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Physics · Ch 6 — Gravitation

SUMMARY

SUMMARY

  • The motion of planets is fully described by Kepler's three laws: (1) every planet orbits the Sun in an ellipse with the Sun at one focus; (2) the radial vector from the Sun to a planet sweeps out equal areas in equal times; (3) T2/a3T^2/a^3 is the same constant for every planet in the solar system.
  • Newton's law of gravitation: F⃗=−Gm1m2r2r^\vec{F}=-\dfrac{Gm_1m_2}{r^2}\hat{r} -- an attractive, inverse-square, action-reaction central force between any two masses; Kepler's laws can be derived starting from this single law.
  • The gravitational field due to a mass mm at distance rr is E⃗=−Gmr2r^\vec{E}=-\dfrac{Gm}{r^2}\hat{r} (a vector, in N/kg); the gravitational potential energy of two masses is U=−Gm1m2rU=-\dfrac{Gm_1m_2}{r} (a scalar, in J); the gravitational potential at distance rr from a mass mm is V=−GmrV=-\dfrac{Gm}{r} (a scalar, in J/kg).
  • g=GMe/Re2g=GM_e/R_e^2 at the surface; gg decreases with increasing altitude and with increasing depth below the surface, and (due to Earth's rotation) is maximum at the poles and minimum at the equator.
  • Escape speed is ve=2gRe≈11.2 km s−1v_e=\sqrt{2gR_e}\approx11.2\ \text{km s}^{-1}, independent of the object's mass or launch direction.
  • A satellite's total mechanical energy, E=−12GMsMeRe+hE=-\dfrac{1}{2}\dfrac{GM_sM_e}{R_e+h}, is always negative, meaning the satellite is permanently bound to the Earth.
  • The heliocentric (Sun-centred) model explains the retrograde motion of planets as a simple consequence of relative motion between Earth and the outer planet, far more simply than Ptolemy's artificial epicycles. …
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