Gravitation

What you'll learn

Universal Law of Gravitation — Newton's formula; what G is.
The difference between G (universal constant) and g (acceleration due to gravity).
Why objects fall with the same acceleration regardless of mass.
Buoyancy and Archimedes' Principle — why things float or sink.

Key concepts

Universal Law of Gravitation (Newton, 1687)

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 = G × m₁ × m₂ / r²

Symbol	Meaning	Value/Unit
F	Gravitational force	N
G	Universal gravitational constant	6.674 × 10⁻¹¹ N·m²/kg²
m₁, m₂	Masses of two objects	kg
r	Distance between centres	m

G is the same everywhere in the universe.
Discovered by Cavendish (1798) experimentally.
This law explains: falling objects, Moon's orbit, tides, planetary motion.

Key relationships:

Double the mass of one object → F doubles.
Double the distance → F reduces to 1/4 (inverse square law).
Triple the distance → F reduces to 1/9.

Why the Moon orbits Earth

Earth's gravity pulls Moon inward (centripetal force).
Moon's tangential velocity prevents it from falling → it orbits.
Same principle: Earth orbits Sun; satellites orbit Earth.

Free fall and acceleration due to gravity (g)

Free fall: motion of an object under gravity alone (no other force).
All objects fall with the same acceleration in vacuum regardless of mass.
- Proof: F = GMm/r² and F = ma → a = GM/r² (m cancels out!)
g = acceleration due to gravity near Earth's surface = 9.8 m/s² ≈ 10 m/s².

g = GM_Earth / R_Earth²

Where M_Earth = 6 × 10²⁴ kg, R_Earth = 6.4 × 10⁶ m.

g varies with location:

Location	g value	Reason
Poles	Slightly higher	Earth is flatter at poles → closer to centre
Equator	Slightly lower	Earth bulges at equator → farther from centre
Mountains	Lower	Greater distance from Earth's centre
Moon	1.63 m/s² (1/6 of Earth)	Moon's mass is much less
Deep mines	Slightly lower	Less mass below

Weight vs Mass

	Mass	Weight
Definition	Amount of matter	Force of gravity on the object
Formula	—	W = mg
Unit	kg	N (Newtons)
Constant?	Yes — same everywhere	Changes with location (varies with g)
Zero gravity	Has mass	Weightless (W = 0)

A 70 kg person on Earth: W = 70 × 9.8 = 686 N.
Same person on Moon: W = 70 × 1.63 = 114 N (1/6 of Earth weight).
In space (g ≈ 0): W ≈ 0 → weightlessness.

Equations of motion under gravity

Using kinematic equations with a = g (downward):

Situation	Equation
Object falling (downward +ve)	v = u + gt; s = ut + ½gt²; v² = u² + 2gs
Object thrown upward	Replace g with −g; at max height, v = 0
Time to reach max height	t = u/g
Max height	H = u²/2g

Buoyancy

Buoyant force (upthrust): upward force exerted by a fluid (liquid or gas) on an object immersed in it.
Direction: always upward (opposite to gravity).
Cause: pressure increases with depth → pressure on bottom face > pressure on top face → net upward force.

Factors affecting buoyant force:

Volume of fluid displaced (greater volume submerged → greater buoyant force).
Density of the fluid (denser fluid → greater buoyant force).
Does NOT depend on the mass or density of the object itself.

Archimedes' Principle

When an object is immersed in a fluid (fully or partially), it experiences an upward buoyant force equal to the weight of the fluid displaced.

Buoyant force = weight of fluid displaced = ρ_fluid × V_displaced × g

Legend: Archimedes (287–212 BC) discovered this while stepping into a bath — shouted "Eureka!" (I have found it!) as he realised displaced water volume equals submerged volume.

Applications:

Ship design: steel is denser than water, but a ship's hull traps air → average density < water → floats.
Submarines: fill ballast tanks with water (increase density → sink); pump out water (decrease density → rise).
Hot air balloon: hot air is less dense than cold air → balloon floats.
Lactometer: measures density of milk (detects adulteration).
Hydrometer: measures density of liquids (battery acid, alcohol).
Life jackets: low-density material → large buoyant force.

Sinking vs floating

Condition	What happens
Object density > fluid density	Sinks (weight > buoyant force)
Object density = fluid density	Neutral buoyancy (floats submerged)
Object density < fluid density	Floats (partially submerged; buoyant force = weight)

Iron: density 7874 kg/m³ > water 1000 kg/m³ → iron sinks.
Wood: density ~500–900 kg/m³ < water → wood floats.
Ice: density 917 kg/m³ < water → ice floats (about 90% submerged).

Quick check

State the Universal Law of Gravitation. Write the formula.
What is the difference between G and g?
Why does a feather and a hammer fall at the same rate in vacuum?
State Archimedes' Principle. Why does a ship float even though it's made of steel?
A 5 kg stone is dropped from a height. Find: (a) its weight (b) velocity after 3 s of free fall (g = 10 m/s²).

Open the Practice tab for graded questions on Gravitation.