Gravitation

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Gravitation — Key Facts for JEE Main • Newton’s law of gravitation: F = G·M·m/r²; G = 6.674×10⁻¹¹ N·m²/kg² (cavity experiment by Cavendish); gravitational force is always attractive, along the line joining centres • Gravitational field intensity: g = F/m = G·M/r² (same form as gravitational acceleration); on Earth’s surface g ≈ 9.8 m/s²; g varies with altitude (g_h = g·[R/(R+h)]²) and latitude (due to Earth’s rotation) • Gravitational potential: V = −G·M/r (negative; zero at infinity); Potential energy: U = −G·M·m/r; Work done = m[ΔV] = m(GM/r_initial − GM/r_final) • Kepler’s 3rd law: T² ∝ r³ (for circular orbits) → T² = (4π²/GM)·r³; also applies to elliptical orbits with r replaced by semi-major axis a • Orbital velocity: v₀ = √(GM/r) = √(gr) for circular orbit (close to Earth, r ≈ R); escape velocity v_e = √(2GM/R) = √2·v₀; satellite velocity at height h: v = √[GM/(R+h)] ⚡ Exam tip: In satellite problems, always check if the orbit is geostationary (T = 24 h, h ≈ 36,000 km, equatorial) — but most JEE Main satellite questions use simpler orbits closer to Earth.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Gravitation — JEE Main / Advanced Study Guide Gravitational field vs acceleration: g = GM/R² at surface; g_h = g·(R/(R+h))² at altitude h << R. Variation with rotation: effective g’ = g − ω²R·cos²λ (at latitude λ); weightlessness at equator when ω²R = g.

Gravitational potential energy: U = −GMm/r (for isolated mass); for extended objects integrate. At height h: U_h = −GMm/(R+h) = −mgR²/(R+h). Change in g with height: Δg/g ≈ −2h/R (for small h, binomial expansion).

Satellite motion: For a satellite in circular orbit: centripetal force = gravitational pull → mv²/(R+h) = GMm/(R+h)² → v = √[GM/(R+h)]. Time period: T = 2π√[(R+h)³/GM] = 2π√[a³/GM] (Kepler’s 3rd law). Angular momentum: L = mvr = m√[GMr] = constant. Energy: KE = ½mv² = GMm/2(R+h); PE = −GMm/(R+h); Total E = −GMm/2(R+h) < 0.

Geostationary satellite: T = 24 h, h ≈ 35,786 km above equator (geosynchronous orbit = same period as Earth); orbital velocity ≈ 3.1 km/s. These are NOT polar satellites.

Gravitational slingshot: Spacecraft gains/loses speed by passing near a planet; momentum transfer. Energy conserved in planet’s frame but changes relative to Sun’s frame.

Solved Example 1: Find orbital speed and period of a satellite at h = 600 km above Earth. R_E = 6400 km, M_E = 6×10²⁴ kg. r = 7000 km = 7×10⁶ m. v = √[6.67×10⁻¹¹×6×10²⁴/7×10⁶] = √[5.72×10⁷] ≈ 7.56 km/s. T = 2π√[r³/GM] = 2π√[(7×10⁶)³/(6.67×10⁻¹¹×6×10²⁴)] = 2π×5840 ≈ 36,700 s ≈ 10.2 h.

Solved Example 2: A body is dropped from height h = R (one Earth radius above surface). Find impact speed (ignore air). Energy: mgh_eff = ΔU = GMm[1/R − 1/(2R)] = GMm/(2R) → v = √(GM/R) = √(gR) = √(10×6.4×10⁶) ≈ 8 km/s.

⚡ Exam tip: For any satellite (circular orbit): v ∝ r⁻¹ᐟ², T ∝ r³ᐟ², KE ∝ r⁻¹ (decreases with orbit height), PE ∝ r⁻¹ (becomes less negative = higher energy with height), Total E ∝ r⁻¹.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer timeline.

Gravitation — Comprehensive JEE Notes Gravitational field of solid sphere: Inside uniform sphere (r < R): M_enc = M·(r³/R³); E(r) = GMr/R³ = g·(r/R). Outside (r > R): E = GM/r² (same as point mass). At centre (r = 0): E = 0 (by spherical symmetry — all forces cancel). For spherical shell: outside → GM/r²; inside (r < R_shell) → 0.

Elliptical orbits (Kepler’s Laws extended): In elliptical orbit, v varies: at perihelion (closest) v_max = √[GM(1+e)/a(1−e)]; at aphelion (farthest) v_min = √[GM(1−e)/a(1+e)]; vis-viva equation: v² = GM[(2/r) − (1/a)]. Areal velocity: dA/dt = L/2m = constant (conservation of angular momentum) — this is the precise form of Kepler’s 2nd law.

Gravitational potential due to ring/sheet/disk: Ring on axis: V = −GM/√(R² + x²); Thin disk: V = −(2GM/a²)[√(a² + x²) − x]; Infinite sheet: V diverges (no convergence). Gravity tunnels: A body dropped through a frictionless tunnel through Earth’s centre would execute SHM with T = 2π√[R/g] ≈ 84 min. At Earth’s centre, ρ_avg = M/(4πR³/3) used.

Black holes & escape velocity: Schwarzschild radius R_s = 2GM/c²; for Sun R_s ≈ 3 km; for Earth R_s ≈ 9 mm. At R_s, even light cannot escape. Gravitational redshift: λ_obs/λ_em = √(1 − R_s/R)⁻¹ᐟ².

Tidal forces: Differential gravitational pull of Moon on Earth creates two tidal bulges → high tide at sub-Lunar and anti-Lunar points. Spring tides (new/full moon): Sun + Moon aligned → higher tides; neap tides (first/third quarter): perpendicular → lower tides. Roche limit: d_R = 2.4R_p(ρ_p/ρ_m)⁽¹ᐟ³⁾ — planet’s tidal force exceeds moon’s self-gravity here.

Pioneer anomaly & flyby anomaly: Both confirmed to be thermal radiation effects (RTG heat), not new physics.

⚡ JEE Advanced tip: Problems involving two massive bodies + satellite often use conservation of energy + angular momentum. For transfer orbits (Hohmann): Δv at perihelion = v₁(√(2r₂/(r₁+r₂)) − 1) and Δv at aphelion = v₂(1 − √(2r₁/(r₁+r₂))). Total Δv determines fuel requirement.

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Sections	Physics, Chemistry, Mathematics
Type	MCQ + Numerical Value (NAT)
Time	3 hours
Marking	+4 correct, −1 wrong (MCQ); +4 correct, 0 wrong (NAT)
Sessions	January + April per year; best score considered

🎯 High-Yield Topics for JEE Main

• Satellite & Orbital Mechanics — 4–6 marks
• Gravitational Field & Potential — 3–4 marks
• Kepler’s Laws — 2–3 marks
• Variation of g (Altitude, Depth, Latitude) — 3 marks
• Escape Velocity — 2–3 marks

📝 Previous Year Question Patterns

• Q: “A satellite is revolving around Earth in a circular orbit. Its orbital velocity is…” [2025 Physics — 3 marks]
• Q: “A body is thrown vertically upward from Earth’s surface. It reaches a maximum height of…” [2024 Physics — 4 marks]
• Q: “Two planets have radii R and 2R. Their densities are same. Ratio of escape velocities…” [2023 Physics — 3 marks]

💡 Pro Tips

• Escape velocity = √2 × orbital velocity for the same planet — remember this relationship
• g varies with depth (g_d = g(1 − d/R)) and height (g_h = g(R/(R+h))²) — but NOT with latitude in simplified problems
• For geostationary orbit: always specify equatorial plane, T = 24 h, h ≈ 36,000 km
• In elliptical orbit problems, use vis-viva equation: v² = GM(2/r − 1/a)

🔗 Official Resources

• NTA JEE Main
• JEE Main Syllabus

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