JEE Physics Notes: Gravitation

Introduction

Gravitation is the force of attraction between any two masses in the universe. It governs the motion of planets, satellites, and even objects on Earth. The concept was first formulated by Isaac Newton in the Universal Law of Gravitation and later refined by Einstein's General Theory of Relativity.

1. Newton’s Law of Universal Gravitation

• Statement: Every mass in the universe attracts every other mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Mathematical Expression: F = G (m₁m₂) / r², where:
  • F = Gravitational force
  • G = Universal Gravitational Constant (6.674 × 10⁻¹¹ Nm²/kg²)
  • m₁, m₂ = Masses of two objects
  • r = Distance between the centers of two masses
• Characteristics:
  • It is always attractive in nature.
  • It acts along the line joining the centers of the two masses.
  • It follows the inverse-square law.

2. Acceleration Due to Gravity (g)

• Definition: The acceleration experienced by a body due to the gravitational pull of Earth.
• Formula: g = G M / R², where M is the mass of Earth and R is the radius of Earth.
• Value of g on Earth: 9.8 m/s².
• Variation of g:
  • With Altitude: g' = g (1 - 2h/R) for small heights.
  • With Depth: g' = g (1 - d/R).
  • With Latitude: g decreases from poles to the equator due to Earth's rotation.

3. Motion of Planets and Satellites

• Kepler’s Laws of Planetary Motion:
  • First Law (Law of Orbits): Planets move in elliptical orbits with the Sun at one focus.
  • Second Law (Law of Areas): The line joining a planet and the Sun sweeps equal areas in equal time intervals.
  • Third Law (Law of Periods): T² ∝ r³, where T is the orbital period and r is the semi-major axis of the orbit.
• Orbital Velocity of a Satellite: v = √(GM/R).
• Escape Velocity: The minimum velocity required to escape from a planet’s gravitational field.
  • Formula: ve = √(2GM/R).
  • For Earth, ve ≈ 11.2 km/s.
• Time Period of a Satellite: T = 2π√(r³/GM).

4. Gravitational Potential and Potential Energy

• Gravitational Potential (V):
  • Potential at a point is the work done per unit mass in bringing a mass from infinity to that point.
  • Formula: V = - GM / r.
• Gravitational Potential Energy (U):
  • Energy possessed by a mass due to its position in a gravitational field.
  • Formula: U = - GMm / r.

5. Satellites

• Types of Satellites:
  • Geostationary Satellite: Orbits Earth in 24 hours, used for communication.
  • Polar Satellite: Moves in a polar orbit, used for weather monitoring and mapping.
• Binding Energy of a Satellite: The energy required to remove a satellite from orbit.
  • Formula: E = - GMm / 2R.

Conclusion

Gravitation plays a vital role in understanding planetary motion, satellite dynamics, and everyday physics. Mastering concepts such as Newton’s law of gravitation, Kepler’s laws, and escape velocity is essential for solving JEE-level problems effectively.