Law of Gravitation Calculator
Law of Universal Gravitation: The Complete Guide
The Law of Universal Gravitation describes how every pair of masses in the universe attracts each other with a force that depends on their masses and the distance between them. From the fall of an apple to the orbits of planets and satellites, this law provides the foundation for celestial mechanics, satellite design, tides, and many engineering calculations. If your calculator above solves gravitational force or acceleration, this guide explains the equations it uses, how to pick variables and units, typical use cases, limitations, and fully worked examples you can follow step-by-step.
Here \(F\) is the magnitude of the attractive force between two point masses \(m_1\) and \(m_2\), separated by a center-to-center distance \(r\). \(G\) is the universal gravitational constant. The force acts along the line joining the two mass centers and is always attractive. When one body is much more massive than the other (e.g., Earth and a small satellite), the smaller body’s motion is predominantly affected.
Core Equations & When to Use Each
- Gravitational force (point masses or spherically symmetric bodies): \( \displaystyle F = G\,\frac{m_1 m_2}{r^2} \). Use to compute mutual attraction between two bodies.
- Gravitational field / acceleration due to a primary of mass \(M\): \( \displaystyle g(r) = \frac{GM}{r^2} \). Use to find the acceleration of a small test mass at distance \(r\) from a spherical mass \(M\).
- Weight near a planet’s surface: \( \displaystyle W = m\,g \), with \( g \approx \frac{GM}{R^2} \) at radius \(R\) (planet’s mean radius).
- Gravitational potential energy (two-body): \( \displaystyle U(r) = -\,\frac{GMm}{r} \). Use for energy-based orbital and escape calculations.
- Escape speed from radius \(r\): \( \displaystyle v_e = \sqrt{\frac{2GM}{r}} \). Speed needed (ignoring atmosphere and rotation) to reach infinite distance with zero residual speed.
- Circular orbital speed: \( \displaystyle v_c = \sqrt{\frac{GM}{r}} \). Use for ideal circular orbits around a spherical primary.
- Orbital period (circular): \( \displaystyle T = 2\pi \sqrt{\frac{r^3}{GM}} \). Use to relate altitude to period for circular orbits.
Variables, Constants & Notation
- \(G\): universal gravitational constant. SI: \( \text{N·m}^2/\text{kg}^2 \). Accepted value \( \approx 6.67430\times10^{-11}\,\text{N·m}^2/\text{kg}^2 \).
- \(m_1, m_2\): interacting masses (kg).
- \(M\): mass of the primary body (e.g., Earth) (kg).
- \(m\): test mass / spacecraft mass (kg).
- \(r\): center-to-center distance between masses (m).
- \(R\): planetary mean radius (m) (e.g., Earth \(R \approx 6.371\times10^6\,\text{m}\)).
- \(g\): local gravitational acceleration (m/s\(^2\)). Near Earth’s surface \(g \approx 9.80665\,\text{m/s}^2\).
- \(U\): gravitational potential energy (J).
- \(v_e\), \(v_c\): escape and circular orbital speeds (m/s).
| Parameter | Typical Value | Notes |
|---|---|---|
| \(G\) | \(6.67430\times10^{-11}\,\text{N·m}^2/\text{kg}^2\) | Use SI consistently to avoid unit errors. |
| Earth mass \(M_\oplus\) | \(5.972\times10^{24}\,\text{kg}\) | Useful for \(g(r)=GM_\oplus/r^2\). |
| Earth radius \(R_\oplus\) | \(6.371\times10^{6}\,\text{m}\) | For near-surface estimates: \(g\approx GM_\oplus/R_\oplus^2\). |
| Sea-level \(g\) | \(\approx 9.80665\,\text{m/s}^2\) | Varies slightly with latitude and elevation. |
How to Calculate Gravitational Quantities (Step-by-Step)
- Identify the bodies and distance: model each as a point mass or a sphere (valid outside a spherically symmetric mass). Determine the center-to-center distance \(r\).
- Choose your goal: force \(F\), acceleration \(g\), potential energy \(U\), escape speed \(v_e\), or orbital speed/period.
- Apply the right equation: \(F = G\frac{m_1 m_2}{r^2}\), \(g=\frac{GM}{r^2}\), \(U=-\frac{GMm}{r}\), \(v_c=\sqrt{\frac{GM}{r}}\), \(v_e=\sqrt{\frac{2GM}{r}}\).
- Keep units consistent (SI): masses in kg, distances in m, \(G\) in N·m\(^2\)/kg\(^2\) yields \(F\) in N, \(g\) in m/s\(^2\).
- Check domain assumptions: outside the mass (for spherical model), negligible atmospheric drag, and no strong third-body effects unless explicitly modeled.
- Sanity-check magnitude: compare results to known values (e.g., \(g\) near Earth \( \sim 9.8 \,\text{m/s}^2 \)).
Your calculator can automate these steps: enter masses and separation to get \(F\); provide a primary mass and radius/altitude to get \(g\), \(v_c\), or \(v_e\), and show clean steps you can reuse.
Worked Examples
Example 1: Force Between Earth and a 1,000 kg Satellite at 400 km Altitude
Given: \( M_\oplus=5.972\times10^{24}\,\text{kg} \), \( m=1000\,\text{kg} \), altitude \(h=400\,\text{km}=4.00\times10^5\,\text{m} \). Earth radius \(R_\oplus=6.371\times10^6\,\text{m}\). Then \( r=R_\oplus+h=6.771\times10^6\,\text{m} \).
Answer: \( \approx 8.7\,\text{kN} \) of gravitational attraction.
Example 2: Local Gravitational Acceleration at 400 km
Given: same \( r=6.771\times10^6\,\text{m} \).
Answer: \( g \) is still substantial (~\(0.89\,g_{\text{surface}}\)); orbital “weightlessness” is due to continuous free fall, not absence of gravity.
Example 3: Circular Orbit Speed at 400 km
Answer: \( v_c \approx 7.67\,\text{km/s} \), which matches typical Low Earth Orbit speeds.
Example 4: Escape Velocity from Earth’s Surface (Ignoring Atmosphere)
Answer: \( v_e \approx 11.2\,\text{km/s} \), a well-known benchmark.
Example 5: Gravitational Potential Energy of a 1,000 kg Satellite at 400 km
Interpretation: Negative \(U\) indicates a bound state; energy must be added to reach \(U\to 0\) at \(r\to\infty\).
How to Interpret Results
- Inverse-square behavior: Doubling distance reduces force and \(g\) by a factor of four. Closer orbits face stronger gravity.
- Negative potential energy: Bound systems have \(U<0\); work must be done to “escape.”
- Orbits as free fall: Satellites are continuously accelerating toward the planet but moving sideways fast enough to miss it—hence sustained orbit.
- Mass proportionality: Force scales with the product \(m_1 m_2\); specific acceleration \(g\) depends only on the primary mass and radius.
Common Use Cases
- Satellite mission design: compute required orbital speeds, periods, and \(\Delta v\) budgets (with additional orbital mechanics).
- Launch analysis: estimate idealized escape speeds or staging requirements (ignoring atmosphere for first-pass estimates).
- Tidal predictions (conceptual): tidal forces arise from differential gravity across an extended body (requires gradient analysis).
- Astrodynamics education: energy and inverse-square intuition for transfers, Hohmann maneuvers (beyond the basic law).
- Geophysics: approximate variations of \(g\) with altitude and latitude (refined models add Earth’s oblateness and rotation).
Assumptions, Limitations & Pitfalls
- Spherical/point-mass assumption: \( F=G\frac{m_1 m_2}{r^2} \) holds exactly for point masses and outside spherically symmetric bodies. Inside a uniform-density sphere, \(F\propto r\) (not \(1/r^2\)). Real planets are oblate and inhomogeneous.
- Atmospheric drag: near-Earth satellites experience drag that decays orbits—basic gravitational equations ignore this.
- Third-body effects: the Sun, Moon, and other planets can significantly perturb orbits; two-body results become approximations.
- Relativistic corrections: close to very massive bodies or in precise timing applications (GPS), General Relativity adds measurable corrections beyond Newton’s law.
- Surface variations: local \(g\) varies with altitude, latitude, and geology; sea-level \(g\) is an average, not a constant everywhere.
- Unit mistakes: keep SI units throughout—mixing km with m or using mass units like kg with forces in lbf will corrupt results.
FAQ: Law of Universal Gravitation
Is gravity always attractive?
Yes, in Newtonian gravity the force is always attractive between positive masses. (General Relativity describes gravity as spacetime curvature, but test masses still accelerate “downhill.”)
Why do astronauts feel “weightless” in orbit if gravity is strong?
They are in continuous free fall. Gravity at Low Earth Orbit is ~\(8.7\)–\(9.0\,\text{m/s}^2\), but the spacecraft and occupants fall together, producing apparent weightlessness.
What is the difference between \(g\) and \(G\)?
\(G\) is the universal constant in the law; \(g\) is the local acceleration due to a particular massive body: \( g(r)=GM/r^2 \).
Does mass cancel out when computing acceleration?
Yes for a small test body in a primary’s field: \( F=G\frac{Mm}{r^2} \) and \( F=ma \Rightarrow a=\frac{GM}{r^2} \), independent of the test mass \(m\).
How accurate is the inverse-square law?
Extremely accurate for most engineering and orbital problems. Deviations arise from non-spherical mass distributions, third-body effects, and relativistic corrections in extreme regimes.
Quick Checklist for Accurate Calculations
- Use SI units consistently (kg, m, s, N) and the current best value of \(G\).
- Measure or compute the center-to-center distance \(r\); add planetary radius to altitude.
- Confirm the two-body assumption is reasonable; note any major perturbations (Moon, Sun) or drag.
- For orbits, choose the correct formula (circular vs. elliptical) and include mission-specific constraints.
- Sanity-check results against known benchmarks: Earth \(g\), LEO speeds (\(\sim 7.8\,\text{km/s}\)), escape speed (\(\sim 11.2\,\text{km/s}\)).
Bottom Line
The Law of Universal Gravitation links masses across the cosmos through a simple inverse-square relationship. With the formulas above, you can compute mutual forces, local gravitational acceleration, orbital speeds, periods, and escape conditions. Pair this guide with the calculator on top of the page to get instant answers and fully explained steps. Whether you’re studying physics, designing a satellite, or estimating how gravity changes with altitude, mastering these fundamentals delivers reliable, decision-ready results.