How It Works
Newton's Law of Universal Gravitation says every pair of objects pulls on each other with a force F = G × m₁ × m₂ / r², where m₁ and m₂ are their masses, r is the center-to-center distance, and G = 6.6726 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant. This calculator solves the equation for any one of three unknowns — force, distance, or one of the masses — given the other three quantities. Inputs accept kg/g/lb for mass, m/km/ft/mi for distance, and N/lbf/dyne for force; the calculator converts to SI internally before computing.
Example Problem
Calculate the gravitational force between Earth (m₁ = 5.972 × 10²⁴ kg) and a 70 kg person standing on its surface (r = 6.371 × 10⁶ m).
- Write Newton's gravitational force formula: F = G × m₁ × m₂ / r².
- Substitute the values: F = (6.6726 × 10⁻¹¹) × (5.972 × 10²⁴) × 70 / (6.371 × 10⁶)².
- Compute the numerator: G × m₁ × m₂ = 6.6726 × 10⁻¹¹ × 5.972 × 10²⁴ × 70 ≈ 2.789 × 10¹⁶ N·m².
- Compute the denominator: r² = (6.371 × 10⁶)² ≈ 4.058 × 10¹³ m².
- Divide: F ≈ 2.789 × 10¹⁶ / 4.058 × 10¹³ ≈ 687 N — the textbook weight of a 70 kg adult at Earth's surface.
Key Concepts
Gravitational force scales linearly with each mass and falls off as the inverse square of the distance, so doubling the distance cuts the force to one-quarter. The gravitational constant G is one of the most precisely measured universal constants and appears unchanged in every gravitational equation derived from Newton's law (Kepler's third law, surface gravity, escape velocity). The force always acts along the line connecting the two centers of mass — never along the surfaces — so r in the formula is the center-to-center distance, not surface-to-surface separation.
Applications
- Spacecraft trajectory planning: computing the gravitational pull from Earth, the Moon, and the Sun simultaneously when designing a mission.
- Astronomy: estimating planetary masses from the gravitational forces measured between celestial bodies and their moons.
- Tidal force analysis: separating the gravitational force on Earth's near and far sides to predict ocean tides.
- Geophysics: detecting subsurface density variations from tiny anomalies in measured surface gravitational force.
- Physics education: demonstrating the inverse-square law and the universality of gravitational interaction.
Common Mistakes
- Using surface-to-surface distance instead of center-to-center distance for r — the formula always needs the distance between centers of mass.
- Confusing the universal constant G (6.6726 × 10⁻¹¹ N·m²/kg², constant everywhere) with the surface gravity g (≈ 9.81 m/s², varies with location).
- Forgetting that r is squared in the denominator — doubling the separation reduces the force by a factor of four, not two.
- Mixing units of mass and distance (e.g., grams with kilometers) without converting — the SI form of the equation requires kg and m for the constant G to apply directly.
- Treating gravitational force as if only the larger mass matters — both masses appear in the formula and contribute equally to the attraction.
Frequently Asked Questions
How do you calculate gravitational force?
Multiply the two masses together and the gravitational constant G (6.6726 × 10⁻¹¹ N·m²/kg²), then divide by the square of the distance between their centers: F = G × m₁ × m₂ / r². Make sure the masses are in kilograms and the distance is in meters before applying the formula.
What is the formula for gravitational force?
F = G × m₁ × m₂ / r², where G is the gravitational constant (6.6726 × 10⁻¹¹ N·m²/kg²), m₁ and m₂ are the two masses in kilograms, and r is the distance between their centers in meters. The result F is in newtons.
What is the gravitational constant G?
G = 6.6726 × 10⁻¹¹ N·m²/kg² is the universal gravitational constant — a fixed value that quantifies how strongly any two masses attract each other. Unlike Earth's surface gravity g (≈ 9.81 m/s²), G is identical anywhere in the universe and appears in every gravitational equation.
Why is gravitational force so weak between everyday objects?
Because G is extremely small (≈ 6.7 × 10⁻¹¹). Two 1 kg masses one meter apart pull on each other with only ~6.7 × 10⁻¹¹ N — about a hundred-billionth of a newton. Gravitational force only becomes appreciable when at least one of the masses is planetary in scale.
How does gravitational force change with distance?
It falls off as the inverse square of the distance. Doubling the separation cuts the force to one-quarter; tripling it cuts the force to one-ninth. This is why orbits weaken rapidly with altitude and why the gravitational influence of distant stars on Earth is negligible.
Does gravitational force depend on the masses of both objects?
Yes — both masses appear in the formula and contribute equally. Earth pulls on you with the same force you pull on Earth (Newton's third law). The reason you accelerate noticeably and Earth doesn't is the enormous difference in mass, not in force.
Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.
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