Gravitation
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Newton's Law & Gravitational Field
Newton's Law of Gravitation
F = G m₁m₂ / r²
G ≈ 6.67 × 10⁻¹¹ Nm²/kg²
G ≈ 6.67 × 10⁻¹¹ Nm²/kg²
Gravitational Field (g)
g = F/m = GM / R²
Also called acceleration due to gravity.
Also called acceleration due to gravity.
Vector Form
F⃗₁₂ = -G (m₁m₂ / |r⃗₁₂|³) r⃗₁₂
Force on m₂ by m₁.
Force on m₂ by m₁.
Variation of g (Height)
gₕ = g / (1 + h/R)²
Approx: gₕ ≈ g(1 - 2h/R) for h≪R
Approx: gₕ ≈ g(1 - 2h/R) for h≪R
Variation of g (Depth)
gₔ = g (1 - d/R)
At center (d=R), g = 0.
At center (d=R), g = 0.
Variation of g (Rotation)
g' = g - Rω²cos²λ
λ = latitude, ω = angular velocity.
λ = latitude, ω = angular velocity.
Gravitational Potential & Energy
Gravitational Potential Energy (U)
U = -G m₁m₂ / r
Always negative. Zero at r = ∞.
Always negative. Zero at r = ∞.
Gravitational Potential (V)
V = U/m = -GM / r
Scalar quantity. Unit: J/kg.
Scalar quantity. Unit: J/kg.
Work Done
W = ΔU = U₂ - U₁
W = GMm (1/r₁ - 1/r₂)
W = GMm (1/r₁ - 1/r₂)
Potential on Earth's Surface
Vₛ = -GM / R = -gR
Using g = GM/R².
Using g = GM/R².
Satellites & Orbital Motion
Orbital Velocity (v₀)
v₀ = √(GM / r)
r = R+h (orbital radius)
r = R+h (orbital radius)
Time Period of Satellite (T)
T = 2πr / v₀ = 2π √(r³ / GM)
Implies T² ∝ r³ (Kepler's 3rd Law)
Implies T² ∝ r³ (Kepler's 3rd Law)
Energy of Satellite (Total)
E = K.E. + P.E. = -GMm / 2r
Total energy is negative (bound system).
Total energy is negative (bound system).
Binding Energy
B.E. = -E = +GMm / 2r
Energy required to free the satellite.
Energy required to free the satellite.
Escape Velocity & Kepler's Laws
Escape Velocity (vₑ)
vₑ = √(2GM / R) = √(2gR)
For Earth, vₑ ≈ 11.2 km/s
For Earth, vₑ ≈ 11.2 km/s
Relation: vₑ and v₀
vₑ = √2 v₀
Escape velocity is ~41.4% greater than orbital velocity near surface.
Escape velocity is ~41.4% greater than orbital velocity near surface.
Kepler's 1st Law (Orbits)
Planets move in elliptical orbits with the Sun at one focus.
Kepler's 2nd Law (Areas)
dA/dt = L / 2m = constant
Areal velocity is constant. Implies conservation of angular momentum.
Areal velocity is constant. Implies conservation of angular momentum.
Kepler's 3rd Law (Periods)
T² ∝ a³
a = semi-major axis. For circles, a=r.
a = semi-major axis. For circles, a=r.
BITSAT Special Shortcuts
Ratio Problems
For g: g₁/g₂ = (M₁/M₂)(R₂/R₁)²
For vₑ: v₁/v₂ = √(M₁R₂ / M₂R₁)
For vₑ: v₁/v₂ = √(M₁R₂ / M₂R₁)
Height vs Depth for g
If g decreases by same amount:
g(1-2h/R) = g(1-d/R) ⇒ d = 2h
g(1-2h/R) = g(1-d/R) ⇒ d = 2h
Exam Hall Quick Checks
Remember satellite energy relations: K.E. = -E and P.E. = 2E. If you find one, you know all three.
For ratio questions, cancel G and other constants immediately to save time.
Escape velocity is independent of the mass of the projected body (m), but depends on the planet's mass (M) and radius (R).
If a satellite's orbit radius increases, its K.E. and speed decrease, but its P.E. and Total Energy increase (become less negative).
Kepler's 2nd law implies that a planet moves fastest when it is closest to the sun (perihelion) and slowest when it is farthest (aphelion).
Satellite Energy Breakdown
| Energy Type | Formula (r = orbital radius) | Relation to Total Energy (E) |
|---|---|---|
Kinetic Energy (K.E.) | GMm / 2r | K.E. = -E |
Potential Energy (P.E.) | -GMm / r | P.E. = 2E |
Total Energy (E) | -GMm / 2r | E |
Binding Energy (B.E.) | GMm / 2r | B.E. = -E |