Gravitational Force Calculator
Newton's law of universal gravitation: F = Gm1m2/r².
About this calculator
Newton's law of universal gravitation says every pair of masses attracts each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers: F = Gm₁m₂/r², where G is the gravitational constant, 6.674×10⁻¹¹ N·m²/kg². This calculator's defaults model a person standing at Earth's surface — Earth's mass (5.972×10²⁴ kg) as m₁, a 70 kg person as m₂, and Earth's radius (6.371×10⁶ m) as the distance — which lets it double as an "acceleration due to gravity" calculator: dividing the force by m₂ gives the familiar surface gravity figure, about 9.8 m/s², without you needing to enter g directly. Beyond force, the calculator derives several related quantities that only depend on the larger mass m₁ and the distance, treating m₂ as a negligible test mass: gravitational potential (energy per unit mass, always negative since gravity is attractive and potential is defined as zero at infinite separation), escape velocity (the speed needed to break free of m₁'s gravity entirely from that distance, √(2Gm₁/r)), and orbital velocity and period for a circular orbit at that exact distance.
Because distance is measured center-to-center, not surface-to-surface, entering a surface altitude directly instead of adding it to the planet's radius is a common error that understates the true separation. Also note this is a two-body, non-relativistic model — it doesn't account for a third body's influence or the extreme regimes where general relativity's corrections become significant, such as near a black hole or for high-precision satellite timing.
Inputs
Results
Gravitational Force (N)
687.37
Acceleration of m₂ (m/s²)
9.82
How to Use This Calculator
- Enter mass 1 (kg) — e.g., Earth = 5.972e24 kg — and mass 2 (kg) — e.g., a person = 70 kg.
- Enter the center-to-center distance (m) between the two masses.
- Read gravitational force (N) and the acceleration of mass 2 (m/s2).
- Review gravitational potential (J/kg), escape velocity (m/s), and orbital velocity (m/s).
- Use orbital period (s) to understand the circular orbit at this distance.
How the result changes with Distance (m)
| Distance (m) | Gravitational Force (N) | Acceleration of m₂ (m/s²) |
|---|---|---|
| 3,185,500 | 2,749.47 | 39.28 |
| 4,778,250 | 1,221.99 | 17.46 |
| 9,556,500 | 305.5 | 4.36 |
| 15,927,500 | 109.98 | 1.57 |
What each input means
- Mass 1 (kg)
- Primary mass (e.g., Earth = 5.972e24 kg).
- Mass 2 (kg)
- Secondary mass (e.g., a person = 70 kg).
- Distance (m)
- Center-to-center distance (Earth radius = 6.371e6 m).
What each result means
- Gravitational Force (N)
- F = Gm₁m₂/r².
- Acceleration of m₂ (m/s²)
- g = F/m₂. Surface gravity if m₂ is a test mass.
- Grav. Potential (J/kg)
- Φ = -Gm₁/r. Potential energy per unit mass.
- Escape Velocity (m/s)
- vₑ = √(2Gm₁/r).
- Orbital Velocity (m/s)
- vₒ = √(Gm₁/r) for circular orbit.
- Orbital Period (s)
- T = 2πr/vₒ for circular orbit.
How this is calculated
Worked example, using the default values
- Identify Input ParametersMass 1 (kg) = 5.972e+24, Mass 2 (kg) = 70, Distance (m) = 6371000 = 3 input(s) provided
- Calculate Gravitational ForceGravitational Force = parseFloat(force.toPrecision(6))687.367 = 687.367
- Calculate Acceleration of m₂Acceleration of m₂ = parseFloat(acceleration2.toPrecision(6))9.81953 = 9.81953
- Calculate Grav. PotentialGrav. Potential = parseFloat(gravPotential.toPrecision(6))-62560200 = -62560200
- Calculate Escape VelocityEscape Velocity = parseFloat(escapeVelocity.toPrecision(6))11185.7 = 11185.7
Engine last updated . Checked against 1 independently-derived test — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why do the default values give a surface gravity of about 9.8 m/s², and how is that computed?
The defaults model a 70 kg person standing at Earth's surface — Earth's mass (5.972×10²⁴ kg) as m1 and Earth's radius (6.371×10⁶ m) as the distance — and the engine derives surface gravity as acceleration2 = force / m2, i.e. Newton's second law applied to the force Newton's law of gravitation predicts. Because m2 cancels out of that division (force is proportional to m2 in the first place), the result comes out to roughly 9.8 m/s² regardless of what mass you actually enter for m2.
Why is gravitational potential always negative?
It's computed as Φ = −Gm1/r, and the negative sign is a deliberate convention: potential is defined as zero at infinite separation, and since gravity is always attractive, moving two masses closer together releases energy, meaning the potential at any finite distance must be lower (more negative) than at infinity. A more negative potential simply indicates a deeper gravitational well, not an error in the calculation.
Do escape velocity and orbital velocity depend on the mass of the second object (m2)?
No — both are derived purely from m1 (the larger, primary mass) and distance: escape velocity is √(2Gm1/r) and orbital velocity is √(Gm1/r). The calculation treats m2 as a negligible test mass, which is why entering a different m2 changes the force and the acceleration outputs but leaves escape velocity, orbital velocity, and orbital period completely unchanged.
Why does the calculator ask for center-to-center distance instead of surface altitude?
Newton's law of gravitation treats both masses as point masses located at their centers, so the r in F = Gm1m2/r² is always the separation between centers, not between surfaces. If you're modeling an object at some altitude above a planet's surface, you need to add that altitude to the planet's radius yourself before entering it — using the altitude alone understates the true distance and inflates the resulting force and gravity figures.
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