Gravitation for JEE: Orbital Motion, Escape Velocity, and Satellites Guide

Escape velocity is the minimum velocity needed to escape a planet's gravitational field. v_esc = √(2GM/R) = √(2gR). For Earth: 11.2 km/s. It is independent of the mass of the escaping object.

For a satellite in orbit, total energy E = -GMm/2r. It is negative because the satellite is bound to the planet. To escape, energy must increase to zero or positive. Binding energy = GMm/2r.

A geostationary satellite orbits at 35,786 km above the equator with a period of 24 hours, appearing stationary from Earth. Its orbital plane is the equatorial plane and direction is west to east.

At altitude h (small compared to R): g' = g(1 - 2h/R). For large h: g' = g(R/(R+h))². Gravitational acceleration decreases with altitude and depth.

Mass is constant everywhere (amount of matter). Weight is the gravitational force: W = mg. It varies with g: weight on Moon = 1/6 weight on Earth. Mass is a scalar, weight is a vector pointing toward the center.

Gravitational field intensity g = F/m = GM/r². It is the force per unit mass at a point in the gravitational field. Its SI unit is N/kg or m/s². It is a vector pointing toward the mass creating the field.

At infinity, gravitational potential energy U = -GMm/r → 0 as r → ∞. The potential energy is always negative for finite distances, representing the bound state. To escape, energy must become zero or positive.

A geostationary satellite must: (1) orbit at 35,786 km above equator (r = 42,164 km from center), (2) have a period of 24 hours, (3) orbit in the equatorial plane, (4) move west to east (same direction as Earth's rotation).

Astronauts feel weightless because both the astronaut and spacecraft are in free fall toward Earth with the same acceleration. They are in continuous free fall around Earth — not because gravity is zero at that altitude.

Polar orbits pass over the poles and can cover the entire Earth's surface (useful for Earth observation). Equatorial orbits are in the equatorial plane. Geostationary satellites must be in equatorial orbits.

g is minimum at the equator and maximum at the poles. This is because: (1) Earth bulges at the equator (r is larger), (2) centrifugal force reduces effective g at the equator. Difference is about 0.5%.

At the center of Earth (r = 0), the gravitational field is zero because mass is symmetrically distributed around the center. However, gravitational potential is minimum (-3GM/2R) at the center.

Kepler's second law states planets sweep equal areas in equal times. This proves angular momentum conservation: L = mvr sinθ = constant. Planets move faster when closer to the Sun (perihelion) and slower when farther (aphelion).

A bound satellite has negative total energy: E = -GMm/2r. The magnitude of E is the binding energy, which is the minimum energy needed to escape. To transfer from a lower to higher orbit, energy must be added.