How strong is the gravitational force?
Type two masses and the distance between their centers. The gravitational force calculator uses Newton’s law of universal gravitation to give the force in newtons and pounds-force, and how fast it pulls each mass.
- Force (N)
- 1.98157 × 10²⁰
The gravitational force between the two masses is 1.98157 × 10²⁰ N (4.45475 × 10¹⁹ lbf).
- Force (lbf)
- 4.45475 × 10¹⁹
- Acceleration of the first mass (m/s²)
- 0.000033181
- Acceleration of the second mass (m/s²)
- 0.00269748
Force (N): 1.98157 × 10²⁰. The gravitational force between the two masses is 1.98157 × 10²⁰ N (4.45475 × 10¹⁹ lbf).
How to calculate
Computes the gravitational force between two masses a distance apart with Newton’s law F = G m₁ m₂ ÷ r², in newtons and pounds-force, with the acceleration each mass feels.
Example with the default inputs (First mass 5,972,000,000,000,000,000,000,000 kg, Second mass 73,460,000,000,000,000,000,000 kg, Distance between centers 384,400 km): The gravitational force between the two masses is 1.98157 × 10²⁰ N (4.45475 × 10¹⁹ lbf).
Method: F = G × m₁ × m₂ ÷ r², G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻².
- The masses are points or spheres (whose mass is spread evenly in shells), and r is the distance between their centers.
- G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2022). Exact unit sizes: 1 lb = 0.45359237 kg; 1 short ton = 907.18474 kg; 1 mi = 1,609.344 m; 1 ft = 0.3048 m.
Worked examples
Each example is checked against the calculator on every build.
- First mass 5,972,000,000,000,000,000,000,000 kg, Second mass 73,460,000,000,000,000,000,000 kg, Distance between centers 384,400 km gives Force (N) 198,157,123,241,918,700,000.Source: OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²), https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation; NIST, CODATA 2022 value: Newtonian constant of gravitation G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², https://physics.nist.gov/cgi-bin/cuu/Value?bg
- First mass 1 kg, Second mass 1 kg, Distance between centers 0.001 km gives Force (N) 0, Acceleration of the first mass (m/s²) 0.Source: OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²), https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation; NIST, CODATA 2022 value: Newtonian constant of gravitation G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², https://physics.nist.gov/cgi-bin/cuu/Value?bg
- First mass 5,972,000,000,000,000,000,000,000 kg, Second mass 70 kg, Distance between centers 6,371 km gives Force (N) 687.39814, Acceleration of the second mass (m/s²) 9.819973.Source: OpenStax, University Physics Volume 1, §13.1 Newton’s Law of Universal Gravitation (F₁₂ = G m₁ m₂ ÷ r²), https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation; NIST, CODATA 2022 value: Newtonian constant of gravitation G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², https://physics.nist.gov/cgi-bin/cuu/Value?bg
How it works
F = G × m₁ × m₂ ÷ r², with G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2022), the masses in kilograms and r in meters. The force shows in newtons and in pounds-force (N ÷ 4.4482216152605). The page also shows the acceleration of each mass: a₁ = F ÷ m₁ and a₂ = F ÷ m₂, in m/s².
Unit sizes: 1 g = 0.001 kg; 1 t = 1,000 kg; 1 lb = 0.45359237 kg; 1 short ton = 907.18474 kg; 1 km = 1,000 m; 1 cm = 0.01 m; 1 mi = 1,609.344 m; 1 ft = 0.3048 m.
Exact arithmetic. Each value is read as the exact decimal you typed, in its unit, and G is taken as exactly 6.6743 × 10⁻¹¹, so the force is an exact fraction until it is rounded for display.
Output format. Every value shows 6 significant figures; values of 10¹⁵ or more, or below 10⁻⁶, show in scientific form (1.98157 × 10²⁰).
When there is no answer. A force beyond the range a computer can hold (the field limits keep it between about 10⁻¹³¹ N and 10¹⁰⁰ N, so this does not happen in practice).
Assumptions
- The masses are points or spheres, and r is measured between their centers.
- Masses 10⁻³⁰ to 10⁴⁰ kg; distance 10⁻¹⁵ to 10³⁰ m.
Worked examples by hand
Earth and the Moon. 6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ × 7.346 × 10²² ÷ (3.844 × 10⁸)² = 2.92807 × 10³⁷ ÷ 1.47763 × 10¹⁷ = 1.98157 × 10²⁰ N.
Two 1 kg masses 1 m apart. F = 6.6743 × 10⁻¹¹ × 1 × 1 ÷ 1² = 6.6743 × 10⁻¹¹ N, and each mass accelerates at 6.6743 × 10⁻¹¹ m/s².
A 70 kg person on Earth’s surface. 6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ × 70 ÷ (6.371 × 10⁶)² = 687.398 N; a = 687.398 ÷ 70 = 9.81997 m/s².
Other questions people ask
What is the formula for gravitational force?
Newton’s law of universal gravitation: F = G × m₁ × m₂ ÷ r², where m₁ and m₂ are the masses in kilograms, r is the distance between their centers in meters, and G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² is the gravitational constant.
What is the gravitational force between the Earth and the Moon?
With m₁ = 5.972 × 10²⁴ kg, m₂ = 7.346 × 10²² kg and r = 384,400 km, F = 6.6743 × 10⁻¹¹ × 5.972 × 10²⁴ × 7.346 × 10²² ÷ (3.844 × 10⁸)² ≈ 1.98 × 10²⁰ N.
Why is gravity between everyday objects so weak?
G is tiny. Two 1 kg masses 1 m apart pull on each other with only 6.67 × 10⁻¹¹ N. Gravity only becomes large when one mass is as big as a planet.
What happens to the force if the distance doubles?
It drops to a quarter, because the distance is squared. Three times the distance gives one ninth of the force.
Does the heavier object pull harder?
No. Both masses feel the same size of force, in opposite directions (Newton’s third law). The lighter one speeds up more: its acceleration is F ÷ m, which the page shows for each mass.
How does this give g on Earth?
Put Earth’s mass and radius in: a 70 kg person at 6,371 km from Earth’s center feels 687 N, and 687 ÷ 70 = 9.82 m/s². The standard value 9.80665 m/s² is a little lower, partly because Earth spins.