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Newton’s Law of Universal Gravitation
Every two objects attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. G = 6.6726 × 10⁻¹¹ N·m²/kg².
F = G × m₁ × m₂ / r²
Kepler’s Third Law
Relates the orbital period of a satellite to its orbital radius and the mass of the central body. Larger orbits have longer periods.
T = √(4π²r³ / GM)
Gravitational Acceleration
The acceleration due to gravity at a distance r from a planet’s center. On Earth’s surface, this gives the familiar 9.81 m/s².
a = GM / r²
Escape Velocity
The minimum speed needed to escape a planet’s gravitational field without further propulsion. Earth’s escape velocity is about 11.2 km/s.
vₑ = √(2GM / R)
How It Works
Newton’s Law of Universal Gravitation states that every two objects attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them (F = Gm₁m₂/r²). The gravitational constant G = 6.6726 × 10⁻¹¹ N·m²/kg². This calculator also covers Kepler’s Third Law (orbital period), gravitational acceleration, and escape velocity — all derived from the same underlying gravitational law.
Example Problem
What is the gravitational force between Earth (5.97 × 10²⁴ kg) and a 70 kg person standing on the surface (r = 6.371 × 10⁶ m)?
- F = G × m₁ × m₂ / r²
- F = 6.67×10⁻¹¹ × 5.97×10²⁴ × 70 / (6.371×10⁶)²
- F ≈ 686 N (about 154 lbf)
When to Use Each Variable
- Solve for Gravitational Force — when you know both masses and the distance between them, e.g., calculating the gravitational pull between Earth and a satellite.
- Solve for Distance — when you know the force and both masses, e.g., finding the orbital altitude where a specific gravitational force acts.
- Solve for Mass — when you know the force, the other mass, and the distance, e.g., estimating a planet's mass from satellite orbital data.
- Solve for Orbital Period — when you know the orbital radius and central body mass, e.g., calculating how long a satellite takes to complete one orbit.
- Solve for Orbital Radius — when you know the desired period and central body mass, e.g., finding the altitude for a geostationary orbit.
- Solve for Gravitational Acceleration — when you know a planet's mass and radius, e.g., finding the surface gravity on Mars or the Moon.
- Solve for Escape Velocity — when you know a planet's mass and radius, e.g., determining the minimum launch speed to leave Earth's gravitational field.
Key Concepts
Newton's Law of Universal Gravitation states that every pair of objects attracts each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The gravitational constant G = 6.6726 × 10⁻¹¹ N·m²/kg² is the same everywhere in the universe. Kepler's Third Law, gravitational acceleration, and escape velocity are all derived from this fundamental law.
Applications
- Aerospace engineering: calculating satellite orbits, launch trajectories, and escape velocities
- Astronomy: estimating planetary masses from the orbits of their moons or satellites
- Geophysics: measuring local variations in gravitational acceleration for mineral and oil exploration
- Space mission planning: determining transfer orbit parameters and gravitational slingshot maneuvers
Common Mistakes
- Confusing G (universal gravitational constant) with g (local gravitational acceleration) — G is constant everywhere, g varies with location
- Using the distance between surfaces instead of centers of mass — Newton's law requires the center-to-center distance
- Forgetting to convert radius to meters — mixing kilometers and meters gives results off by factors of 10³ or 10⁶
- Assuming g = 9.81 m/s² at all altitudes — gravitational acceleration decreases with the square of the distance from Earth's center
Frequently Asked Questions
What is the gravitational constant G?
G is a fundamental constant (6.6726 × 10⁻¹¹ N·m²/kg²) that sets the strength of gravity. Unlike g (9.81 m/s² on Earth), G is the same everywhere in the universe.
What is Earth’s escape velocity?
Earth’s escape velocity is about 11.2 km/s (roughly 25,000 mph). Any object launched at this speed or faster will not fall back to Earth, ignoring air resistance.
How does gravity change with altitude?
Gravitational acceleration decreases with the square of the distance from a planet’s center. At the altitude of the ISS (~400 km), g is about 8.7 m/s² — roughly 89% of the surface value.
Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.
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Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.