Universal gravitation
The idea
The same pull that drops an apple holds the Moon in orbit — Newton's law of universal gravitation makes that one statement quantitative: every pair of masses attracts with force F = Gm₁m₂/r², where G = 6.67 × 10⁻¹¹ N·m²/kg² and r is the center-to-center distance. You already know weight as the local pull of gravity; this law explains where weight comes from and how it weakens with distance.
The inverse-square structure is the heart of the model: double the distance and the force drops to one quarter; triple it and the force falls to one ninth. The familiar g = 9.8 m/s² is just GM/r² evaluated at Earth's surface, so g itself shrinks with altitude. Because G is so tiny, gravity between everyday objects is negligible — it takes a planet-sized mass for the force to matter.
The misconception worth attacking head-on: astronauts do not float because they have escaped Earth's gravity. At the International Space Station's altitude gravity is still nearly nine tenths of its surface strength. Astronauts float because station and occupants are all in continuous free fall around the Earth, falling toward it while moving sideways fast enough to keep missing it.
Worked example
The International Space Station orbits about 400 km above Earth's surface. Using Earth's mass M = 5.97 × 10²⁴ kg, Earth's radius 6.38 × 10⁶ m, and G = 6.67 × 10⁻¹¹ N·m²/kg², find the gravitational field strength g at the station's altitude and compare it with the surface value.
- Build the correct distance first: r is measured from Earth's center, so r = 6.38 × 10⁶ + 0.40 × 10⁶ = 6.78 × 10⁶ m. Using altitude alone is the classic setup error.
- Write the field strength as g = GM/r², which is the gravitational force per kilogram of orbiting mass.
- Compute the numerator: GM = 6.67 × 10⁻¹¹ × 5.97 × 10²⁴ = 3.98 × 10¹⁴ m³/s².
- Compute the denominator: r² = (6.78 × 10⁶)² = 4.60 × 10¹³ m².
- Divide: g = 3.98 × 10¹⁴ / 4.60 × 10¹³ ≈ 8.7 m/s² — about 88% of the surface value of 9.8 m/s².
- Interpret the result: gravity at the station is barely reduced, so the crew's weightlessness must come from free fall, not from any absence of gravitational pull.
Answer. Gravitational field strength at the station's altitude is about 8.7 m/s², still roughly 88% of its surface value.
Check your understanding
- Why does doubling the distance between two masses cut the gravitational force to a quarter rather than a half?
- How would you explain to a friend why astronauts float even though gravity at orbital altitude is nearly full strength?
- What two quantities about a planet would you need to predict the surface gravity of a world you have never visited?
- Why is gravitational attraction between two people in a room utterly negligible while Earth's pull on each is obvious?
Build the foundations first
Universal gravitation builds on these concepts. If any feel shaky, start there.