Learning Objectives
- State and apply Newton's universal law of gravitation
- Derive the acceleration due to gravity and its variations
- Understand gravitational potential energy and escape velocity
- Learn Kepler's laws of planetary motion
- Study orbital mechanics and satellite motion
Key Concepts
Newton's Law of Universal Gravitation
F = Gm₁m₂/r²
G = 6.674 × 10⁻¹¹ N·m²/kg² (universal gravitational constant)
The force is always attractive, acts along the line joining centres, and follows the inverse square law.
Gravitational force obeys the principle of superposition: net force on a particle is the vector sum of forces due to all other particles.
Acceleration Due to Gravity (g)
On Earth's surface: g = GM/R² ≈ 9.8 m/s²
Variation with altitude: g_h = g(1 - 2h/R) for h << R, or g_h = GM/(R+h)² exactly.
Variation with depth: g_d = g(1 - d/R). At Earth's centre (d = R), g = 0.
Variation with latitude: g_φ = g - Rω²cos²φ. Maximum at poles (φ = 90°), minimum at equator (φ = 0°).
Effect of shape: Earth is flattened at poles (R_pole < R_equator), so g is greater at poles.
Gravitational Potential Energy
U = -GMm/r (taking U = 0 at infinity)
Near Earth's surface: ΔU = mgh (for small heights).
Gravitational PE is always negative, indicating a bound system.
Gravitational Potential
V = -GM/r (potential at distance r from mass M)
V = U/m (potential energy per unit mass). g = -dV/dr.
Escape Velocity
Minimum velocity needed to escape from a gravitational field: v_e = √(2GM/R) = √(2gR)
For Earth: v_e ≈ 11.2 km/s. Escape velocity is independent of the mass and direction of the projectile.
Kepler's Laws
- 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 intervals of time. dA/dt = L/2m = constant. This implies planets move faster when closer to the Sun (at perihelion).
- Third Law (Law of Periods): T² ∝ a³, where T is the orbital period and a is the semi-major axis. T² = (4π²/GM)a³.
Orbital Velocity and Satellites
Orbital velocity: v₀ = √(GM/r) = √(gR²/r)
For orbit near surface: v₀ = √(gR) ≈ 7.9 km/s
Relation: v_e = √2 × v₀
Time period: T = 2πr/v₀ = 2π√(r³/GM)
Energy of orbiting satellite: KE = GMm/2r, PE = -GMm/r, Total E = -GMm/2r
Geostationary orbit: T = 24 hours, height ≈ 36,000 km above equator, orbits in equatorial plane.
Summary
Newton's law of gravitation describes the universal attractive force between masses. The acceleration due to gravity varies with altitude, depth, and latitude. Gravitational potential energy is negative for bound systems. Escape velocity (11.2 km/s for Earth) is independent of mass. Kepler's three laws describe planetary orbits, area sweeping, and the period-radius relationship. Satellites have specific orbital velocities and energies.
Important Terms
- Gravitational Constant (G): 6.674 × 10⁻¹¹ N·m²/kg²
- Escape Velocity: Minimum speed to escape gravitational pull
- Orbital Velocity: Speed needed to maintain a circular orbit
- Geostationary Satellite: Satellite with 24-hour period over equator
- Perihelion: Closest point to Sun in orbit
- Aphelion: Farthest point from Sun in orbit
Quick Revision
- F = Gm₁m₂/r²; g = GM/R²
- g varies: with altitude g_h = g(1-2h/R), with depth g_d = g(1-d/R)
- U = -GMm/r; V = -GM/r
- v_e = √(2gR) ≈ 11.2 km/s; v₀ = √(gR) ≈ 7.9 km/s
- Kepler: elliptical orbits, equal area in equal time, T² ∝ a³
- Satellite energy: E = -GMm/2r (half the PE)