Gravitation — Essential NEET Facts — Summary

Newton's law: $F = Gm_1 m_2 / r^2$ ; $G = 6.674 \times 10^{-11}$ N $m^{2}$ $kg^{-2}$ ; $[M^{-1} L^3 T^{-2}]$ .
Surface gravity: $g = GM/R^2 \approx 9.8$ m/ $s^{2}$ ; $[M^0 L^1 T^{-2}]$ .
g at altitude (exact): $g' = gR^2/(R+h)^2$ — inverse-square decrease.
g at altitude (approx, $h \ll R$ ): $g' \approx g(1 - 2h/R)$ .
g at depth: $g' = g(1 - d/R)$ — linear decrease; zero at Earth's centre.
g at latitude: $g_\text{eff} = g - R\omega^2\cos^2\lambda$ ; max at pole, min at equator.
Kepler I: Elliptical orbit, Sun at one focus.
Kepler II: Equal areas in equal times; $dA/dt = L/(2m) =$ const; fastest at perihelion.
Kepler III: $T^2 \propto r^3$ ; $T^2 = (4\pi^2/GM)\,r^3$ .
Gravitational PE: $U = -GMm/r$ ; negative = bound; zero at infinity.
Gravitational potential: $V = -GM/r$ J/kg; $[M^0 L^2 T^{-2}]$ .
Escape velocity: $v_e = \sqrt{2GM/R} = \sqrt{2gR} \approx 11.2$ km/s; independent of body mass and angle.
Orbital velocity: $v_0 = \sqrt{GM/r} \approx 7.9$ km/s (near surface); $[M^0 L^1 T^{-1}]$ .
Relation: $v_e = \sqrt{2}\, v_0$ .
Satellite energies: $KE = GMm/2r$ ; $PE = -GMm/r$ ; $E = -GMm/2r$ ; ratio $1:{-2}:{-1}$.
Rule: $|PE| = 2KE$ ; total energy negative = bound; $E \geq 0$ = escape.
Geostationary: $T = 24$ h; $r \approx 42{,}164$ km from centre; equatorial; west-to-east.
Altitude vs depth comparison: At $h = R$ : $g' = g/4$ . At $d = R/2$ : $g' = g/2$ . Depth is gentler.
Escape velocity scaling: $v_e' = v_e \sqrt{M'/M_E}\cdot\sqrt{R_E/R'}$ ; handle square root carefully.
Trap: Escape velocity is the same regardless of the mass of the escaping body. Only energy needed ( $\frac{1}{2}mv_e^2$ ) changes with mass.$