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Part of ME-06 — Gravitation

Gravitation — Quick Review (10 Sentences)

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  1. Newton's law of gravitation states F=Gm1m2/r2F = Gm_1 m_2/r^2, where G=6.674×10−11G = 6.674 \times 10^{-11} N m2m^{2} kg−2kg^{-2} has the dimensional formula [M−1L3T−2][M^{-1} L^3 T^{-2}].
  2. Surface gravity g=GM/R2≈9.8g = GM/R^2 \approx 9.8 m/s2s^{2}; with altitude it decreases by the inverse-square law g′=gR2/(R+h)2g' = gR^2/(R+h)^2, and with depth it decreases linearly as g′=g(1−d/R)g' = g(1 - d/R).
  3. At Earth's centre the depth formula gives g′=0g' = 0; at height h=Rh = R above the surface, g′=g/4g' = g/4.
  4. Kepler's Second Law (equal areas in equal times) means a planet moves fastest at perihelion and slowest at aphelion.
  5. Kepler's Third Law states T2∝r3T^2 \propto r^3, expressed as T2=(4π2/GM) r3T^2 = (4\pi^2/GM)\,r^3.
  6. Gravitational potential energy is U=−GMm/rU = -GMm/r (negative, bound system; zero at infinity), and gravitational potential is V=−GM/rV = -GM/r (J/kg).
  7. Escape velocity ve=2GM/R≈11.2v_e = \sqrt{2GM/R} \approx 11.2 km/s for Earth; it is independent of the mass and direction of the projected body.
  8. Orbital velocity v0=GM/r≈7.9v_0 = \sqrt{GM/r} \approx 7.9 km/s near Earth's surface, and the fundamental relation is ve=2 v0v_e = \sqrt{2}\,v_0.
  9. For a satellite at radius rr: KE=GMm/2rKE = GMm/2r, PE=−GMm/rPE = -GMm/r, E=−GMm/2rE = -GMm/2r, with the ratio KE:PE:E=1:−2:−1KE:PE:E = 1:{-2}:{-1} and ∣PE∣=2KE|PE| = 2KE.
  10. A geostationary satellite has T=24T = 24 h, orbits at r≈42,164r \approx 42{,}164 km from Earth's centre in the equatorial plane from west to east, and appears stationary to any ground observer.

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