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APEX CLASS ESTD 2018

IITJEE: Variation of Gravity (g), Escape/Orbital Velocity & Kepler’s Laws

By

Ashish Kumar Bahekar

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APEX CLASS • IITJEE Gravitation & Orbital Mechanics Lab

Variation of Gravity (g), Escape/Orbital Velocity & Kepler’s Laws

Explore Inside-vs-Outside Earth Gravity Graphs, Satellite Energy Ratios, and Kepler’s Elliptical Orbits!

🛰️ Select an Orbital Mechanics or Kepler’s Law Concept:
Earth (M,R) 1. Circle: v = v_o = 7.9 km/s 2. Ellipse (v_o < v < v_e) 3. Parabola: v = v_e = 11.2 km/s (Escapes!) Orbital (v_o) vs. Escape Speed (v_e) • Close Orbit: v_o = √(GM/R) = √(gR) = 7.9 km/s • Escape Speed: v_e = √(2GM/R) = √(2gR) = 11.2 km/s • Universal Ratio: v_e = √2 × v_o (+41.4% speed!) • Interstellar Speed (when launched at v > v_e): v_∞ = √(v² – v_e²) (Hyperbolic Path!) 🚀 JEE Trap: Escape velocity v_e = √(2GM/R) is INDEPENDENT of projectile mass ‘m’ and launch angle θ! +Energy 0 -Energy Orbital Radius (r) → K = +GMm / (2r) (> 0) Total E = -GMm / (2r) = -K Potential U = -GMm / r = 2E Bound Orbit Energy Identities (Same as Bohr!) E_total = -K = U / 2 = -GMm / (2r) • Binding Energy (To Escape): BE = +GMm / (2r) • As Orbit Radius r INCREASES (Higher Orbit): Speed v & Kinetic Energy K DECREASE, while Potential U & Total Energy E INCREASE! ⚠️ Atmospheric Drag Paradox: If air drag makes a satellite lose Total Energy E (more negative), it drops to smaller r and SPEEDS UP (K = -E)! Sun (Focus) Perihelion (r_min) v_max! Aphelion (r_max) v_min ← Major Axis 2a = r_min + r_max → 🏆 Kepler’s 3 Laws & JEE Shortcuts 1. Law of Orbits: r_min = a(1 – e), r_max = a(1 + e) 2. Law of Areas (Angular Momentum L Conserved!): dA / dt = L / (2m) = Const ⇒ v_max · r_min = v_min · r_max 3. Speed & KE Ratio (Perihelion vs Aphelion): v_max / v_min = (1 + e) / (1 – e) | K_max / K_min = [(1+e)/(1-e)]² 4. Law of Periods: T² = (4π² / GM) a³ (T² ∝ a³!) 🪐 Central Force Rule: Because Gravitational Force points directly toward the Sun, Torque τ = r × F = 0 → Angular Momentum L is CONSTANT! 1. Geostationary Satellite (Communication) • Orbit Plane: Equatorial Plane (Concentric) • Time Period: T = 24 Hours (West → East) • Height above Surface: h ≈ 36,000 km (≈ 5.6 R_e) • Distance from Center: r ≈ 42,400 km (≈ 6.6 R_e) • Orbital Speed: v_o ≈ 3.08 km/s 2. Polar Satellite (Remote Sensing / Weather) • Orbit Plane: North-South Polar Axis (90° to Equator) • Time Period: T ≈ 100 Minutes (Scans whole Earth!) • Low Altitude: h ≈ 500 km to 800 km • Close-Earth Satellite (h ≪ R_e): T = 2π √(R_e / g) • Close Orbit Period T_0 = 84.6 Minutes (5075 s)! 📡 NCERT Statement Trap: A Geostationary satellite CANNOT be placed directly over Mumbai or Delhi—it must lie in the Equatorial plane (0° latitude)!
Orbital vs Escape Speed
v_e = √2 v_o (11.2 vs 7.9)
Using GM = g R² Substitution
Satellite Energy Law
E = -K = U / 2
E = -GMm / (2r) [Bound System]
Areal Velocity (dA/dt)
dA / dt = L / (2m)
v_max / v_min = (1 + e) / (1 – e)
Kepler’s 3rd Law
T₁ / T₂ = (r₁ / r₂)³/²
If r becomes 4× → T becomes 8×!
🌍 Surface Gravity g = GM / R² = (4/3)πGρR. Select Variation Factor:
Center (g = 0!) R = 6400 km Solid Uniform Earth (ρ) g(r) r = R (Surface: 9.8 m/s²) Distance r → Inside: g ∝ r g_in = g(1 – d/R) g_max = GM / R² Outside: g ∝ 1 / r² 🎯 Mass vs Density Trap: If two planets have same RADIUS R, g ∝ M; if they have same DENSITY ρ, use g = (4/3)πGρR ⇒ g ∝ R! 1. Variation with Height (h above Surface) Exact (Any h): g_h = g / (1 + h/R)² Small h (h ≪ R, < 5%): g_h ≈ g (1 – 2h/R) • At h = R (6400 km): g_h = g / 4 = 2.45 m/s² • % Drop for small h: Δg/g × 100 = 2 × (h/R × 100)% 2. Variation with Depth (d below Surface) Exact for ALL Depths d: g_d = g (1 – d/R)! Same g at Small Height h & Depth d: d = 2 h! • Exact General Relation (Valid for ANY h & d): 1 – d/R = 1 / (1 + h/R)² ⚠️ Do NOT use g(1 – 2h/R) when h = R/2 or h = R! Use the binomial approximation ONLY when h is less than 300 km (~5% of R)! North Pole (λ = 90°: g’ = g) Equator (λ = 0°) λ Apparent Gravity due to Rotation (ω) g’ = g – ω² R cos²λ 1. At Poles (λ = 90°, cos90° = 0): g_pole = g (No Effect!) 2. At Equator (λ = 0°, cos0° = 1): g_eq = g – ω² R (Minimum!) 3. Weightlessness at Equator (g_eq = 0): ω_new = √(g / R) = 17 × ω_present (Day = 84.6 mins!) 🌐 If Earth suddenly stops rotating (ω → 0), apparent weight increases at the Equator by mω²R, while weight at the Poles stays unchanged! 1. Max Height H When Launched at v < v_e Never use h = u²/(2g) when u is comparable to v_e! H = R / [ (v_e / v)² – 1 ] = u²R / (2gR – u²) • Example: Fired upward at u = √(gR) = v_e / √2: H = R / [ 2 – 1 ] = R (Reaches height R = 6400 km!) 2. Potential & Escape from Earth’s Center • Surface Potential: V_s = -GM / R • Center Potential: V_c = 1.5 V_s = -3GM / (2R) • Escape Speed from Center of Earth (Tunnel): v_e(center) = √(-2 V_c) = √(3GM/R) = √1.5 × v_e! 🕳️ Tunnel Through Earth SHM: A ball dropped into ANY chord or diameter tunnel through Earth executes SHM with T = 2π √(R/g) = 84.6 mins!
Surface Gravity Forms
g = GM/R² = (4/3)πGρR
Check if M or ρ is constant!
Small Height vs Depth
g(1 – 2h/R) vs g(1 – d/R)
Equal when Depth d = 2h
Work to Lift m to Height h
ΔU = mgh / (1 + h/R)
At h = R: ΔU = ½ mgR (Not mgR!)
Gravitational Self-Energy
U_solid = -⅗ (GM² / R)
Hollow Shell: U_shell = -½ (GM²/R)
🔥 Instant Electrostatics ↔ Gravitation Formula Translator:
Replace Charge q → Mass m and Constant k = 1/(4πε₀) → -G! Every formula for Hollow Sphere, Solid Sphere, Ring on Axis, and Potential Energy in Electrostatics works identically in Gravitation!
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