You're Calculating Orbital Mechanics and Need to Know the Gravitational Pull Between Earth and a Satellite
Every mass attracts every other mass through gravity. Planets orbit stars, moons orbit planets, and satellites orbit Earth. The force depends on both masses and the distance between them. Newton's law of universal gravitation describes this with elegant simplicity: F = G × m₁ × m₂ / r². This calculator computes that force instantly, whether you're studying astrophysics, designing satellite orbits, or understanding why the Moon pulls on Earth's oceans.
What This Calculator Does
This calculator applies Newton's law of universal gravitation to find the force between any two masses. You enter the two masses (in kilograms) and the distance between them (in meters), and it instantly computes the gravitational force in newtons. It also shows you how this force compares to familiar weights and accelerations, so you understand the magnitude.
How to Use This Calculator
Mass 1 (m₁): Enter the first mass in kilograms (kg). This could be Earth (5.972 × 10²⁴ kg), the Sun (1.989 × 10³⁰ kg), or any object.
Mass 2 (m₂): Enter the second mass in kilograms. For a satellite orbiting Earth, this might be 1000 kg.
Distance (r): Enter the distance between the centers of mass in meters (m). For Earth and satellite, this is Earth's radius (6,371 km) plus orbital altitude. For the Earth-Sun system, it's about 149.6 million km (1 AU).
The calculator shows the gravitational force in newtons and compares it to everyday forces (like the weight of an object on Earth).
The Formula Behind the Math
Newton's law of universal gravitation states:
F = G × m₁ × m₂ / r²
Where:
The force is always attractive (pulls the masses toward each other) and is equal in magnitude but opposite in direction on each mass (Newton's third law).
The gravitational acceleration (acceleration due to gravity) caused by mass m₁ at distance r is:
g = G × m₁ / r²
This is why surface gravity on Earth (g ≈ 9.81 m/s²) is different from surface gravity on the Moon (g ≈ 1.62 m/s²)-the Moon is less massive and smaller.
Worked Example:
Find the gravitational force between the Earth and a 1000 kg satellite in low Earth orbit (altitude 400 km above the surface).
This is equivalent to the weight of about 886 kg on Earth's surface. The satellite is in "free fall" around Earth-this 8,690 N force continuously pulls it toward Earth, but because the satellite is also moving forward, it orbits rather than falls straight down.
Our calculator does all of this instantly, but now you understand exactly what it's computing.
Orbital Mechanics and Satellite Design
Satellites must travel at a specific velocity (orbital velocity) to stay in orbit at a given altitude. At low Earth orbit (400 km altitude), orbital velocity is about 7,660 m/s (27,600 km/h). The gravitational force provides the centripetal acceleration needed to keep the satellite moving in a circle. A too-slow satellite falls to Earth; a too-fast one escapes. NASA engineers use gravitational calculations constantly.
Planetary and Lunar Orbits
The Moon orbits Earth at about 384,400 km away. The gravitational force between Earth and Moon is about 2.0 × 10²⁰ newtons. This same force, by Newton's third law, pulls Earth toward the Moon-creating tides in Earth's oceans. The Moon's gravity is about 1/6 of Earth's, so an object that weighs 100 pounds on Earth weighs about 17 pounds on the Moon.
Binary Stars and Exoplanet Detection
Two stars orbiting each other in a binary system experience mutual gravitational attraction. Their orbital periods and separation depend on their masses and the gravitational constant. Astronomers detect exoplanets by watching the "wobble" a planet causes in its star's motion-the star and planet orbit their common center of mass, and gravity governs the motion.
Tips and Things to Watch Out For
Distance is measured between centers of mass, not surfaces. For two spheres, measure from the center of one to the center of the other. If a satellite is 400 km above Earth's surface, the distance is Earth's radius (6,371 km) plus 400 km. This is critical for accuracy.
The force decreases with the square of distance. Double the distance, and the force becomes 1/4 as strong. Triple it, and the force becomes 1/9 as strong. Small changes in distance (like satellite altitude) cause significant force changes.
G is very small. The gravitational constant (6.674 × 10⁻¹¹) is tiny. This is why gravity is weak compared to electromagnetic forces. Two electrons repel each other with 10⁴² times more force than they attract gravitationally. Gravity only dominates at astronomical scales where masses are enormous.
Gravitational force is always attractive. Unlike electric forces (which can repel), gravity only pulls. Two masses always attract each other, never repel.
Don't confuse gravitational force with weight. Weight is the force of gravity on an object: W = m × g. Weight is measured in newtons (or pounds-force). Gravitational force (from this calculator) is the force between two objects. On Earth's surface, the weight of a 1 kg object is about 9.81 N (the gravitational force between Earth and that object).
Frequently Asked Questions
Why is the gravitational constant G so small?
By definition, G is what it is (6.674 × 10⁻¹¹ N⋅m²/kg²). The small value reflects that gravity is incredibly weak at everyday scales. Two 1 kg objects separated by 1 meter experience a gravitational force of only 6.674 × 10⁻¹¹ newtons-utterly undetectable without incredibly sensitive equipment. Gravity only becomes significant when masses are large (like planets) or when small objects are very close (like two dust grains in space).
How strong is the Moon's gravity?
Surface gravity on the Moon is about 1.62 m/s², which is 1/6 of Earth's (9.81 m/s²). This is because the Moon is less massive and has a smaller radius. An object that weighs 100 pounds on Earth weighs about 17 pounds on the Moon.
Do heavier objects fall faster due to stronger gravity?
No. The gravitational force on a heavier object is larger (F = m × g), but the object's inertia is also larger. The acceleration (a = F / m = m × g / m = g) is the same for all objects. In a vacuum, all objects fall at the same rate, regardless of mass. This is why Galileo's famous experiment (dropping balls from the Tower of Pisa) was important.
What's the difference between mass and weight?
Mass (measured in kg) is the amount of matter in an object-it's the same everywhere. Weight (measured in newtons or pounds-force) is the gravitational force on that mass: W = m × g. On Earth, g ≈ 9.81 m/s², so a 1 kg object weighs 9.81 N. On the Moon, g ≈ 1.62 m/s², so the same 1 kg object weighs 1.62 N. The mass didn't change; the weight did.
Can I calculate gravity inside the Earth?
The formula F = G × m₁ × m₂ / r² applies outside a sphere. Inside, gravity decreases linearly with distance from the center (assuming uniform density). At Earth's center, gravity is zero (the entire mass surrounds you equally). For this, use more advanced formulas that account for Earth's layered structure.
What's escape velocity?
Escape velocity is the speed needed to leave a planet's gravity well entirely, never returning. It depends on the planet's mass and radius: v = √(2 × G × M / R). For Earth, it's about 11.2 km/s. For the Moon, it's about 2.4 km/s. Rockets don't need to reach escape velocity to go to orbit; they need only reach orbital velocity.
Related Calculators
Use our Acceleration Calculator to find how fast an object accelerates under gravity. The Kinetic Energy Calculator computes the kinetic energy of orbiting objects. The Potential Energy Calculator finds gravitational potential energy. For more physics concepts, explore our Pressure and Density Calculators.