Gravitational force calculator

A gravitational force calculator works out the pull between two masses with Newton's law of universal gravitation: F = Gm₁m₂/r², where r is the distance between their centres and G = 6.674 × 10⁻¹¹ N m²/kg². Our Gravitational Force Calculator takes the two masses and the distance, gives the force in newtons, and shows every step of the working.

It opens on the Earth and the Moon, whose pull on each other is about 1.98 × 10²⁰ N. Every value goes in as a number times a power of ten, so huge planets and tiny forces are easy to type.

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Gravitational Force Calculator

Calculate Newton's gravitational force

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How to use the gravitational force calculator

1. Type Mass 1 and Mass 2. Each box has two parts: the number, then the power of ten after ×10. For 70 kg, type 70 and 0, or 7 and 1.
2. Type the Radius: the distance between the two centres, not the gap between the surfaces.
3. Choose units under the inputs if you need them. The mass unit can be grams, kilograms, tonnes, pounds or solar masses, and it applies to both masses. Distance can be anything from nanometres to astronomical units.
4. Read the force, and the same force in kN, MN, dynes and pounds-force.
5. Follow the working: the formula, the numbers put in, the top and bottom of the fraction, and the answer.

A graph shows how the force falls as the distance grows, with your distance marked. For practice, Practice and help has a Random problem button that sets gravitation questions with worked solutions, and the formula sheet lists the law, the value of G and the law rearranged for r.

The gravitational force formula

Both masses feel the same size of force, pulling them towards each other: the Earth pulls the Moon exactly as hard as the Moon pulls the Earth. This is Newton's third law. The calculator uses G = 6.6743 × 10⁻¹¹ N m²/kg², the value NIST lists. For a round body such as a planet, measure r from its centre, so on the surface r is the planet's radius.

Worked examples you can check in the calculator

Example 1: two people

Two people of mass 70 kg and 60 kg stand 1.0 m apart. How hard do they pull on each other?

Type 7 ×10¹, 6 ×10¹ and 1 ×10⁰. The calculator gives 2.80321 × 10⁻⁷ N, far too small to notice, which is why you only feel gravity from something as massive as a planet.

Example 2: the Earth and the Moon

The Earth has mass 5.972 × 10²⁴ kg, the Moon 7.35 × 10²² kg, and their centres are about 3.844 × 10⁸ m apart:

These are the values the calculator opens on, and it gives 1.98157 × 10²⁰ N. That force keeps the Moon in its orbit.

Example 3: your weight from the law of gravitation

A 70 kg person stands on the Earth's surface. The Earth's mean radius is 6.371 × 10⁶ m, so that is r:

The calculator gives 687.3981 N. That is the person's weight: 687 ÷ 70 = 9.82 N/kg, the gravitational field strength g at the surface. Our gravitational field strength guide explains g = GM/r² in full.

How the force changes with distance: the inverse square law

The force depends on 1/r², so it falls quickly with distance. Double the distance between the centres and the force drops to a quarter; triple it and it drops to a ninth. Move the 70 kg person to twice the Earth's radius from its centre, r = 1.2742 × 10⁷ m, and the calculator gives 171.8495 N, a quarter of 687 N.

Here x is the distance from the Earth's centre in Earth radii, starting at the surface, and y is the force in newtons. The curve never reaches zero: gravity has no range limit, it just gets weaker. Check the pattern:

Finding a mass or a distance by hand

Rearrange F = Gm₁m₂/r² for whatever you need. The calculator's formula sheet shows the version for r.

Example 4: how far apart?

Two masses of 5,000 kg and 2,000 kg pull on each other with a force of 1.0 × 10⁻⁶ N. How far apart are their centres?

To check, type 5 ×10³, 2 ×10³ and 25.83 ×10⁰ into the calculator. It gives 1.00036 × 10⁻⁶ N, so the distance is right.

Example 5: the mass of a planet

On the surface of Mars, a 2.0 kg object weighs 7.42 N. Mars has a radius of 3.3895 × 10⁶ m. Find the mass of Mars:

NASA gives the mass of Mars as 6.42 × 10²³ kg, so the answer is close; the small gap comes from rounding the weight. This is how the masses of planets are found: from the pull they produce.

The force of gravity on other planets and the Sun

Keep the person's 70 kg and change the planet's mass and radius, and the calculator gives their weight there. With the Sun and the Earth, 1.989 × 10³⁰ kg and 5.972 × 10²⁴ kg at 1.496 × 10¹¹ m, it gives 3.54240 × 10²² N, the pull that holds the Earth in orbit. For the speed needed to leave a planet altogether, use the Escape Velocity Calculator. For orbits, the Kepler's Third Law Calculator links the period to the radius, and for energy in a gravitational field, use the Gravitational Potential Calculator.

Common mistakes

• Using the gap between surfaces. r runs from centre to centre. On a planet's surface, r is the planet's radius.
• Forgetting to square r. The force depends on r², so leaving the square out gives an answer that is wrong by a factor of r.
• Using kilometres. G is in N m²/kg², so masses must be in kilograms and r in metres, unless you choose the units in the calculator.
• Missing the power of ten. Each box in the calculator needs both parts. Type 0 in the ×10 box for an ordinary number.
• Mixing up G and g. G is the universal constant, 6.674 × 10⁻¹¹ N m²/kg². g is the field strength at one place, about 9.81 N/kg on the Earth's surface.
• Thinking the bigger mass pulls harder. The force on each mass is the same size. The smaller mass just accelerates more.

Practise gravitation in a short timed paper

Newton's law of gravitation comes up in College Physics (Calculus Based) practice, and in A-level Physics: Year 2 in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics.

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