Gravitation

Newton's Universal Law of Gravitation states that every mass attracts every other mass with a force F = Gm_1 $m_{2}$ / $r^{2}$, where G = 6.674 x $10^{-11}$ N $m^{2} \; kg^{-2}$ is the universal gravitational constant with dimensional formula [$M^{-1} \; L^{3} \; T^{-2}$], $m_{1}$ and $m_{2}$ are masses (kg), and r is the distance between their centres (m).
The acceleration due to gravity at Earth's surface is g = GM/$R^{2}$ [$M^{0} \; L^{1} \; T^{-2}$] (m/$s^{2}$), where M is Earth's mass and R is Earth's radius. At the surface, g is approximately 9.8 m/$s^{2}$.

The value of g varies with position.
With altitude h: g' = $gR^{2}$/(R + h)² (exact), or g' = g(1 - 2h/R) for h << R (inverse-square decrease).
With depth d: g' = g(1 - d/R), which is a linear decrease reaching zero at the centre (d = R).
With latitude $\lambda$: $g_{eff}$ = g - R $\omega$² $cos^{2}$($\lambda$), maximum at the poles ($\lambda$ = 90 degrees, cos = 0) and minimum at the equator ($\lambda$ = 0 degrees, cos = 1). The key distinction is that g decreases linearly with depth but by inverse-square with altitude — the formulas are not the same.

Kepler's three laws govern planetary motion. The First Law states planets orbit in ellipses with the Sun at one focus.
The Second Law (law of areas) states the radius vector sweeps equal areas in equal times, meaning areal velocity dA/dt = L/(2m) = constant.
The Third Law states $T^{2}$ is proportional to $r^{3}$: $T^{2}$ = ($4\pi^{2}$/GM)$r^{3}$, where T is the orbital period and r is the semi-major axis.

Gravitational potential energy U = -GMm/r [$M^{1} \; L^{2} \; T^{-2}$] (J). The negative sign indicates a bound system; U = 0 at infinity. Gravitational potential V = -GM/r [$M^{0} \; L^{2} \; T^{-2}$] (J/kg) is the PE per unit mass.

Escape velocity $v_{e}$ = $\sqrt{2gR}$ = $\sqrt{2GM/R}$ [$M^{0} \; L^{1} \; T^{-1}$] (m/s), approximately 11.2 km/s for Earth. Crucially, escape velocity is independent of the mass of the projected body and the angle of projection — it depends only on the planet's mass M and radius R.
Orbital velocity for a satellite at radius r: $v_{0}$ = $\sqrt{GM/r}$ = $\sqrt{gR^{2}/r}$ [$M^{0} \; L^{1} \; T^{-1}$] (m/s), approximately 7.9 km/s near Earth's surface.
The relation between them: $v_{e}$ = $\sqrt{2}$ x $v_{0}$.

For a satellite at orbital radius r: KE = GMm/(2r), PE = -GMm/r, Total energy E = -GMm/(2r). The relation |PE| = 2KE holds, and total energy is negative (bound orbit). If E becomes zero or positive, the satellite escapes.

A geostationary satellite has period T = 24 hours, orbital radius r = 42,164 km from Earth's centre (about 35,786 km above the surface), orbits in the equatorial plane from west to east, and appears stationary relative to the ground.

Solved Numerical 1: At height h = R: g' = $gR^{2}$/(R + R)² = $gR^{2}$/($4R^{2}$) = g/4 = $\frac{10}{4}$ = 2.5 m/$s^{2}$.
At depth d = R/2: g' = g(1 - R/(2R)) = g(1 - $\frac{1}{2}$) = g/2 = 5.0 m/$s^{2}$. So g at depth R/2 (5 m/$s^{2}$) is greater than g at height R (2.5 m/$s^{2}$). The depth formula gives a gentler decrease than the altitude formula.

Solved Numerical 2: Planet with mass M' = $4M_{E}$, radius R' = $2R_{E}$.
Escape velocity: $v_{e}$ = $\sqrt{2GM/R}$.
$v_{e}$' = $\sqrt{2G(4M_E}$/($2R_{E}$)) = $\sqrt{2 x 2GM_E/{R}_{E}}$ = $\sqrt{2}$ x $\sqrt{2GM_E/{R}_{E}}$ = $\sqrt{2}$ x $v_{e}$(Earth) = 1.414 x 11.2 = 15.84 km/s.

Solved Numerical 3: Satellite at h = R (so r = 2R), mass m = 200 kg (R = 6400 km = 6.4 x $10^{6}$ m, g = 10 m/$s^{2}$).
Orbital velocity: $v_{0}$ = $\sqrt{gR^{2}/r}$ = $\sqrt{gR^{2}/(2R}$) = $\sqrt{gR/2}$ = $\sqrt{10 x 6.4 x 10^{6} / 2}$ = $\sqrt{3.2 x 10^{7}}$ = 5657 m/s = 5.66 km/s.
Time period: T = $2\pi$ r/$v_{0}$ = $2\pi$(2 x 6.4 x $10^{6}$)/5657 = $2\pi$ x 12.8 x $10^{6}$ / 5657 = 14,225 s = 3.95 hours.
Total energy: E = -GMm/(2r) = -$gR^{2}$ m/(2 x 2R) = -gRm/4 = -(10)(6.4 x $10^{6}$)(200)/4 = -3.2 x $10^{9}$ J = -3.2 GJ.

The key testable concept is the variation of g with altitude (inverse-square) versus depth (linear), and the relationship $v_{e}$ = $\sqrt{2}$ x $v_{0}$, with escape velocity being independent of the projected body's mass.