G = 6.67 × 10⁻¹¹ N m²/kg² (universal constant)
Vector form: F₁₂ = –G m₁ m₂ / r² · r̂

Field intensity (g): Force per unit mass: g = F/m = GM/r² (direction towards M)

Gravitational potential (V): V = –GM/r (zero at infinity)
U = m V = –GMm/r

Relation: g = –dV/dr

For spherical shell: Inside (r < R) → g = 0, V = –GM/R (constant).
For solid sphere: Inside → g ∝ r, V = –GM(3R² – r²)/(2R³).

Escape velocity: v_esc = √(2GM/R) = √(2gR) ≈ 11.2 km/s (Earth)

Orbital velocity (circular): v_o = √(GM/r) = √(gR²/r)

For near-Earth orbit (r ≈ R): v_o ≈ 7.9 km/s

Time period: T = 2πr / v_o = 2π √(r³/GM)

Total energy of satellite: E = –GMm/(2r) = –K (KE = –E, PE = 2E)

g with height: g_h = g(1 – 2h/R) for h << R

g with depth: g_d = g(1 – d/R)

g with rotation: g_eff = g – ω²R cos²λ (λ = latitude)

Potential energy (near Earth): U = mgh (valid only for small h)

True U = –GMm/r; gravitational binding energy of a sphere = –(3/5) GM²/R.

💡 NEET TIPS & SHORTCUTS

• For circular orbits, v ∝ 1/√r, T ∝ r³/², KE ∝ 1/r, PE ∝ –1/r.
• If orbital radius increases, speed decreases but time period increases.
• Escape velocity is √2 times orbital velocity.
• At the center of Earth, g = 0 but gravitational potential is minimum (most negative).

⚠️ COMMON MISTAKES

• Using g = GM/r² for points inside Earth – only valid outside.
• Confusing gravitational potential energy with potential (V = U/m).
• Forgetting that satellites in lower orbits have higher speeds.
• Applying Kepler's 3rd law for non-circular orbits without using semi-major axis.

Newton's law: F = GMm/r²

Gravitational field: g = GM/r²

Potential: V = –GM/r

Orbital velocity: v_o = √(GM/r)

Escape velocity: v_esc = √(2GM/r)

Time period: T = 2π√(r³/GM)