Essential 15-Point Revision List

1. Newton's law: F = $Gm_{1}$m_{2}/$r^{2}$; G = $6.674 \times 10^{-11}$N$m^{2}$$kg^{-2}$; [G] =$M^{-1}$L^{3}$T^{-2}$

2. Surface gravity: g = GM/$R^{2}$ ≈ 9.8 m/$s^{2}$; substitute GM = g$R^{2}$ in all problems

3. g at altitude (exact): g' = g$R^{2}$/(R+h)^{2} — inverse-square decrease; at h = R: g' = g/4

4. g at altitude (approx): g' ≈ g(1 − 2h/R) — valid only for h ≪ R

5. g at depth: g' = g(1 − d/R) — linear decrease; g' = 0 at Earth's centre (d = R)

6. g at latitude: g_eff = g − Rω^{2}co$s^{2}$λ; maximum at poles; minimum at equator

7. Kepler's laws: Ellipse + Sun at focus; Equal areas = angular momentum conservation; $T^{2}$ ∝ $r^{3}$

8. Gravitational PE: U = −GMm/r (always negative); U = 0 at ∞; $\Delta U$ ≈ mgh near surface

9. Escape velocity: v_e = $\sqrt{2gR}$ = $\sqrt{2GM/R}$ ≈ 11.2 km/s; independent of body mass and direction

10. Orbital velocity: v_{0} = $\sqrt{GM/r}$ = √(g$R^{2}$/r) ≈ 7.9 km/s near surface; v_e = √2 × v_{0}

11. Satellite KE: KE = GMm/(2r) — always positive

12. Satellite PE: PE = −GMm/r — always negative; |PE| = 2KE

13. Satellite total E: E = −GMm/(2r) — always negative (bound orbit); E = −KE

14. Energy ratio: KE : PE : E = 1 : (−2) : (−1) — the Virial theorem for circular orbits

15. Geostationary orbit: T = 24 h; r ≈ 42,164 km from centre; equatorial plane; west to east

Critical Values to Memorize