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Gravitational Force Calculator

Apply Newton's law of universal gravitation, F = G·m₁·m₂/r², to find the force between two masses, one of the masses, or their distance.

Level: Grades 11–12

Printable worksheet

Gravitational Force Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Solve for

Measured center to center. The Earth–Moon average is about 384,400 km.

Result

Gravitational force

1.9816e+20 N

Gravitational force is 1.9816e+20 N.

In kilonewtons
1.9816e+17 kN
In pounds-force
4.4549e+19 lbf
Acceleration of object 2 toward object 1
0.0027 m/s²0.000275 g
Acceleration of object 1 toward object 2
3.3181e-5 m/s²

See the step-by-step solution ↓

Step-by-step solution

  1. Use the approximate mass of Earth for object 1

    m₁=5.9722e+24 kg

  2. Use the approximate mass of the Moon for object 2

    m₂=7.346e+22 kg

  3. Change distance to m, the standard unit the formula uses (1 km = 1,000 m)

    384,400 km×1,000=384,400,000 m

  4. Multiply G by both masses and divide by the distance squared, because the pull grows with mass and fades with distance

    6.6743e-11×5.9722e+24×7.346e+22384,400,0002=1.981638e+20 N

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how strongly two objects, like Earth and the Moon, pull on each other through gravity. Every pair of masses pulls on each other. Newton's law of universal gravitation gives the size of that pull: F = G × m₁ × m₂ ÷ r². Here m₁ and m₂ are the two masses, and r is the distance between their centers. G is the gravitational constant, a fixed number that sets how strong gravity is. Doubling either mass doubles the force. Doubling the distance cuts it to a quarter, which is why it is called an inverse-square law.

This calculator finds the force, the second mass or the separation, given the other values. You can type in any masses or pick approximate values for Earth, the Moon and the Sun. Alongside the main answer, it shows how quickly each object speeds up as it is pulled toward the other. That is how the same law explains both falling apples and orbiting moons.

Where it's used

Why this math matters outside the classroom:

  • Earth and Moon. Astronomers find Earth and the Moon pull on each other with a force of about 2 × 10²⁰ N.
  • Inverse square. Students see that doubling the distance between two masses cuts their attraction to a quarter.
  • Everyday masses. Two 70 kg people 1 m apart attract each other with only about 3.3 × 10⁻⁷ N, far too little to notice.
  • Satellite orbits. Satellite engineers find the pull on a spacecraft at its orbital height, which sets the speed it must keep.
  • Weighing the Earth. Measuring G in the lab, as Cavendish did, let scientists work out Earth's mass from its surface gravity.

Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSA.CED.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose whether to solve for the force, the mass of object 2 or the distance.
  2. For each object, pick a preset body or choose "Enter a mass" and type the mass with its unit.
  3. Enter the center-to-center distance and, if needed, the known force.
  4. Choose the output unit and read the result and the accelerations below it.

What the inputs mean

Object 1
The presets use rounded published masses: Earth ≈ 5.9722 × 10²⁴ kg, Moon ≈ 7.346 × 10²² kg, Sun ≈ 1.9885 × 10³⁰ kg. Use "Enter a mass" if you need a different value.
Distance between centers (r)
Always measure between the centers of mass. For an object on the surface of a planet, that is the planet's radius, not zero. The average Earth–Moon distance is about 384,400 km.
Gravitational force (F)
The force each object exerts on the other. By Newton's third law the two pulls are equal in size and opposite in direction.

Formula and method

F = G × m₁ × m₂ ÷ r²
m₂ = F × r² ÷ (G × m₁)
r = √(G × m₁ × m₂ ÷ F)
G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²
a₂ = G × m₁ ÷ r²

F is the pulling force, m₁ and m₂ are the two masses, and r is the distance between their centers. G is the gravitational constant. a₂ is the acceleration of object 2, meaning how fast its speed grows each second as object 1 pulls on it. The calculator converts masses to kilograms and the distance to meters, so the force comes out in newtons (N).

G is the CODATA 2018 recommended value, the figure scientists agree to use. It is one of the least precisely known fundamental constants, uncertain in roughly the fifth significant figure.

Dividing the force by one object's mass gives the acceleration of that object toward the other. Put in Earth's mass and radius, and the acceleration of a small object comes out close to the familiar 9.8 m/s², the surface gravity.

Worked example

How hard do Earth and the Moon pull on each other at their average separation of 384,400 km?

  1. Convert the distance to meters, the unit the formula expects: 384,400 km × 1,000 = 3.844 × 10⁸ m.
  2. Put the numbers into Newton's law: F = 6.6743 × 10⁻¹¹ × 5.9722 × 10²⁴ × 7.346 × 10²² ÷ (3.844 × 10⁸)².
  3. Work it out: F ≈ 1.9816 × 10²⁰ N.
  4. Divide the force by the Moon's mass to see how fast it is pulled toward Earth. That acceleration is G × m_Earth ÷ r² ≈ 0.0027 m/s².

Result: Earth and the Moon attract each other with a force of about 1.98 × 10²⁰ N.

Understanding the result

Gravity is extremely weak between everyday objects. Two 1 kg masses 1 m apart pull on each other with only 6.67 × 10⁻¹¹ N. That is why you only notice the force when at least one mass is the size of a planet.

The accelerations differ even though the forces are equal. A small object falling toward Earth speeds up quickly. Earth moves toward it too, but far too little to measure.

Assumptions and limitations

  • Both bodies are point masses or spheres with symmetric mass distribution, so the distance between centers is all that matters.
  • Other bodies nearby are ignored.
  • Newtonian gravity applies, meaning gravity is weak and speeds are well below the speed of light.
  • Irregular or very close bodies need an integration over their shapes rather than a single center-to-center distance.
  • Near very dense objects such as neutron stars or black holes, general relativity is needed.
  • The preset masses are rounded reference values, not precise ephemeris data.

Frequently asked questions

Why does a 1 kg mass weigh about 9.82 N with this formula but 9.80665 N in other calculators?

9.80665 m/s² is standard gravity, a fixed conventional value. Newton's law with Earth's mean radius gives about 9.82 m/s², and real surface gravity also varies with latitude, altitude and Earth's rotation, from about 9.78 to 9.83 m/s².

Does the heavier object pull harder?

No. The force on each object is exactly the same size. The heavier object just speeds up less, because the same force is spread over more mass.

What happens to the force if I double the distance?

It drops to one quarter, because the distance is squared in the denominator. Triple the distance and the force falls to one ninth.

Can I use this for a person standing on Earth?

Yes. Choose Earth for object 1, enter the person's mass for object 2, and use Earth's radius (about 6,371 km) as the distance. The force is the person's weight.

How this calculator is checked

Method: Newton's law of universal gravitation; G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2018). Body masses are rounded published reference values. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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