Ch 05: Gravitation | Exam Fusion Prep

⭐ 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)

Orbital Vel. (km/s)

Gravitation = Force of attraction between any two bodies in the universe
Proponent: Isaac Newton
Universal Gravitational Constant G discovered by Cavendish (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
Discovered by: 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

📉 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	Effective mass of Earth reduces
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: g decreases linearly.

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

Law 1 = Orbits → Elliptical (NOT circular)
Law 2 = Areas → Equal area in equal time → Areal speed is constant
Law 3 = Periods → T² ∝ a³
Sun is at focus of the ellipse (not centre)
Law 2 is a consequence of Conservation of Angular Momentum

🛰️ 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

24 Hours

Height from Surface

36,000 km

Direction

West → East

Appears stationary relative to Earth's surface
Period = Earth's rotation period = 24 hours
Height = 36,000 km above Earth's surface
Rotates in the equatorial plane
Used for: TV broadcast, weather monitoring, communication
Examples: INSAT series (India)

⚠️ Exam Trap — Geostationary vs Polar

Geostationary → 36,000 km height, 24 hr 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 (Vₑ) = 11.2 km/s = √2 × 8 ≈ 11.31 km/s
If satellite speed increased by √2 times (~41%), it escapes Earth's gravity
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
If velocity < 11.2 km/s → object falls back to Earth
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-Frequency BPSC/BSSC Exam Points

Value of G = 6.67 × 10⁻¹¹ Nm²/kg² — discovered by Cavendish (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 = 36,000 km, period = 24 hours
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.	36,000 km
Geostationary Period	24 hours

📝 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

G (universal constant) ≠ g (varies with location)
Mass ≠ Weight. Mass never changes; weight changes with g.
Kepler's orbits are elliptical, NOT circular
Escape velocity does NOT depend on mass of the projectile
Satellite in orbit → weightless (apparent weight = 0), NOT massless
Geostationary satellite period = 24 hrs, height = 36,000 km (not 3,600 km!)
g is maximum at poles → NOT at equator

⚡ Chapter 05: Gravitation