Gravitation

Gravitation is one of the four fundamental forces of nature and governs the motion of planets, satellites, and falling objects. Newton's Law of Universal Gravitation states that every two masses attract each other with a force F = GMm/$r^{2}$, where G = 6.674 x 10^{-11} N*$\frac{m^{2}}{kg^{2}}$ is the universal gravitational constant. This is an inverse-square law and acts along the line joining the centres of the two masses.

Gravitational Field (g-field) at a point is the force per unit test mass: E = -GM/$r^{2}$ (directed toward the source mass). On Earth's surface, g = $\frac{\text{GM}_{E}}{R_{E}^{2}}$ = 9.8 m/$s^{2}$.

Variation of g:

• With height h above the surface: $g_{h}$ = g(1 - 2h/$R_{E}$) for h << $R_{E}$, or $g_{h}$ = $\text{GM}_{E}$/($R_{E}$ + h)² exactly.
• With depth d below the surface: $g_{d}$ = g(1 - d/$R_{E}$), assuming uniform density. At the centre (d = $R_{E}$), g = 0.
• With latitude: $g_{eff}$ = g - $R_{E}$$\omega$²$cos^{2}$($\lambda$).
  Maximum at poles ($\lambda$ = 90), minimum at equator ($\lambda$ = 0).

Gravitational Potential at distance r from mass M: V = -GM/r. Potential is always negative (zero at infinity). Gravitational potential energy of a two-mass system: U = -GMm/r.

Escape Velocity: The minimum velocity needed to escape a gravitational field: $v_{e}$ = $\sqrt{2GM/R}$ = $\sqrt{2gR}$.
For Earth, $v_{e}$ = 11.2 km/s. Note: escape velocity is independent of the mass and direction of the projected body.

Orbital Velocity for a satellite at height h: $v_{o}$ = $\sqrt{GM/(R+h}$).
For a near-surface orbit (h << R): $v_{o}$ = $\sqrt{gR}$ = $\frac{v_{e}}{\sqrt{2}}$ = 7.9 km/s.
The relation $v_{e}$ = $\sqrt{2}$ * $v_{o}$ is frequently tested.

Kepler's Laws:

1. Law of Orbits: Planets move in ellipses with the Sun at one focus.
2. Law of Areas: The radius vector sweeps equal areas in equal times (dA/dt = L/(2m) = constant). This is conservation of angular momentum.
3. Law of Periods: $T^{2}$ is proportional to $a^{3}$, where a is the semi-major axis.
   For circular orbits: $T^{2}$ = 4*$\pi$²*$r^{3}$/(GM).

Satellite Energy: For a satellite in circular orbit of radius r: KE = GMm/(2r), PE = -GMm/r, Total Energy = -GMm/(2r). Note that |Total Energy| = KE = |PE|/2. Binding energy = +GMm/(2r).

Geostationary Orbit: T = 24 hours, height approximately 36,000 km above Earth's surface, orbits in the equatorial plane with the same angular velocity as Earth's rotation.

The key problem-solving concept is: always use the correct sign for gravitational PE (negative), distinguish between g at height (inverse-square drop) and g at depth (linear drop), and remember that escape velocity = $\sqrt{2}$ times orbital velocity.

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