Newton’s Law of Universal Gravitation
Published in 1687 in the Principia Mathematica, Newton’s law states that every particle of matter attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
This single law explains falling apples, orbiting planets, ocean tides, and the structure of galaxies. It remained the best description of gravity for over 200 years until Einstein’s general relativity (1915) provided a more accurate framework for extreme conditions.
Newton’s Law of Universal Gravitation
The Gravitational Force Equation
Where G = 6.674 × 10⁻¹¹ N·m²/kg², m₁ and m₂ are the two masses in kg, and r is the centre-to-centre distance in metres.
Find Mass
Find Distance
Field Strength
The Inverse-Square Law
Gravitational force decreases with the square of distance. If you double the separation, the force drops to one-quarter. Triple the distance, and the force becomes one-ninth.
| Distance (r) | Force (relative) | Explanation |
|---|---|---|
| r | F | Reference |
| 2r | F/4 | 2² = 4 |
| 3r | F/9 | 3² = 9 |
| 10r | F/100 | 10² = 100 |
| ½r | 4F | (1/2)² = 1/4 → ×4 |
Gravitational Field Strength (g)
The gravitational field strength at a distance r from a mass M tells you the acceleration a test mass would experience. On Earth’s surface (M = 5.972 × 10²&sup4; kg, r = 6,371 km): g ≈ 9.81 m/s².
This is why W = mg works: weight is the gravitational force on mass m in a field of strength g.
Escape and Orbital Velocity
Escape Velocity
Orbital Velocity
Escape velocity is the minimum speed to leave a body’s gravitational influence without further thrust. Orbital velocity is the speed for a stable circular orbit at radius r.
Escape velocity is always √2 ≈ 1.414 times the orbital velocity at the same radius.
| Body | vₑₛₐ (km/s) | vₒₓₕ at surface (km/s) |
|---|---|---|
| Moon | 2.38 | 1.68 |
| Earth | 11.19 | 7.91 |
| Mars | 5.03 | 3.55 |
| Jupiter | 59.5 | 42.1 |
| Sun | 617.5 | 436.7 |
Solar System Gravity Data
| Body | Mass (kg) | Radius (km) | Surface g (m/s²) |
|---|---|---|---|
| Mercury | 3.301 × 10²³ | 2,440 | 3.70 |
| Venus | 4.867 × 10²&sup4; | 6,052 | 8.87 |
| Earth | 5.972 × 10²&sup4; | 6,371 | 9.81 |
| Moon | 7.342 × 10²² | 1,737 | 1.63 |
| Mars | 6.417 × 10²³ | 3,390 | 3.72 |
| Jupiter | 1.898 × 10²&sup7; | 69,911 | 24.79 |
| Saturn | 5.683 × 10²&sup6; | 58,232 | 10.44 |
| Sun | 1.989 × 10³&sup0; | 695,700 | 274 |
Source: NASA Planetary Fact Sheet (2024).
How to Use the Calculator
- Select a mode: find force, mass, distance, field strength, escape velocity, or orbital velocity.
- Enter masses, distance, or force with units.
- Click Calculate.
- Review the result, step-by-step solution, and extras.
Example Calculations
Earth–Moon gravitational force
F = 6.674×10⁻¹¹ × 5.972×10²⁴ × 7.342×10²² / (3.844×10⁸)² ≈ 1.98 × 10²⁰ N
Earth surface gravity
g = 6.674×10⁻¹¹ × 5.972×10²⁴ / (6.371×10⁶)² ≈ 9.82 m/s²
Earth escape velocity
v = √(2 × 6.674×10⁻¹¹ × 5.972×10²⁴ / 6.371×10⁶) ≈ 11,186 m/s ≈ 11.2 km/s
Common Mistakes
- Using surface-to-surface distance instead of centre-to-centre distance.
- Forgetting to square r in the denominator.
- Confusing G (gravitational constant) with g (gravitational acceleration).
- Using the wrong value of G (it’s 6.674 × 10⁻¹¹, a very small number).
- Not converting km to m when computing with SI units.
Accuracy and Limitations
Newton’s law of gravitation is an excellent approximation for most scenarios. It breaks down for very strong gravitational fields (near black holes), very high speeds (near the speed of light), or extreme precision (Mercury’s perihelion precession). General relativity is needed in these cases. This calculator treats all bodies as point masses.
FAQ
What is Newton's law of universal gravitation?›
Every particle of matter attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: F = Gm₁m₂/r².
What is the value of the gravitational constant G?›
G = 6.674 × 10⁻¹¹ N·m²/kg² (CODATA 2018). It was first measured by Henry Cavendish in 1798 using a torsion balance.
Why don't we feel gravity from nearby objects?›
G is incredibly small (6.674×10⁻¹¹). Two 100 kg people 1 m apart attract each other with only about 0.67 μN — billions of times weaker than their weight. Only astronomical-mass objects produce noticeable gravity.
What is the difference between g and G?›
G is the universal gravitational constant (6.674×10⁻¹¹), the same everywhere. g is gravitational field strength (acceleration due to gravity) at a specific location, e.g. g = 9.81 m/s² on Earth’s surface. g = GM/r².
What is escape velocity?›
The minimum speed to escape a body’s gravitational pull without further propulsion: v = √(2GM/r). For Earth’s surface, it’s about 11.2 km/s.
How fast does the ISS orbit?›
The ISS orbits at about 7,660 m/s (27,600 km/h) at an altitude of ~408 km (orbital radius ~6,779 km from Earth’s centre). Its orbital period is about 92 minutes.
Is gravity a force or curvature of spacetime?›
In Newtonian physics, gravity is a force (F = Gm₁m₂/r²). In general relativity, gravity is the curvature of spacetime caused by mass-energy. For everyday and engineering calculations, the Newtonian model is accurate.
Does the gravitational force depend on the medium between objects?›
No. Unlike electromagnetic forces, gravity is not affected by intervening materials. It cannot be shielded or screened.
Sources
Author & technical reviewer
Manish Kumar
PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.
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