Gravitation — Engineering & Real-world Applications — Summary

Satellite communication: Geostationary satellites at $r \approx 42{,}164$ km appear fixed in the sky, enabling continuous TV broadcast, weather monitoring, and GPS augmentation. Their 24-hour period is dictated directly by Kepler's third law.
GPS constellation: GPS satellites orbit at $\sim 20{,}200$ km altitude, not geostationary. Their precisely known orbital periods (from $T^2 \propto r^3$ ) enable nanosecond-level timing and metre-level positioning.
Space launch windows: Escape velocity of 11.2 km/s sets the minimum fuel requirement for interplanetary missions. Multi-stage rockets and gravity assists (using a planet's orbital velocity) reduce the fuel cost.
Gravitational variation for geophysics: Variation of g with depth ( $g' = g(1-d/R)$ ) is used in gravity surveys to detect subsurface density anomalies — oil and mineral deposits cause measurable local g deviations.
Weightlessness in orbit: Astronauts are not outside gravity's reach. At the ISS ( $h \approx 400$ km), $g' \approx 8.7$ m/ $s^{2}$ . Weightlessness occurs because the ISS and its occupants are in the same free fall (orbital acceleration $= g'$ ). This is an application of satellite orbital mechanics.
Latitude correction in precision instruments: Weighing systems and pendulum clocks must be corrected for latitude. A clock calibrated at the equator runs slightly slower at the poles because $g_\text{pole} > g_\text{equator}$ , and pendulum period $T \propto 1/\sqrt{g}$ .
Tidal forces: Differential gravitational pull ( $\Delta F \propto 1/r^3$ for a finite-size body) of the Moon and Sun on Earth's oceans produces tides — an extension of Newton's inverse-square law.
Black holes and escape velocity: When the escape velocity of a compact object equals the speed of light ( $c$ ), not even light can escape — this defines the Schwarzschild radius: $r_s = 2GM/c^2$ . This is a direct consequence of the escape velocity formula.
Rocket staging: The orbital velocity relation $v_0 = \sqrt{GM/r}$ shows that lower orbits (smaller $r$ ) require higher speeds. This determines the staging profile of launch vehicles.
Kepler III in exoplanet detection: By measuring an exoplanet's orbital period $T$ and semi-major axis $r$ , astronomers determine the host star's mass using $M = 4\pi^2 r^3 / GT^2$ .