Concept Core
From Newton's law to satellites β the universe in one chapter.
Newton's Law of Universal Gravitation
Every mass attracts every other mass with force:
F = GMm/rΒ² (G = 6.674Γ10β»ΒΉΒΉ NΒ·mΒ²/kgΒ²)
Force is along the line joining the two masses. It is attractive, conservative, and central. It obeys Newton's 3rd law β both masses experience equal, opposite forces.
Acceleration Due to Gravity
At Earth's surface: g = GM/RΒ² β 9.8 m/sΒ²
Variation with height h above surface:
g_h = g(1 β 2h/R) (h << R)
Variation with depth d below surface:
g_d = g(1 β 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 = βGMm/r (negative, zero at infinity)
At surface: U = βGMm/R = βmgR
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 = β with zero KE):
v_e = β(2GM/R) = β(2gR) β 11.2 km/s
Independent of mass of the escaping object. No angle dependence β same in any direction.
Orbital Velocity & Satellites
For circular orbit at radius r from Earth's centre:
v_o = β(GM/r) = β(gRΒ²/r)
At surface (r=R): v_o = β(gR) β 7.9 km/s. Note: v_e = β2 Γ v_o
Time period: T = 2Οr/v_o = 2Οβ(rΒ³/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. β Planet moves fastest at perihelion (closest), slowest at aphelion (farthest).
3rd Law (TΒ² β rΒ³): TΒ² = 4ΟΒ²rΒ³/GM. The ratio TΒ²/rΒ³ is constant for all planets around the same star.
Formula Vault
Every gravitation formula for EAPCET.
Newton's Gravity
F = GMm/rΒ²
G = 6.674Γ10β»ΒΉΒΉ NΒ·mΒ²/kgΒ²
g at Surface
g = GM/RΒ²
g β 9.8 m/sΒ² at Earth's surface
g at Height h
g_h = g(1β2h/R)
Valid for h << R
g at Depth d
g_d = g(1βd/R)
g=0 at Earth's centre
Gravitational PE
U = βGMm/r
Negative; zero at infinity
Escape Velocity
v_e = β(2gR) = β(2GM/R)
β11.2 km/s from Earth
Orbital Velocity
v_o = β(GM/r) = β(gRΒ²/r)
v_e = β2 Γ v_o at surface
Orbital Period
T = 2Οβ(rΒ³/GM)
Kepler's 3rd: TΒ²βrΒ³
Kepler's 3rd Law
TΒ²/rΒ³ = 4ΟΒ²/GM = const
Same for all planets of a star
Binding Energy
BE = GMm/2r
Energy to remove satellite from orbit
Worked Examples
5 problems β gravity variation, escape velocity, satellites, and Kepler.
EasyFind g at height h = R above Earth's surfaceβΎ
Find the value of g at height h = R (one Earth radius above surface). g at surface = 9.8 m/sΒ².
1
At height h, exact formula: g_h = gΓRΒ²/(R+h)Β²
2
h = R: g_h = gΓRΒ²/(2R)Β² = g/4 = 9.8/4 = 2.45 m/sΒ²
β g at height R = g/4 = 2.45 m/sΒ²
EasyEscape velocity of a planet with half Earth's radius and same massβΎ
A planet has mass = M (Earth's mass) and radius = R/2. Find its escape velocity.
1
v_e = β(2GM/R_planet) = β(2GM/(R/2)) = β(4GM/R) = 2β(GM/R) = 2 Γ 11.2 = 22.4 km/s
β v_e = 2 Γ Earth's escape velocity = 22.4 km/s
MediumFind orbital velocity and time period of satellite at height R above surfaceβΎ
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 = β(gRΒ²/r) = β(gRΒ²/2R) = β(gR/2) = v_surface/β2 = 7.9/β2 β 5.6 km/s
3
T = 2Οr/v = 2Ο(2R)/v_o. Using T = 2Οβ(rΒ³/GM): T = 2Οβ(8RΒ³/GM) = 2β2 Γ T_surface
β v_o = 5.6 km/s, T = 2β2 Γ 90 min β 4 hours
EAPCET LevelPlanet A has T = 8 years, planet B has T = 1 year. Find ratio of their orbital radii.βΎ
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: TΒ² β rΒ³ β (T_A/T_B)Β² = (r_A/r_B)Β³
2
(8/1)Β² = (r_A/r_B)Β³ β 64 = (r_A/r_B)Β³
β r_A/r_B = 4 (Planet A is 4Γ farther from the star)
Trap QuestionEscape velocity depends on the angle of launch β True or False?βΎ
An object is launched vertically at escape velocity. Would it also escape if launched at 45Β°? β οΈ Common misconception.
1
The trap: Students think angle matters because the vertical component must overcome gravity.
2
Escape velocity is derived from energy conservation: Β½mv_eΒ² = GMm/R. Energy is a scalar β no direction.
3
The same minimum KE = GMm/R is needed regardless of launch direction.
4
At 45Β°, the object escapes along a parabolic path, but the minimum speed required is still v_e = β(2gR).
β False β escape velocity is the same in any direction (energy, not force condition)
Mistake DNA
4 gravitation errors from EAPCET distractor analysis.
π
Using g_h = g(1β2h/R) for Large Heights
This approximation is valid only when h βͺ R. For h = R or more, use the exact formula g_h = gΒ·RΒ²/(R+h)Β².
β Wrong
g at h=R:
g(1β2R/R) = g(β1) = βg β
(gives negative g!)
β Correct
g_h = gRΒ²/(R+h)Β²
= gRΒ²/4RΒ² = g/4 β
Always use exact
for large h
The approximation (1β2h/R) is a first-order Taylor expansion valid for h/R βͺ 1. For h = R, the exact formula gives g/4, not βg.
π
Confusing Escape Velocity and Orbital Velocity
v_escape = β2 Γ v_orbital at the same radius. They are related but different.
β Wrong
v_e β 7.9 km/s (orbital) β
v_o β 11.2 km/s (escape) β
(swapped!)
β Correct
v_o = β(GM/R) β 7.9 km/s β
v_e = β(2GM/R) β 11.2 km/s β
v_e = β2 Γ v_o
Orbital velocity keeps a satellite in circular orbit. Escape velocity breaks free from gravity entirely. v_escape/v_orbital = β2 β 1.41.
β¬οΈ
Thinking g Increases as You Go Deeper Into Earth
g decreases linearly from surface to centre as depth increases β not increases.
β Wrong
Deeper = more gravity
because more mass
below β
β Correct
g_d = g(1βd/R) β
g decreases to zero
at Earth's centre β
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.
πͺ
Applying Kepler's 3rd Law to Different Stars
TΒ²/rΒ³ = constant holds only for planets orbiting the SAME central body. Different stars have different constants.
β Wrong
Planet of star A and
planet of star B:
TΒ²/rΒ³ same for both β
β Correct
TΒ²/rΒ³ = 4ΟΒ²/GM_star β
Different stars β
different constants β
The constant in Kepler's 3rd law is 4ΟΒ²/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 (2019β2024)
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 = β2 Γ v_orbital
Exam Strategy
- For g variation: use exact formula g_h = gRΒ²/(R+h)Β² for height questions. Use approximate g(1β2h/R) only when explicitly told h βͺ R.
- Escape velocity questions often change planet mass or radius β use v_e = β(2GM/R) = β(2gR) and substitute what changed.
- Kepler's 3rd Law: (Tβ/Tβ)Β² = (rβ/rβ)Β³. Take the ratio β no need to know G or M of the star.
- Satellites: orbital KE = GMm/2r, PE = βGMm/r, Total E = βGMm/2r. Total energy is negative and half the magnitude of PE.
- Geostationary satellite: T = 24 hours (same as Earth's rotation), r β 42,000 km, v_o β 3.07 km/s. Fixed point in sky.
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