🌍 Gravitation

1. Kepler's Laws of Planetary Motion

Law of Orbits: Every planet revolves around the sun in an elliptical orbit with the sun at one of its foci.
Law of Areas: The line joining the planet to the sun sweeps out equal areas in equal intervals of time. (Areal velocity is constant - implies conservation of angular momentum).
Law of Periods: The square of the time period of revolution is proportional to the cube of the semi-major axis of the ellipse. (T² ∝ R³)

2. Newton's Universal Law of Gravitation

Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

F = G (m₁ m₂) / r² G = 6.67 × 10⁻¹¹ N m²/kg² (Universal Gravitational Constant)

3. Acceleration Due to Gravity (g)

The acceleration produced in a freely falling body. On Earth's surface, g = GM / R² ≈ 9.8 m/s².

Variation of 'g'

1. With Altitude (h): g' = g(1 - 2h/R) [for h << R]

2. With Depth (d): g' = g(1 - d/R)

Note: 'g' is maximum at poles and minimum at the equator due to Earth's equatorial bulge and rotation.

4. Escape Velocity & Satellites

Escape Velocity (ve): Minimum velocity required by a body to escape Earth's gravitational field forever. ve = √(2GM/R) = √(2gR) ≈ 11.2 km/s.

Orbital Velocity (vo): Velocity required to put a satellite into orbit. vo = √(GM/R) = √(gR) ≈ 7.9 km/s.

Relation: ve = √2 vo