Chapter 8 — Gravitation

Fig 1 — Educational Diagram: Kepler's Laws of Planetary Motion

Johannes Kepler formulated three empirical laws describing planetary motion around the Sun based on Tycho Brahe's observations:

• First Law (Law of Orbits): All planets move in elliptical orbits with the Sun located at one of the two foci of the ellipse.
• Second Law (Law of Areas): The line joining any planet to the Sun sweeps out equal areas in equal intervals of time. This is a direct consequence of the conservation of angular momentum under a central force field (dA/dt = L / (2m) = constant).
• Third Law (Law of Periods): The square of the time period of revolution (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its elliptical orbit: T² ∝ a³ or T² / a³ = constant.

Fig 2 — Educational Diagram: Universal Law of Gravitation & Gravitational Constant

Newton's Law of Universal 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 their centers.

Vector formulation: F₁₂ = - G (m₁ m₂ / r²) r̂₁₂. The force is always attractive, acts along the line joining the centers of the masses, and obeys Newton's third law (F₁₂ = - F₂₁).

Universal Gravitational Constant (G): G is a fundamental physical constant measured by Henry Cavendish using a torsion balance. G = 6.67430 × 10⁻¹¹ N m² kg⁻² (SI units) with dimensions [M⁻¹ L³ T⁻²].

Fig 3 — Educational Diagram: Acceleration due to Gravity (g) & its Variation

Acceleration due to gravity on Earth's surface: g = G M_E / R_E², where M_E is Earth's mass and R_E is Earth's mean radius (~6371 km). Standard value of g = 9.8 m/s².

Variation with Altitude (Height h): g_h = G M_E / (R_E + h)² = g / (1 + h/R_E)². For small height h << R_E, expanding binomially gives g_h ≈ g (1 - 2h / R_E).

Variation with Depth (d): Assuming uniform Earth density ρ, g_d = g (1 - d / R_E). At the center of the Earth (d = R_E), g = 0.

Variation with Latitude (Rotation of Earth): Due to Earth's rotation at angular speed ω, effective g at latitude λ is g' = g - ω² R_E cos²λ. g is maximum at the poles (λ = 90°) and minimum at the equator (λ = 0°).

Fig 4 — Educational Diagram: Gravitational Potential Energy & Potential

Gravitational Potential Energy (U): The work done in bringing a mass m from infinity to a distance r from a mass M under the action of gravitational force: U(r) = - G M m / r. The negative sign indicates that the gravitational force is attractive.

Gravitational Potential (V): The gravitational potential energy per unit mass at a point in space: V(r) = U(r) / m = - G M / r. SI unit: J/kg.

Relation with Gravitational Field (E): E = - dV/dr = - G M / r² r̂.

Fig 5 — Educational Diagram: Escape Speed & Earth Satellites

Escape Speed (v_e): The minimum speed required for a projectile to escape Earth's gravitational influence completely without further propulsion. By energy conservation: ½ m v_e² - G M_E m / R_E = 0 ⇒ v_e = √(2 G M_E / R_E) = √(2 g R_E). For Earth, v_e ≈ 11.2 km/s.

Orbital Speed of Satellite (v_o): For a satellite of mass m orbiting at radius r = R_E + h: Centripetal force = Gravitational force ⇒ m v_o² / r = G M_E m / r² ⇒ v_o = √(G M_E / r) = √[G M_E / (R_E + h)]. Near Earth's surface (h << R_E), v_o = √(g R_E) ≈ 7.92 km/s.

Time Period of Satellite (T): T = 2π r / v_o = 2π r^(3/2) / √(G M_E) = 2π √[(R_E + h)³ / (g R_E²)]. Near surface, T ≈ 84.6 minutes.

Energies of Orbiting Satellite:

• Kinetic Energy: K = ½ m v_o² = G M_E m / (2r)
• Potential Energy: U = - G M_E m / r
• Total Energy: E = K + U = - G M_E m / (2r)
• Binding Energy = - E = G M_E m / (2r)

Geostationary vs. Polar Satellites:

• Geostationary Satellite: Orbits in the equatorial plane from west to east with a time period T = 24 hours at an altitude h ≈ 35,800 km. Appears stationary relative to Earth; used for telecommunication and weather monitoring.
• Polar Satellite: Orbits in a north-south plane passing over the poles at lower altitude (h ≈ 500-800 km, T ≈ 100 min); used for remote sensing, environmental scanning, and military reconnaissance.