2.1 Newton's Law of Gravitation

Newton's law: $F = Gm_1 m_2 / r^2$. $G = 6.674 \times 10^{-11}$ N $m^{2}$ $kg^{-2}$, dimensional formula $[M^{-1} L^3 T^{-2}]$. Surface gravity: $g = GM/R^2 \approx 9.8$ m/$s^{2}$, dimensional formula $[M^0 L^1 T^{-2}]$. G is measured by Cavendish's torsion balance experiment; it is a universal constant, independent of the medium and the nature of masses.

2.2 Variation of g

Four effects modify the apparent or actual value of g:

• With altitude (exact): $g' = gR^2/(R+h)^2$. At $h = R$: $g' = g/4$.
• With altitude (approximate, $h \ll R$): $g' \approx g(1 - 2h/R)$.
• With depth: $g' = g(1 - d/R)$. Linear; zero at centre.
• With latitude: $g_\text{eff} = g - R\omega^2\cos^2\lambda$. Poles have maximum g; equator has minimum.

Key comparison: for the same fractional penetration, depth gives a higher g than altitude.

2.3 Kepler's Laws

• First: Elliptical orbits, Sun at one focus.
• Second: Equal areas in equal times; areal velocity $dA/dt = L/(2m) =$ const; fastest at perihelion.
• Third: $T^2 = (4\pi^2/GM)\, r^3$; used to compare orbital periods.

2.4 Gravitational PE and Potential

• Gravitational PE: $U = -GMm/r$ (J); $[M^1 L^2 T^{-2}]$; negative sign = bound state; $U \to 0$ as $r \to \infty$.
• Gravitational potential: $V = -GM/r$ (J/kg); $[M^0 L^2 T^{-2}]$; PE per unit mass.

2.5 Escape and Orbital Velocities

• Escape velocity: $v_e = \sqrt{2GM/R} \approx 11.2$ km/s. Independent of body mass and projection angle.
• Orbital velocity: $v_0 = \sqrt{GM/r} \approx 7.9$ km/s near surface.
• Key relation: $v_e = \sqrt{2}\, v_0$.

2.6 Satellite Energy

Ratio $KE : PE : E_{total} = 1 : {-2} : {-1}$. Total energy negative = bound orbit. If energy ≥ 0, escape occurs.

2.7 Geostationary Satellites

$T = 24$ h; $r \approx 42{,}164$ km; equatorial, west-to-east orbit. Used for communication and weather.