Concept Core
From Newton's law to satellites — the universe in one chapter.
Newton's Law of Universal Gravitation
Every mass attracts every other mass with force:
F = GMm/r² (G = 6.674×10⁻¹¹ N·m²/kg²)
Force is along the line joining the two masses. It is attractive, conservative, and central. It obeys Newton's 3rd law — both masses experience equal, opposite forces.
Acceleration Due to Gravity
At Earth's surface: g = GM/R² ≈ 9.8 m/s²
Variation with height h above surface:
g_h = g(1 − 2h/R) (h << R)
Variation with depth d below surface:
g_d = g(1 − d/R)
At centre: g = 0. g decreases both above and below surface.
Gravitational Potential Energy
PE of mass m at distance r from Earth's centre:
U = −GMm/r (negative, zero at infinity)
At surface: U = −GMm/R = −mgR
The negative sign means work must be done against gravity to move mass away from Earth.
Escape Velocity
Minimum speed to escape Earth's gravity entirely (reach r = ∞ with zero KE):
v_e = √(2GM/R) = √(2gR) ≈ 11.2 km/s
Independent of mass of the escaping object. No angle dependence — same in any direction.
Orbital Velocity & Satellites
For circular orbit at radius r from Earth's centre:
v_o = √(GM/r) = √(gR²/r)
At surface (r=R): v_o = √(gR) ≈ 7.9 km/s. Note: v_e = √2 × v_o
Time period: T = 2πr/v_o = 2π√(r³/GM)
Kepler's Three Laws
1st Law (Ellipse): Planets move in ellipses with the Sun at one focus.
2nd Law (Equal Areas): Line from Sun to planet sweeps equal areas in equal times. → Planet moves fastest at perihelion (closest), slowest at aphelion (farthest).
3rd Law (T² ∝ r³): T² = 4π²r³/GM. The ratio T²/r³ is constant for all planets around the same star.
Formula Vault
Every gravitation formula for EAPCET.
Newton's Gravity
F = GMm/r²
G = 6.674×10⁻¹¹ N·m²/kg²
g at Surface
g = GM/R²
g ≈ 9.8 m/s² at Earth's surface
g at Height h
g_h = g(1−2h/R)
Valid for h << R
g at Depth d
g_d = g(1−d/R)
g=0 at Earth's centre
Gravitational PE
U = −GMm/r
Negative; zero at infinity
Escape Velocity
v_e = √(2gR) = √(2GM/R)
≈11.2 km/s from Earth
Orbital Velocity
v_o = √(GM/r) = √(gR²/r)
v_e = √2 × v_o at surface
Orbital Period
T = 2π√(r³/GM)
Kepler's 3rd: T²∝r³
Kepler's 3rd Law
T²/r³ = 4π²/GM = const
Same for all planets of a star
Binding Energy
BE = GMm/2r
Energy to remove satellite from orbit
Worked Examples
5 problems — gravity variation, escape velocity, satellites, and Kepler.
EasyFind g at height h = R above Earth's surface▾
Find the value of g at height h = R (one Earth radius above surface). g at surface = 9.8 m/s².
1
At height h, exact formula: g_h = g×R²/(R+h)²
2
h = R: g_h = g×R²/(2R)² = g/4 = 9.8/4 = 2.45 m/s²
✓ g at height R = g/4 = 2.45 m/s²
EasyEscape velocity of a planet with half Earth's radius and same mass▾
A planet has mass = M (Earth's mass) and radius = R/2. Find its escape velocity.
1
v_e = √(2GM/R_planet) = √(2GM/(R/2)) = √(4GM/R) = 2√(GM/R) = 2 × 11.2 = 22.4 km/s
✓ v_e = 2 × Earth's escape velocity = 22.4 km/s
MediumFind orbital velocity and time period of satellite at height R above surface▾
Find orbital speed and period of a satellite orbiting at height h = R above Earth's surface.
1
Orbital radius r = R + R = 2R
2
v_o = √(gR²/r) = √(gR²/2R) = √(gR/2) = v_surface/√2 = 7.9/√2 ≈ 5.6 km/s
3
T = 2πr/v = 2π(2R)/v_o. Using T = 2π√(r³/GM): T = 2π√(8R³/GM) = 2√2 × T_surface
✓ v_o = 5.6 km/s, T = 2√2 × 90 min ≈ 4 hours
EAPCET LevelPlanet A has T = 8 years, planet B has T = 1 year. Find ratio of their orbital radii.▾
Two planets orbit the same star. Planet A has period T_A = 8 years, planet B has T_B = 1 year. Find r_A/r_B.
1
Kepler's 3rd Law: T² ∝ r³ → (T_A/T_B)² = (r_A/r_B)³
2
(8/1)² = (r_A/r_B)³ → 64 = (r_A/r_B)³
✓ r_A/r_B = 4 (Planet A is 4× farther from the star)
Trap QuestionEscape velocity depends on the angle of launch — True or False?▾
An object is launched vertically at escape velocity. Would it also escape if launched at 45°? ⚠️ Common misconception.
1
The trap: Students think angle matters because the vertical component must overcome gravity.
2
Escape velocity is derived from energy conservation: ½mv_e² = GMm/R. Energy is a scalar — no direction.
3
The same minimum KE = GMm/R is needed regardless of launch direction.
4
At 45°, the object escapes along a parabolic path, but the minimum speed required is still v_e = √(2gR).
✓ False — escape velocity is the same in any direction (energy, not force condition)
Mistake DNA
4 gravitation errors from EAPCET distractor analysis.
📏
Using g_h = g(1−2h/R) for Large Heights
This approximation is valid only when h ≪ R. For h = R or more, use the exact formula g_h = g·R²/(R+h)².
❌ Wrong
g at h=R:
g(1−2R/R) = g(−1) = −g ✗
(gives negative g!)
✓ Correct
g_h = gR²/(R+h)²
= gR²/4R² = g/4 ✓
Always use exact
for large h
The approximation (1−2h/R) is a first-order Taylor expansion valid for h/R ≪ 1. For h = R, the exact formula gives g/4, not −g.
🚀
Confusing Escape Velocity and Orbital Velocity
v_escape = √2 × v_orbital at the same radius. They are related but different.
❌ Wrong
v_e ≈ 7.9 km/s (orbital) ✗
v_o ≈ 11.2 km/s (escape) ✗
(swapped!)
✓ Correct
v_o = √(GM/R) ≈ 7.9 km/s ✓
v_e = √(2GM/R) ≈ 11.2 km/s ✓
v_e = √2 × v_o
Orbital velocity keeps a satellite in circular orbit. Escape velocity breaks free from gravity entirely. v_escape/v_orbital = √2 ≈ 1.41.
⬆️
Thinking g Increases as You Go Deeper Into Earth
g decreases linearly from surface to centre as depth increases — not increases.
❌ Wrong
Deeper = more gravity
because more mass
below ✗
✓ Correct
g_d = g(1−d/R) ✓
g decreases to zero
at Earth's centre ✓
Going deeper, the outer shell above contributes no net gravity (shell theorem). Only the mass of the sphere below the depth matters, and that decreases.
🪐
Applying Kepler's 3rd Law to Different Stars
T²/r³ = constant holds only for planets orbiting the SAME central body. Different stars have different constants.
❌ Wrong
Planet of star A and
planet of star B:
T²/r³ same for both ✗
✓ Correct
T²/r³ = 4π²/GM_star ✓
Different stars →
different constants ✓
The constant in Kepler's 3rd law is 4π²/GM where M is the mass of the central body. Only compare planets of the same star.
Chapter Intelligence
Gravitation connects directly to circular motion, energy, and SHM.
EAPCET Weightage (2019–2024)
Escape/orbital velocity~8 g variation with height/depth~6 Gravitational PE & binding energy~4 Satellites & geostationary orbit~3 High-Yield PYQ Patterns
g at height nR above surfaceEscape velocity from another planetKepler's 3rd: ratio of r from TOrbital velocity at given heightg at depth dv_e = √2 × v_orbital
Exam Strategy
- For g variation: use exact formula g_h = gR²/(R+h)² for height questions. Use approximate g(1−2h/R) only when explicitly told h ≪ R.
- Escape velocity questions often change planet mass or radius — use v_e = √(2GM/R) = √(2gR) and substitute what changed.
- Kepler's 3rd Law: (T₁/T₂)² = (r₁/r₂)³. Take the ratio — no need to know G or M of the star.
- Satellites: orbital KE = GMm/2r, PE = −GMm/r, Total E = −GMm/2r. Total energy is negative and half the magnitude of PE.
- Geostationary satellite: T = 24 hours (same as Earth's rotation), r ≈ 42,000 km, v_o ≈ 3.07 km/s. Fixed point in sky.