04 // Interactive Formula Compendium // PHYSICSF = G · (m₁ · m₂) / r²
Newton's Law of Universal Gravitation, Orbital & Escape Velocity Sheet
Newton's Law of Universal Gravitation states that any two point masses m₁ and m₂ separated by center-to-center distance r attract each other with force F = G m₁ m₂ / r², where G = 6.67430 × 10⁻¹¹ N·m²/kg².
Formula Reference
Main Formula
F = G · (m₁ · m₂) / r²
Rearranged Forms
g(r) = G · M / r² (Gravitational Field Strength)
v_orbit = √(G · M / r), v_escape = √(2 G · M / r) = √2 · v_orbit
T² = (4π² / GM) · r³ (Kepler's Third Law of Orbital Periods)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| F | Gravitational Attraction Force (F) | N |
| m1 | Primary Mass m₁ (kg) | kg |
| m2 | Secondary Mass m₂ (kg) | kg |
| r | Center-to-Center Distance r (m) | m |
Inputs & Parameters
[kg]
[kg]
[m]
Calculated Result (Gravitational Attraction Force (F))
F =8,694.005N
Formula Notes & Background
- By spherical symmetry (Newton's Shell Theorem), a spherically symmetric body of mass M attracts external particles as if all its mass were concentrated at its center.
- Setting gravitational force G M m / r² equal to centripetal force m v² / r cancels m and yields circular orbital speed v_orbit = √(GM/r).
- Setting total mechanical energy ½mv² − GMm/r = 0 yields escape velocity v_escape = √(2GM/r).
Step-by-Step Examples
#1: Calculate the gravitational pull on a 1,000 kg satellite in Low Earth Orbit at altitude 400 km (r = 6.771 × 10⁶ m).
M_Earth = 5.972 × 10²⁴ kgm₂ = 1,000 kgr = 6.771 × 10⁶ m
F = (6.6743×10⁻¹¹ × 5.972×10²⁴ × 1000) / (6.771×10⁶)²
→ F = 8,693 N (local gravity g = 8.69 m/s², 88.6% of surface gravity!)
#2: Find the circular orbital velocity and escape velocity at that same 400 km orbit.
GM_Earth = 3.986 × 10¹⁴ m³/s²r = 6.771 × 10⁶ m
v_orbit = √(3.986×10¹⁴ / 6.771×10⁶) = 7,672 m/s; v_escape = √2 × 7,672
→ v_orbit = 7.67 km/s, v_escape = 10.85 km/s
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Frequently Asked Questions
Why do astronauts on the ISS feel weightless if Earth's gravity at 400 km altitude is still 88.6% of surface gravity?+
Because both the station and the astronauts are in continuous free-fall around the Earth with identical centripetal acceleration g = v²/r.
What happens to gravitational force if the distance r between two masses is tripled?+
By the inverse-square law (1/r²), tripling the distance reduces the gravitational force to 1/3² = 1/9 of its original value.
How was the universal gravitational constant G first measured in a lab?+
Henry Cavendish measured G in 1798 using a delicate torsion-balance fiber suspending lead spheres to weigh the Earth.