided the required statistical assumptions are reasonable.
8. When should correlation coefficients be used?
Use them when inputs share meaningful systematic effects. Examples include a common calibration source or common alignment method. Leave them at zero for reasonably independent measurements.
9. Why can the calculator reject a correlation set?
Some combinations do not represent a physically possible correlation matrix. They can create negative calculated variance. Reassess the measurement relationships and use correlations supported by evidence.
10. When is first-order propagation less suitable?
It becomes less reliable for very large uncertainties, strongly nonlinear behavior, truncated distributions, or poorly characterized inputs. In those cases, simulation methods such as Monte Carlo analysis may be more appropriate.
11. How should I report the final result?
Report F with its unit, combined or expanded uncertainty, coverage factor, and relevant confidence interpretation. Also record input units, measurement conditions, data sources, and any assumed correlations.