Gravitational Force Calculator
Calculate the gravitational attraction force between two masses using Newton's law of universal gravitation. Enter the two masses, the distance between their centres, and get the force in newtons, pound-force and more.
Find the Attraction Between Two Masses (Newton's Law)
Gravitational Force Formula
Where:
- \(F\) — gravitational force between the two bodies (N)
- \(G\) — gravitational constant, \(6.674 \times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}\)
- \(m_1\) — mass of the first body (kg)
- \(m_2\) — mass of the second body (kg)
- \(r\) — distance between the centres of the two bodies (m)
This is Newton's law of universal gravitation: every pair of masses attracts each other with a force proportional to the product of the masses and inversely proportional to the square of the distance between them.
Gravitational Force – Calculation Example
Example: find the gravitational pull between the Earth (\(m_1 = 5.972 \times 10^{24}\) kg) and the Moon (\(m_2 = 7.348 \times 10^{22}\) kg), whose centres are on average \(r = 384\,400\) km apart.
\[ F = 6.674 \times 10^{-11} \cdot \frac{5.972 \times 10^{24} \times 7.348 \times 10^{22}}{(3.844 \times 10^{8})^2} \]
\[ F = \frac{2.93 \times 10^{37}}{1.478 \times 10^{17}} \approx 1.98 \times 10^{20}\ \mathrm{N} \]
Result: the Earth and the Moon attract each other with a force of about \(1.98 \times 10^{20}\) N, or roughly \(4.46 \times 10^{19}\) lbf — the default values of the calculator, so you can reproduce this result instantly.
How to use the calculator
- Enter Mass 1 (m₁) and pick its unit: kg, g, tonne, lb, Earth masses or Solar masses. Use scientific notation for large values, e.g.
5.972e24. - Enter Mass 2 (m₂) with its unit in the same way.
- Enter the Distance between centers (r) and choose the unit: m, km, cm, ft, in or mi. The distance must be measured from the centre of one body to the centre of the other.
- Read the results: the gravitational force is shown in newtons, kilonewtons, meganewtons, pound-force and kilogram-force, together with the acceleration each body experiences and a comparison that puts the force on a human scale.
What is gravitational force?
Gravitational force is the attractive interaction between any two objects that have mass. Newton's law of universal gravitation, published in 1687, states that the force grows with the product of the two masses and shrinks with the square of the distance between them. Double the distance and the force falls to a quarter; halve it and the force becomes four times stronger. Unlike friction or tension, gravity is never repulsive — it always pulls the two bodies toward each other along the straight line joining their centres.
The constant \(G\) sets the strength of gravity. It is remarkably small: \(6.674 \times 10^{-11}\), which is why gravity between everyday objects is almost imperceptible. Two people standing one metre apart attract each other with a force of roughly \(10^{-7}\) N — billions of times weaker than their weight. Gravity only becomes dominant at planetary scales, where enormous masses compensate for the tiny constant.
The result is symmetric: both bodies feel exactly the same force, but not the same acceleration. By Newton's second law, \(a = F/m\), so the lighter body accelerates much more. In the Earth–Moon example the force on both bodies is \(1.98 \times 10^{20}\) N, but the Moon accelerates toward Earth about 81 times more than Earth accelerates toward the Moon. The calculator shows both accelerations in the results table.
If you need to connect this force to motion, the Kinematic Equations Solver, the Centripetal Force Calculator (for orbits) and the Conservation of Energy Calculator are natural next steps.
Assumptions, limitations and practical tips
- Point masses or spheres only. The formula is exact for point masses and for spherically symmetric bodies. For irregular shapes at close range, treat the result as an approximation.
- Distance is measured centre-to-centre. For a satellite orbiting Earth, \(r\) is the orbital radius from Earth's centre, not the altitude above the surface. Add Earth's radius (≈ 6371 km) to the altitude.
- The inverse-square law dominates. Gravity never reaches exactly zero, but it fades fast: at 10 times the distance the force is only 1% of the original value.
- Everyday masses give tiny forces. If the result looks negligibly small for two cars or two people, that is correct — gravity really is the weakest fundamental interaction.
- Use scientific notation for planetary masses (e.g.
7.348e22), or simply switch the unit to Earth masses or Solar masses and enter a plain number.
Reference values
| Body | Mass (kg) | Gravitational attraction with Earth |
|---|---|---|
| Moon | \(7.348 \times 10^{22}\) | \(\approx 1.98 \times 10^{20}\) N |
| Sun | \(1.989 \times 10^{30}\) | \(\approx 3.54 \times 10^{22}\) N |
| Person (70 kg) at Earth's surface | 70 | \(\approx 686\) N (their weight) |
To explore other forces and their interplay, try the Friction Force Calculator or the Force Calculator.
Frequently Asked Questions
How do I calculate gravitational force by hand?
Write down the masses in kilograms and the centre-to-centre distance in metres, then apply \( F = G m_1 m_2 / r^2 \) with \( G = 6.674 \times 10^{-11} \). Multiply the two masses, multiply by G, and divide by the square of the distance. Keep everything in SI units and the result comes out directly in newtons; convert other units before you start to avoid mistakes.
Why is the gravitational constant 6.674×10⁻¹¹ and not 9.81?
These are two different numbers. \( G = 6.674 \times 10^{-11}\ \mathrm{m^3 kg^{-1} s^{-2}} \) is the universal gravitational constant — it is the same everywhere in the universe and describes the strength of gravity per unit mass. The value 9.81 m/s² is \( g \), the acceleration you experience at Earth's surface, which already has Earth's mass and radius plugged into the formula.
What is the difference between gravitational force and weight?
Weight is just gravitational force in a specific situation: the pull a planet exerts on an object at (or near) its surface. To find your weight on Earth, enter your mass, Earth's mass (5.972×10²⁴ kg) and the distance \( r \) equal to Earth's radius (≈ 6,371,000 m) — not the distance to the surface. A 70 kg person gets about 686 N, which matches \( mg \) with \( g \approx 9.81 \) m/s².
How do I calculate surface gravity on the Moon or another planet?
For a body of mass \( M \) and radius \( R \), surface gravity is \( g = G M / R^2 \). In this calculator, set Mass 1 to the planet, Mass 2 to any small test mass (say 1 kg), and the distance to the planet's radius in metres. The acceleration shown for the test mass is the surface gravity: about 1.62 m/s² for the Moon and 3.71 m/s² for Mars.
What happens to the force if I double one mass or halve the distance?
The force scales linearly with each mass but with the square of the distance. Doubling one mass doubles the force; doubling both masses makes it four times stronger. Halving the distance makes the force four times larger, while doubling the distance makes it four times smaller — this inverse-square behaviour is the same rule that governs light and radio-wave intensity.
Can I use this calculator to find a satellite's orbital speed or period?
Indirectly, yes. For a circular orbit, gravity supplies the centripetal force: \( G m_1 m_2 / r^2 = m_2 v^2 / r \), so \( v = \sqrt{G m_1 / r} \) and the period is \( T = 2\pi r / v \). Use the orbital radius (Earth's radius plus altitude) in this calculator to get the acceleration, then apply these formulas. The Centripetal Force Calculator helps with the intermediate step.
Does gravity ever reach zero, and can the force be negative?
No on both counts. Gravitational force decreases with the square of the distance but never becomes exactly zero — even distant galaxies pull on you, just immeasurably weakly. And gravity is always attractive, so Newton's formula only ever gives a positive magnitude; the direction is along the line joining the centres. If the calculator rejects your input, it is because a mass or distance of zero makes the formula undefined.