Gravitational Force Calculator — Newton's Law of Universal Gravitation
Gravitational force governs everything from the orbit of the Moon to the trajectory of spacecraft. The free gravitational force calculator on PublicSoftTools lets you compute F = Gm₁m₂/r² for any two masses and distance — including planetary-scale values using scientific notation.
What Is Newton's Law of Universal Gravitation?
Published in 1687, Newton's law states that every mass in the universe attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them:
F = G × m₁ × m₂ / r²
Where G = 6.674 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant, m₁ and m₂ are the two masses in kilograms, and r is the distance between their centers in meters. The result F is in Newtons.
How to Use the Calculator
- Open the gravitational force calculator.
- Enter Mass 1 in kilograms. Scientific notation is accepted — Earth's mass is 5.972e24 kg.
- Enter Mass 2 in kilograms. The Moon's mass is 7.342e22 kg.
- Enter the distance r between the centers of the two masses in meters. The Earth-Moon distance is 3.844e8 m.
- The gravitational force appears instantly in Newtons, formatted in scientific notation for large values.
Worked Examples
| System | m₁ | m₂ | r | Force |
|---|---|---|---|---|
| Earth–Moon | 5.972×10²⁴ kg | 7.342×10²² kg | 3.844×10⁸ m | ~1.98×10²⁰ N |
| Earth–Sun | 5.972×10²⁴ kg | 1.989×10³⁰ kg | 1.496×10¹¹ m | ~3.54×10²² N |
| Two 1 kg balls, 1 m apart | 1 kg | 1 kg | 1 m | 6.674×10⁻¹¹ N |
| Surface gravity check | 5.972×10²⁴ kg | 1 kg | 6.371×10⁶ m | ~9.81 N |
The last row is a sanity check: the gravitational force between Earth and a 1 kg object at Earth's surface should equal 9.81 N — matching the standard g = 9.81 m/s².
The Inverse-Square Law
The r² denominator means gravitational force follows an inverse-square law. Double the distance and the force drops to one-quarter. Triple the distance and it drops to one-ninth. This is why:
- Satellites in low Earth orbit (400 km up) experience almost the same gravity as on the surface — they feel weightless only because they are in free fall.
- At the Moon's distance, Earth's gravity is about 0.0027 m/s² — enough to keep the Moon in orbit but far weaker than surface gravity.
- The speed required to orbit at a given altitude scales with √(1/r): higher orbits require less speed.
The Gravitational Constant G
G = 6.674 × 10⁻¹¹ N·m²/kg² is one of the fundamental constants of nature. It was first measured by Henry Cavendish in 1798 using a torsion balance — two small lead balls attracted to two larger ones. The measurement was so delicate that it took until 1798 to accomplish, more than a century after Newton published the law.
G is dimensionally small, which is why gravity between everyday objects is undetectable. Two 1 kg masses 1 m apart exert only 6.674 × 10⁻¹¹ N of force on each other — about 100 billion times weaker than the weight of a grain of sand.
Common Questions
Why is gravity so weak between ordinary objects?
Because G is extremely small. Gravity only becomes significant when at least one mass is astronomical — like a planet or star. At human scales, electromagnetic and contact forces (friction, normal force) are billions of times stronger than gravity.
Can I use centimeters instead of meters?
The calculator uses SI units: kilograms for mass and meters for distance. If your distance is in centimeters, divide by 100 before entering. The result is always in Newtons.
Does this formula work for objects with non-uniform mass distributions?
The formula applies exactly when both objects are point masses (or perfect spheres with uniform density). For real objects with irregular shapes, the formula is still a good approximation when the distance r is much larger than the objects' sizes.
Calculate Gravitational Force
Enter any two masses and separation distance to compute the gravitational force between them.
Open Gravitational Force Calculator