Gravitation Notes

Gravitation

• Gravitation is the force of attraction between any two objects in the universe.

• It is a universal force: it acts everywhere, on all objects, and at all times.

The earth attracts (or pulls) all the objects towards its centre. The force with which the earth pulls the objects towards
it is called the gravitational force of earth or gravity (of earth). It is due to the gravitational force of earth that all the
objects fall towards the earth when released from a height.

The gravitational force of earth (or gravity of earth) is responsible for holding the atmosphere above the earth; for the
rain falling to the earth; and for the flow of water in the rivers. It is also the gravitational force of earth (or gravity of
earth) which keeps us firmly on the ground (and we do not float here and there).

Every Object in the Universe Attracts Every Other Object

According to Newton, every object in this universe attracts every other object with a certain force. The force with
which two objects attract each other is called gravitational force (or gravity). The gravitational force between two
objects acts even if the two objects are not connected by any means.

UNIVERSAL LAW OF GRAVITATION

Now, according to the universal law of gravitation:

Where:

• F = Force of attraction

• m1, m2 = Masses of two objects

• r= Distance between their centers

• G= Universal Gravitational Constant

G = 6.7×10−11 Nm2kg−2
where G is a constant known as “Universal Gravitational Constant”.

Since the gravitational force between two bodies is inversely proportional to the square of the distance between them,
therefore,

if we double the distance between two bodies, the gravitational force becomes one-fourth and

if we halve the distance between two bodies, then the gravitational force becomes four times.

Important Points

• The force is attractive in nature.

• The value of GGG is constant everywhere in the universe.

• Gravitation is a very weak force compared to other fundamental forces (like electrostatic force).

Force of Gravity

If an object of mass mmm is placed at the surface of the Earth:

Free fall

The falling of a body (or object) from a height towards the earth under the gravitational force of earth (with no other
forces acting on it) is called free fall.
 Variation on Earth’s surface:

• On Earth’s surface, g varies with location: greater at poles, smaller at equator → here it increases or

decreases depending on where you go.

• With height (upwards) → g always decreases.

• With depth (inside Earth) → g always decreases.

• The acceleration due to gravity does not depend on the mass of the body, all the bodies (whether heavy or light)

fall with the same acceleration towards the surface of the earth.
 Equations of Motion for Freely Falling Bodies

For uniform acceleration → equations are:

1. v = u + at

2. s = ut + ½ at2

3. 2as = v2 – u2

For freely falling bodies:

• Replace a with g (acceleration due to gravity).

• Replace s with h (height).

Thus, equations become:

1. v = u + gt

2. h = ut+ ½ gt2

3. 2gh = v2 – u2

Important Points

• Falling vertically downward: velocity increases → g=+9.8 m/s2

• Thrown vertically upward: velocity decreases → g=−9.8 m/s2

• Free fall from height: initial velocity u = 0

• Thrown upward: final velocity at highest point v = 0

• Time of rise = time of fall (for the same height).

Special Case: Object dropped from rest (u = 0)

1. 𝑣 = 𝑔 · 𝑡
2. ℎ = (1/2) · 𝑔 · 𝑡²
3. 𝑣² = 2 · 𝑔 · ℎ

Motion when thrown upwards (a = –g):

1. 𝑣 = 𝑢 – 𝑔 · 𝑡
2. ℎ = 𝑢 · 𝑡 – (1/2) · 𝑔 · 𝑡²
3. 𝑣² = 𝑢² – 2 · 𝑔 · ℎ

Maximum Height (H):

𝐴𝑡 𝑣 = 0,
𝐻 = 𝑢² / (2 · 𝑔)
Time of Ascent:

𝑡 = 𝑢/𝑔

Time of Descent (from height H):

𝑡 = √(2𝐻 / 𝑔)

Total Time of Flight (up and down):

𝑇 = (2 · 𝑢) / 𝑔
3. Mass, Weight & Free Fall

Mass

• Mass is the amount of matter contained in a body.

• It is a scalar quantity.

• Unit: kilogram (kg)

• Measured using a beam balance.

• Constant everywhere in the universe (does not change with location).

Weight

• Weight is the force with which Earth attracts a body towards its center.

• Formula:

W = mg
Where:

• m = mass of the body

• g = acceleration due to gravity

• Unit: Newton (N)

• Weight is a vector quantity.

• Measured using a spring balance.

• Weight varies with location because g varies.

In the interplanetary space, where g = 0 the weight of a body becomes zero and we feel true weightlessness.
Thus, the weight of a body can be zero.

Relationship between Mass and Weight

• Mass remains the same everywhere, but weight changes with g.

Mass Weight

Amount of matter in a body Gravitational force on a body

Constant everywhere Changes with location

Scalar quantity Vector quantity

SI Unit: kg SI Unit: N

Measured with beam balance Measured with spring balance

Thrust

• Thrust is the force acting on a surface in a direction perpendicular (normal) to it.

• Example: When you press your hand against a wall, the force you apply is thrust.

• SI Unit: Newton (N).

Pressure

• Pressure is the thrust per unit area.

• Formula:
Force or Thrust or Weight
Pressure =
Area

F
P=
A

• Where:
F = normal force (thrust)
A = area on which the force is applied

• SI Unit: Pascal (Pa)

(1 Pascal = 1 Newton / 1 m²)
Factors Affecting Pressure

1. Force (Thrust): Greater force → greater pressure (if area is constant).

2. Area: For the same force, smaller area → greater pressure.

Examples in Daily Life

• A sharp knife cuts better than a blunt one (smaller area → more pressure).

• Camels walk easily on sand because they have broad feet (larger area → less pressure on sand).

• Nails and pins have sharp tips to apply greater pressure with the same force.

Pressure in Fluids

• Fluids (liquids and gases) exert pressure in all directions.

• Pressure in a fluid increase with depth because of the weight of the fluid above.

Section 6: Archimedes’ Principle

Statement

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

Explanation

• When an object is placed in a fluid:

o Fluid at the bottom of the object exerts more pressure than the fluid at the top.

o This pressure difference creates an upward force on the object.

• This upward force is called Buoyant Force (FB).

According to Archimedes’ Principle:

Buoyant Force = Weight of displaced fluid

Applications of Archimedes’ Principle

1. Floating of ships: Ships are made hollow so they displace a large volume of water, providing enough upward
buoyant force to keep them afloat.

2. Submarines: Submarines change their depth by filling tanks with water (to sink) or air (to rise).

3. Hydrometer: Works on Archimedes’ principle to measure the density of liquids.

4. Hot air balloons: Rise upward because hot air inside the balloon is lighter (less dense) than cold air outside.
 Condition for Floating

• A body floats in a fluid if:

Weight of body ≤ Buoyant Force
i.e., Weight of body ≤ Weight of displaced fluid

• If Weight of body > Buoyant Force → Body sinks.

• If Weight of body = Buoyant Force → Body floats.

Density

• Density is the mass per unit volume of a substance.

• Formula:
𝐌𝐚𝐬𝐬
Density (ρ) =
𝐕𝐨𝐥𝐮𝐦𝐞
𝐦
• ρ=
𝐯

• SI Unit: kg/m³ (kilogram per cubic metre)

• Common unit in lab work: g/cm³

Examples:

• Density of water = 1000 kg/m³ = 1 g/cm³

• Density of iron ≈ 7800 kg/m³

Substances with density greater than water sink, while those with density less than water float.

Relative Density

• Relative Density (R.D.) is the ratio of the density of a substance to the density of water.

• It has no unit (dimensionless quantity) because it is a ratio.

Formula:
𝐃𝐞𝐧𝐬𝐢𝐭𝐲 𝐨𝐟 𝐒𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞
Relative Density =
𝑫𝒆𝒏𝒔𝒊𝒕𝒚 𝒐𝒇 𝑾𝒂𝒕𝒆𝒓

𝛒(𝐬𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞)
R.D. =
𝛒(𝐰𝐚𝐭𝐞𝐫)

Since Density = Mass ÷ Volume,

𝐦𝐚𝐬𝐬 𝐨𝐟 𝐬𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞 𝐯𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 𝐰𝐚𝐭𝐞𝐫 𝐦𝐬 𝐯w
R.D. = 𝐱 or 𝐱
𝐯𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 𝐬𝐮𝐛𝐬𝐭𝐚𝐧𝐜𝐞 𝐦𝐚𝐬𝐬 𝐨𝐟 𝐰𝐚𝐭𝐞𝐫 𝐯𝐬 𝐦w

Since density of water = 1 g/cm³,

Relative Density = Numerical value of density in g/cm³
Examples

1. Density of gold = 19.3 g/cm³

⇒ Relative Density = 19.3 ÷ 1 = 19.3
(Gold is 19.3 times denser than water).

2. Density of cork = 0.25 g/cm³

⇒ Relative Density = 0.25
(Cork floats on water because its R.D. < 1).

Applications of Relative Density

1. To compare whether substances will sink or float in water.

o R.D. > 1 → substance sinks.

o R.D. < 1 → substance floats.

2. Used in designing ships, submarines, and other floating bodies.

3. Hydrometer measures relative density of liquids (like milk, acid, etc.).

Concept Formula Symbols / Terms SI Units

F = force, F → Newton (N),

Universal Law of m₁ & m₂ = masses, m → kg,

F = G·m₁·m₂ / r²
Gravitation r = distance, r → m,

G = universal gravitational constant G → N·m²/kg²

Value of G G = 6.67 × 10⁻¹¹ Universal constant N·m²/kg²

Acceleration due to M = mass of Earth,

g = GM / R² g → m/s²
Gravity R = radius of Earth

u = initial velocity,
v = u + g·t u, v → m/s
Equations of Motion v = final velocity,
h = u·t + ½·g·t² t→s
under Gravity t = time,
v² – u² = 2·g·h h→m
h = displacement (height)

Mass m = (constant, no formula) Amount of matter kg

Weight W = m·g m = mass, g = gravity W → Newton (N)

P → Pascal (Pa) =
Pressure P=F/A F = thrust (force), A = area
N/m²

Density ρ=m/V m = mass, V = volume ρ → kg/m³ (or g/cm³)

Relative Density (by No unit

RD = ρ(substance) / ρ(water) ρ(water) = 1000 kg/m³ = 1 g/cm³
density) (dimensionless)