Gravitation

Why This Chapter Matters

Gravitation appears in JEE every year — 4-8 marks. Kepler's laws, orbital velocity, escape velocity, gravitational potential energy, and satellite motion are all tested.

Core Concepts

1. Newton's Law of Gravitation

F = GMm/r² (G = 6.67×10⁻¹¹ N·m²/kg²)

Gravitational field: g = GM/r²

On Earth surface: g₀ = GM_E/R_E² ≈ 9.8 m/s²

At height h: g = g₀(R/(R+h))² → for h<

At depth d: g = g₀(1 - d/R)

2. Gravitational Potential Energy

Between two masses: U = -GMm/r (negative — bound system)

Near Earth surface: U = mgh (taking surface as reference)

At infinity: U = 0

Total energy of satellite: E = KE + U = -GMm/2r (negative → bound orbit)

3. Kepler's Laws

1st Law: Planets orbit in ellipses with Sun at one focus

2nd Law: Line joining planet to Sun sweeps equal areas in equal times (conservation of angular momentum)

3rd Law: T² ∝ r³ (T = period, r = semi-major axis)

T²/r³ = 4π²/GM (same for all satellites around a body)

4. Orbital Velocity

For circular orbit at radius r from Earth's center:

v_orbital = √(GM/r) = √(g₀R²/r)

At surface: v_orbital = √(g₀R) ≈ 7.9 km/s (first cosmic velocity)

Orbital period: T = 2πr/v = 2π√(r³/GM)

Geostationary orbit: T = 24h, r ≈ 42,000 km from center (36,000 km above surface)

5. Escape Velocity

Minimum speed to escape gravitational pull completely:

v_escape = √(2GM/R) = √(2g₀R) ≈ 11.2 km/s

Note: v_escape = √2 × v_orbital (at same radius)

PYQs

2024: Satellite at height R above Earth surface (R = Earth radius). Orbital speed?

r = R + R = 2R. v = √(GM/2R) = √(gR/2) = (1/√2)√(gR)

2023: By what % does the weight of a person decrease when taken to height R/2 above Earth surface?

g' = g₀(R/(R+R/2))² = g₀(2/3)² = 4g₀/9

Decrease = (g₀ - 4g₀/9)/g₀ = 5/9 ≈ 55.6%

2022: Two satellites in orbits of radii r and 4r. Ratio of their periods?

T ∝ r^(3/2). T₂/T₁ = (4r/r)^(3/2) = 4^(3/2) = 8. So T₂ = 8T₁.

Revision Notes

GRAVITATION: F = GMm/r²  g = GM/r²
VARIATION: g ∝ 1/r² outside; g decreases with depth (linearly)

KEPLER'S 3RD: T² ∝ r³  |  T²/r³ = 4π²/GM

ORBITAL: v_orb = √(GM/r)  |  T = 2π√(r³/GM)
ESCAPE: v_esc = √(2GM/R) = √2 × v_orb

ENERGY of satellite: E = -GMm/2r (negative)
KE = GMm/2r | PE = -GMm/r

GEOSTATIONARY: T=24h, 36000 km above surface, 42000 km from center