General Science · Physics

⚡ Chapter 05: Gravitation

SSC / Railway / UPSC / BPSC · Exam Fusion Prep  |  Newton's Law · Kepler's Laws · Satellites · Escape Velocity

⭐ Gravitation — At a Glance
9.8
g on Earth (m/s²)
6.67×10⁻¹¹
G (Nm²/kg²)
11.2
Escape Vel. Earth (km/s)
8
Orbital Vel. (km/s)
  • Gravitation = Force of attraction between any two bodies in the universe
  • Proponent: Isaac Newton
  • Determination of G is conventionally credited to Cavendish’s torsion-balance experiment (1798)
  • Key topics: Newton's Law · g variation · Kepler's Laws · Satellites · Escape Velocity
📌 Key Formulas
F = G(m₁m₂) / r²Newton's Gravitational Force
g = GMₑ / Rₑ²Gravitational Acceleration
Vₒ = √(gR)Orbital Velocity
Vₑ = √(2gR)Escape Velocity
🗂️ Chapter Topics
  • Newton's Universal Law of Gravitation
  • Gravitational Constant (G) vs Acceleration (g)
  • Change in value of g (altitude, depth, latitude)
  • Kepler's 3 Laws of Planetary Motion
  • Mass vs Weight
  • Satellites & Geostationary Orbit
  • Escape Velocity
🍎 Newton's Universal Law of Gravitation
📘 Definition

Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

F ∝ m₁ × m₂ / r² Proportional form
F = G × (m₁ × m₂) / r² Full formula with G
🔢 Gravitational Constant (G)
  • Symbol: G
  • Value: 6.67 × 10⁻¹¹ Nm²/kg²
  • Torsion-balance measurement credited to: Henry Cavendish (1798)
  • Universal constant — same everywhere in universe
  • Does NOT change with location or mass
⚠️ Exam Trap

G (Gravitational Constant) ≠ g (Gravitational Acceleration). G is universal & fixed; g changes with location.

📐 Gravitational Acceleration (g)
  • Symbol: g
  • Standard value: 9.8 m/s² (approx. 10 m/s²)
  • Formula: g = GMₑ / Rₑ²
  • Does NOT depend on mass of the falling object
  • Depends on: mass of Earth & distance from centre
Earth
9.8 m/s²
Moon
1/6 of Earth
Centre of Earth
0
📉 Change in Value of g

The value of g is NOT constant — it varies with position on/in Earth.

Location g Value Reason
Poles Maximum Closer to Earth's centre (Earth is oblate)
Equator Minimum Farthest from Earth's centre + rotation effect
Going Up (altitude) Decreases Distance from centre increases
Going Down (depth) Decreases Only enclosed mass contributes; linear depth relation assumes uniform density
Centre of Earth Zero (0) Equal pull from all directions
Moon's Surface 1/6 of Earth Moon has less mass & radius
⚠️ Exam Trap — Poles vs Equator
  • g is maximum at poles — frequently asked!
  • g is minimum at equator
  • Weight of a body at poles > weight at equator (because W = mg)
  • At centre of Earth, g = 0, so Weight = 0 but Mass ≠ 0
📘 Key Trick

g ∝ 1/r² → As distance from centre increases (going up), g decreases.


At depth d in a uniform-density spherical Earth model: g = g₀(1 − d/R).


Moon: g_moon = g_earth / 6 → Weight on Moon = 1/6 of Earth weight

🪐 Kepler's Laws of Planetary Motion

Johannes Kepler gave 3 laws describing how planets orbit the Sun.

Law 1
Law of Orbits

Every planet moves in an elliptical orbit around the Sun, with the Sun at one of the two foci of the ellipse.

Law 2
Law of Areal Speed (Equal Areas)

The imaginary line joining the planet to the Sun sweeps equal areas in equal time intervals. → Planet moves faster when closer to the Sun (perihelion) and slower when farther (aphelion).

Law 3
Law of Period of Revolution (Harmonic Law)

The square of the orbital period (T²) of a planet is proportional to the cube of the semi-major axis (a³) of its orbit.

T² ∝ a³Kepler's Third Law Formula
⚠️ Exam Trap — Kepler's Laws
🛰️ Satellites
📘 Definition

Bodies revolving around a planet are called satellites. Moon is a natural satellite of Earth. INSAT, GPS etc. are artificial satellites.

  • Orbital Velocity (nearest to Earth's surface): Vₒ = √(gR)
  • Value: 8 km/s (approx.)
  • A satellite in orbit is in a state of free fall
  • Satellite feels weightlessness (apparent weight = 0)
🌐 Geostationary Satellite
Time Period
23 h 56 min (approximately 24 h)
Height from Surface
35,786 km (approximately 36,000 km)
Direction
West → East
  • Appears stationary relative to Earth's surface
  • Period = Earth's rotation period = 23 h 56 min (approximately 24 h)
  • Height = 35,786 km (approximately 36,000 km) above Earth's surface
  • Circular, prograde orbit in the equatorial plane
  • Used for: TV broadcast, weather monitoring, communication
  • Examples: INSAT series (India)
⚠️ Exam Trap — Geostationary vs Polar
  • Geostationary → 35,786 km (approximately 36,000 km) height, 23 h 56 min period, equatorial orbit
  • Polar satellite → Low orbit (~500–800 km), passes over poles, used for spy/remote sensing
🚀 Escape Velocity
📘 Definition

The minimum velocity with which a body, when thrown upward from Earth's surface, can escape the gravitational field of Earth permanently (without further propulsion).

Vₑ = √(2gR)Escape Velocity Formula
11.2 km/s
Escape Vel. — Earth
2.35 km/s
Escape Vel. — Moon
√2 × Vₒ
Vₑ in terms of Vₒ
🔗 Relation: Escape Velocity vs Orbital Velocity
Vₑ = √2 × VₒEscape Velocity = √2 times Orbital Velocity
  • Orbital Velocity (Vₒ) = 8 km/s
  • Escape velocity ≈ 11.2 km/s; √2 × 7.9 ≈ 11.2 km/s (use unrounded values in calculations)
  • At a given radius, escape speed is √2 times local circular-orbit speed; sufficient impulse can change a circular orbit to an escape trajectory
  • Moon has lower escape velocity → atmosphere not retained (gases escape)
⚠️ Exam Trap
  • Escape velocity does NOT depend on mass of the object thrown
  • Escape velocity depends on mass and radius of the planet
  • Below local escape speed, the unpowered trajectory is gravitationally bound; it may orbit or intersect Earth depending on direction and angular momentum
  • If velocity = 11.2 km/s → object just escapes Earth
  • If velocity > 11.2 km/s → object leaves with kinetic energy to spare
⚖️ Mass vs Weight
Property Mass Weight
Definition Intrinsic/inherent property of a body (amount of matter) Force of attraction on object towards Earth's centre
Symbol m W
Formula — W = mg
Unit kg (kilogram) Newton (N)
Type Scalar Vector
Changes? Never changes Changes with g (location)
At Earth's centre Same Zero (g=0)
On Moon Same 1/6 of Earth weight
W = m × gWeight Formula — W in Newtons, m in kg, g in m/s²
⚠️ Exam Trap — Mass vs Weight
  • Mass is same everywhere; Weight changes with location
  • In space (far from all bodies), weight = 0 but mass ≠ 0
  • "Weightlessness" means apparent weight is zero — NOT that mass is zero
  • A spring balance measures Weight; a beam balance measures Mass
🎯 High-Yield Exam Points
  • G ≈ 6.67 × 10⁻¹¹ Nm²/kg²; Cavendish’s torsion-balance experiment (1798)
  • Value of g = 9.8 m/s² (use 10 m/s² for calculations)
  • g is maximum at poles, minimum at equator
  • g = 0 at centre of Earth
  • g on Moon = 1/6 of Earth's g
  • Escape velocity of Earth = 11.2 km/s
  • Escape velocity of Moon = 2.35 km/s
  • Orbital velocity = 8 km/s
  • Vₑ = √2 × Vₒ
  • Geostationary satellite → height = 35,786 km (approximately 36,000 km), period = 23 h 56 min (approximately 24 h)
  • Kepler's Law 1 = Ellipse, Law 2 = Equal Areas, Law 3 = T² ∝ a³
  • Weight = 0 at centre of Earth (g=0), but Mass remains same
🔢 Important Values Quick Table
Quantity Value
G 6.67 × 10⁻¹¹ Nm²/kg²
g (Earth) 9.8 m/s²
g (Moon) ~1.63 m/s² (1/6 of Earth)
g (Centre) 0
Orbital Velocity 8 km/s
Escape Vel. Earth 11.2 km/s
Escape Vel. Moon 2.35 km/s
Geostationary Ht. 35,786 km (approximately 36,000 km)
Geostationary Period 23 h 56 min (approximately 24 h)
📝 Formula Summary
F = G(m₁m₂)/r²Newton's Gravitational Law
g = GMₑ/Rₑ²Gravitational Acceleration
W = mgWeight
T² ∝ a³Kepler's 3rd Law
Vₒ = √(gR)Orbital Velocity
Vₑ = √(2gR) = √2 × VₒEscape Velocity
⚠️ Most Common Exam Traps
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