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Formulas & Dimensional Analysis
Every formula you need — verified dimensionally, tagged by exam level. Searchable. No gaps.
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Newton's Law of Gravitation
F = Gm₁m₂/r²
[F] = [G][M]²[L]⁻² → G = [M⁻¹L³T⁻²]
Vector form: F⃗ = -Gm₁m₂r̂/r² (attractive)
Acceleration due to Gravity
g = GM/R²
[g] = [M⁰L¹T⁻²]
g = 9.8 m/s² on Earth's surface. Also: g = 4πGρR/3
g at Height h
g_h = g(1 − 2h/R) [h<<R]
g_h = GM/(R+h)² [exact]
h<<R: use approx. h~R: use exact. Always verify which one applies.
g at Depth d
g_d = g(1 − d/R)
Linear decrease. At d=R (centre): g=0. At d=R/2: g=g/2.
g due to Earth's Rotation
g' = g − ω²R·cos²λ
λ = latitude. At equator: g' = g − ω²R. At pole: g' = g (no change).
Kepler's Third Law
T² = 4π²a³/GM
a = semi-major axis. For circular: a = r. Ratio: T₁²/T₂² = r₁³/r₂³
Escape Velocity
Vₑ = √(2GM/R) = √(2gR)
For Earth: Vₑ ≈ 11.2 km/s. Vₑ = √2 × v_orbital(surface)
Escape Velocity (density form)
Vₑ = R√(8πGρ/3)
Use when density is given instead of mass. Critical for planet comparison problems.
Orbital Velocity
v₀ = √(GM/r) = √(gR²/r)
r = R+h. Near Earth surface: v₀ ≈ 7.9 km/s. Independent of satellite mass.
Orbital Time Period
T = 2π√(r³/GM) = 2πr/v₀
Near Earth: T ≈ 84 min. Geostationary: T = 24h → r ≈ 42,000 km
Satellite Total Energy
E = −GMm/2r
KE = +GMm/2r | PE = −GMm/r
|KE| = |E|, PE = 2E, KE = −½PE. Total energy is always negative.
Gravitational Potential Energy
U = −GMm/r
Reference at ∞. Always negative. Near surface: ΔU = mgh (for small h)
Gravitational Potential
V = −GM/r
[V] = [M⁰L²T⁻²] = J/kg
U = mV. At surface: V_s = −GM/R. Relation: g = −dV/dr
Potential Inside Solid Sphere
V = −GM(3R²−r²)/2R³
At centre (r=0): V_c = −3GM/2R = 3/2 × V_surface. JEE Advanced level.
Field Inside Solid Sphere
g_inside = GMr/R³
Increases linearly from centre. At r=R: g = GM/R². Shell contributes nothing (shell theorem).
Binding Energy (Satellite)
BE = GMm/2r = −E_total
Minimum energy to remove satellite from orbit to infinity. Also = mVₑ²/2 for surface.
Angular Momentum of Satellite
L = mv₀r = m√(GMr)
Conserved in orbit. Kepler's 2nd law: dA/dt = L/2m = constant
Weight on Moon vs Earth
g_moon = g_earth/6
Due to smaller mass and radius. Weight on Moon = 1/6 × weight on Earth.
Dimensional Analysis
Verify every formula. Dimension problems appear in JEE Main directly.
| Quantity | Symbol | Formula | Dimensions | SI Unit |
|---|---|---|---|---|
| Gravitational Constant | G | Fr²/m₁m₂ | [M⁻¹L³T⁻²] | N·m²·kg⁻² |
| Gravitational Field Intensity | g or I | GM/r² | [M⁰L¹T⁻²] | m/s² or N/kg |
| Gravitational Potential | V | −GM/r | [M⁰L²T⁻²] | J/kg |
| Gravitational Potential Energy | U | −GMm/r | [M¹L²T⁻²] | J |
| Orbital Velocity | v₀ | √(GM/r) | [M⁰L¹T⁻¹] | m/s |
| Escape Velocity | Vₑ | √(2GM/R) | [M⁰L¹T⁻¹] | m/s |
| Time Period | T | 2π√(r³/GM) | [M⁰L⁰T¹] | s |
| Angular Momentum | L | mvr | [M¹L²T⁻¹] | kg·m²/s |
Exam Insight: Dimensional Analysis Questions
JEE Main asks: "Find dimension of G/g" → [M⁻¹L³T⁻²]/[LT⁻²] = [M⁻¹L²]. Also: "G has same dimensions as ___ × ___" → common trap question. Know all dimensions cold.
Interactive Calculators
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🔭 Gravitational Force
🚀 Escape Velocity
🛸 Orbital Parameters
Key Constants & Values
6.674×10⁻¹¹
G (N·m²/kg²)
Universal Gravitational Constant
9.8
g_Earth (m/s²)
Surface acceleration (equator)
11.2
Vₑ Earth (km/s)
Escape velocity from Earth
7.9
v₀ Surface (km/s)
Near-Earth orbital speed
5.972×10²⁴
Mₑ (kg)
Mass of Earth
6.371×10⁶
Rₑ (m)
Mean radius of Earth
36,000
h_geo (km)
Geostationary orbit altitude
84 min
T near Earth
Orbital period (near surface)