Gravitational Fields

Gravitational fields govern the attraction between masses, shaping the motion of planets, stars, and satellites. Rooted in Newton’s Law of Universal Gravitation and expanded by Einstein’s general relativity, they are central to physics and astronomy. This MathMultiverse guide explores gravitational force, field strength, potential, orbital and escape velocities, with detailed examples, formulas, and applications in satellite orbits and astrophysics, enhanced with interactive visualizations.

Newton’s Law and Key Formulas

Newton’s Law of Universal Gravitation:

\[ F = G \frac{m_1 m_2}{r^2} \]

Where:

  • \( F \): Gravitational force (N)
  • \( G \): Gravitational constant (\( 6.674 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \))
  • \( m_1, m_2 \): Masses (kg)
  • \( r \): Distance between centers (m)

Related Formulas:

  • Gravitational Field Strength:
    \[ g = \frac{G M}{r^2} \]
  • Gravitational Potential:
    \[ V = -\frac{G M}{r} \]
  • Orbital Velocity:
    \[ v = \sqrt{\frac{G M}{r}} \]
  • Escape Velocity:
    \[ v_e = \sqrt{\frac{2 G M}{r}} \]

Gravitational Force vs. Distance

Force decreases with the square of distance.

Examples

Earth-Moon Gravitational Force

Force between Earth (\( 5.972 \times 10^{24} \, \text{kg} \)) and Moon (\( 7.342 \times 10^{22} \, \text{kg} \)), \( r = 3.844 \times 10^8 \, \text{m} \):

\[ F = (6.674 \times 10^{-11}) \frac{(5.972 \times 10^{24})(7.342 \times 10^{22})}{(3.844 \times 10^8)^2} \approx 1.98 \times 10^{20} \, \text{N} \]

Earth’s Surface Field Strength

\( M = 5.972 \times 10^{24} \, \text{kg} \), \( r = 6.371 \times 10^6 \, \text{m} \):

\[ g = (6.674 \times 10^{-11}) \frac{5.972 \times 10^{24}}{(6.371 \times 10^6)^2} \approx 9.82 \, \text{m/s}^2 \]

Gravitational Potential (Earth)

At Earth’s surface:

\[ V = -\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{6.371 \times 10^6} \approx -6.255 \times 10^7 \, \text{J/kg} \]

Orbital Velocity (Satellite)

At \( r = 6.771 \times 10^6 \, \text{m} \):

\[ v = \sqrt{\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{6.771 \times 10^6}} \approx 7672 \, \text{m/s} \]

Escape Velocity (Earth)

From Earth’s surface:

\[ v_e = \sqrt{\frac{2 \cdot (6.674 \times 10^{-11}) \cdot (5.972 \times 10^{24})}{6.371 \times 10^6}} \approx 11185 \, \text{m/s} \]

Applications

Satellite Orbits

Orbital velocity at 400 km above Earth (\( r = 6.771 \times 10^6 \, \text{m} \)):

\[ v \approx 7672 \, \text{m/s} \]

Planetary Motion (Mars)

Force on a 1000 kg probe at Mars (\( M = 6.417 \times 10^{23} \, \text{kg} \), \( r = 3.396 \times 10^6 \, \text{m} \)):

\[ F \approx 3712 \, \text{N} \]

Black Hole Field

Field strength at \( 1.5 \times 10^9 \, \text{m} \) from a black hole (\( M = 1.989 \times 10^{30} \, \text{kg} \)):

\[ g \approx 5.9 \, \text{m/s}^2 \]

Tidal Forces

Force difference across Earth’s diameter (Moon):

\[ \Delta F \approx 1.27 \times 10^{19} \, \text{N} \]

Spacecraft Escape (Moon)

Escape velocity from Moon (\( M = 7.342 \times 10^{22} \, \text{kg} \), \( r = 1.737 \times 10^6 \, \text{m} \)):

\[ v_e \approx 2375 \, \text{m/s} \]