Gravitation
'I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve.' — Isaac Newton
1. Chapter Overview
GRAVITATION is the UNIVERSAL force of attraction between all matter. This chapter begins with NEWTON'S LAW OF GRAVITATION, explores how g varies with altitude/depth/latitude, introduces GRAVITATIONAL POTENTIAL ENERGY, ESCAPE VELOCITY, and ORBITAL MECHANICS, and concludes with KEPLER'S LAWS of planetary motion.
2. Newton's Universal Law of Gravitation
- Statement: Every point mass attracts every other point mass with a force DIRECTLY proportional to the product of their masses and INVERSELY proportional to the SQUARE of the distance between them.
- Formula: F = GMm/r² (where G = 6.67 × 10⁻¹¹ N·m²/kg²)
- G is the UNIVERSAL GRAVITATIONAL CONSTANT (same everywhere in the universe)
- The force ALONG the line joining the two masses
Vector Form
- F₁₂ = -Gm₁m₂/r² × r̂₁₂ (the negative sign indicates attraction)
- The force on m₁ due to m₂ is TOWARD m₂
3. Acceleration Due to Gravity (g)
Variation with Height (Altitude)
- g_h = g₀(1 — 2h/R) for h << R (near surface)
- g decreases as height INCREASES
Variation with Depth
- g_d = g₀(1 — d/R) for depth d below surface
- At the centre of Earth (d = R): g = 0
- g decreases LINEARLY with depth
Variation with Latitude (Rotation of Earth)
- g_λ = g₀ — Rω²cos²λ (where λ = latitude)
- Maximum at POLES (λ = 90°): g_pole = g₀
- Minimum at EQUATOR (λ = 0°): g_equator = g₀ — Rω²
- Earth's rotation causes about 0.034 m/s² reduction at equator
Key Comparison
| Location | g (m/s²) | Reason |
|---|---|---|
| Surface | 9.8 | Standard value |
| Pole | 9.83 | No centrifugal effect |
| Equator | 9.78 | Maximum centrifugal effect |
| Mountain top | < 9.8 | Increased distance from centre |
| Deep mine | < 9.8 | Reduced effective mass |
4. Gravitational Field and Potential
- Gravitational Field E_g = F/m = GM/r² (force per unit mass)
- Gravitational Potential V_g = -GM/r (work per unit mass to bring from infinity)
- Relation: E_g = -dV_g/dr
- Gravitational POTENTIAL ENERGY: U = -GMm/r (U = 0 at infinity, NEGATIVE for bound systems)
Worked Problem
Q: Calculate the gravitational potential at a point 2R above Earth's surface. (M = 6×10²⁴ kg, R = 6400 km) A: Distance from centre = R + 2R = 3R. V = -GM/3R = -(6.67×10⁻¹¹×6×10²⁴)/(3×6.4×10⁶) = -2.08×10⁷ J/kg
5. Escape Velocity
- Definition: The MINIMUM velocity needed for an object to escape Earth's gravitational pull (reach infinity with KE = 0)
- Formula: v_esc = √(2GM/R) = √(2gR)
- For Earth: v_esc = √(2×9.8×6.4×10⁶) = 11.2 km/s
- Independent of mass of the escaping object
Comparison
| Celestial Body | Escape Velocity |
|---|---|
| Earth | 11.2 km/s |
| Moon | 2.4 km/s |
| Sun | 618 km/s |
| Jupiter | 59.5 km/s |
| Black Hole | > c (speed of light) |
6. Satellites — Orbital Motion
Orbital Velocity
- v₀ = √(GM/r) (for a circular orbit)
- Near Earth's surface: v₀ ≈ √(gR) ≈ 7.92 km/s
Time Period
- T = 2πr/v₀ = 2π√(r³/GM)
- Geostationary Satellite: Orbits above EQUATOR at distance where T = 24 hours
- Height ≈ 36,000 km above surface
- Used for COMMUNICATION (appears stationary from Earth)
Energy of an Orbiting Satellite
- KE = GMm/2r
- PE = -GMm/r
- Total Energy = KE + PE = -GMm/2r (NEGATIVE, meaning bound system)
- To increase orbit radius (r ↑): Total energy increases (less negative), KE decreases, PE increases
7. Kepler's Laws of Planetary Motion
First Law (Law of Orbits)
- Each planet moves in an ELLIPTICAL orbit with the Sun at ONE FOCUS
Second Law (Law of Areas)
- The line joining planet to Sun sweeps EQUAL areas in EQUAL intervals of time
- Consequence: v_areal = L/2m = CONSTANT → Angular momentum is CONSERVED
Third Law (Law of Periods)
- T² ∝ a³ (where a = semi-major axis)
- T²/T₁² = a³/a₁³
- For a circular orbit: T² = (4π²/GM)a³
Worked Problem
Q: If Earth's orbital period is 1 year at radius R, what is the period of a planet at 4R? A: Using Kepler's third law: T²/T_E² = (4R/R)³ = 64. T = 8 years.
8. Common Mistakes
- g is NOT the same as G: G is universal constant (6.67×10⁻¹¹), g is acceleration = GM/R²
- Weightlessness ≠ zero gravity: Astronauts are in FREE FALL (g ≈ 8.7 m/s² at ISS height), but they feel weightless because the spacecraft is also accelerating at the same rate
- g decreases with height AND depth: But for different reasons
- Satellite KE is half of |PE|: For a circular orbit, KE = |PE|/2 = |E|
- Escape velocity from Earth is 11.2 km/s, NOT 7.92 km/s: 7.92 km/s is orbital velocity near surface
9. CBSE Exam Focus
- Newton's law of gravitation — derivation of g (3-mark)
- Variation of g with height, depth, latitude (5-mark)
- Escape velocity derivation (3-mark)
- Orbital velocity, time period of satellite (5-mark)
- Kepler's laws — statements and derivations
- Energy of orbiting satellite — binding energy
10. Key Formulas
- F = GMm/r², G = 6.67×10⁻¹¹ N·m²/kg²
- g = GM/R² (surface), g_h = g(1 — 2h/R), g_d = g(1 — d/R)
- v_esc = √(2GM/R) = √(2gR)
- v₀ = √(GM/r)
- T = 2π√(r³/GM)
- U = -GMm/r, E_total = -GMm/2r
- T² ∝ a³ (Kepler's third law)
11. Self-Test (5+ Q&A)
Q1: Find g at a height equal to R above Earth's surface. A: Distance = 2R. g' = GM/(2R)² = (GM/R²)/4 = g/4 = 2.45 m/s².
Q2: What is the orbital velocity of a satellite at height 300 km? (R = 6400 km, M = 6×10²⁴ kg) A: r = 6400 + 300 = 6700 km = 6.7×10⁶ m. v₀ = √(GM/r) = √(6.67×10⁻¹¹×6×10²⁴/6.7×10⁶) = √(5.97×10⁷) = 7.73 km/s.
Q3: Why is g zero at the centre of the Earth? A: At the centre, mass of Earth surrounds the point equally in all directions. The NET gravitational force is zero because forces from all sides cancel.
Q4: Calculate escape velocity on the Moon (g_m = 1.62 m/s², R_m = 1740 km). A: v_esc = √(2gR) = √(2×1.62×1.74×10⁶) = √(5.64×10⁶) = 2.37 km/s.
Q5: State Kepler's second law. Which conservation law does it correspond to? A: Kepler's second law: The line joining planet to Sun sweeps equal areas in equal times. It corresponds to the CONSERVATION OF ANGULAR MOMENTUM.
12. Conclusion
Gravitation connects TERRESTRIAL physics (g, free fall) with CELESTIAL physics (planetary motion). Newton showed that the SAME force makes an apple fall and holds the Moon in orbit. The concepts of orbital mechanics — escape velocity, geostationary satellites, Kepler's laws — have direct practical applications in space exploration and satellite technology.
