CLASS 9 SCIENCE NOTES: CHAPTER - GRAVITATION

GRAVITATION

Gravitation is the force of attraction between any two objects in the universe because of their masses.

Newton’s Universal Law of Gravitation

Every object in the universe attracts every other object with a force.

Let there are two objects of masses m1 and m2 and the distance between the object is r, then according to universal law of attraction. This force of attraction:

is directly proportional to the product of their masses (F ∝ m₁ x m₂); and
is inversely proportional to the square of the distance between their centers (F ∝ 1 / r²)

Mathematical form:

F = G × (m₁ × m₂) / r²

G = Universal Gravitational Constant = 6.67 × 10⁻¹¹ N·m²/kg²

PROOF FOR THE UNIT OF GRAVITATIONAL CONSTANT

Now,

G = F x d² / (M x m)

G = (Newton x Meter²) / (Kilogram x Kilogram)

G = (Newton Meter²) / (Kilogram²)

G = Nm²/kg²

So, SI Unit of G is Nm²/kg²

Importance of the Universal Law of Gravitation

It is the force that keeps us (and all objects) on Earth.
It explains the motion of the Moon around the Earth.
It explains the motion of the Earth around the Sun and other planetary motions.
It helps explain ocean tides caused by the Moon (and Sun).
It is used to understand orbits of satellites and spacecraft.

Free Fall and Acceleration due to Gravity (g)

Free fall: When an object moves only under the influence of gravity (no air resistance), it is said to be in free fall.

Acceleration due to gravity (g): During free fall the acceleration produced in the object is called g.

Acceleration due to Gravity

According to Newton's law of gravitation:

G = gravitational constant.
M = mass of the Earth.
m = mass of the object.
R = Earth's radius.

F = (G · M · m) / R²

The force on an object due to gravity:

F = m · g

{ g = acceleration due to gravity }

Equating Forces:

m · g = (G · M · m) / R²

g = (G · M) / R²

Relation to G (Earth): g = G × (M / R²)

M = mass of the Earth (≈ 6 × 10²⁴ kg)
R = radius of the Earth (≈ 6.4 × 10⁶ m)
Substituting values gives g ≈ 9.8 m/s² at Earth's surface.

Difference between G and g

Universal Gravitational Constant (G)	Acceleration due to Gravity (g)
Constant value for the whole universe.	Value of acceleration caused by Earth at a location; varies with place.
G = 6.67 × 10⁻¹¹ N·m²/kg²	g ≈ 9.8 m/s² on Earth's surface (varies slightly)
Used in force formula F = G m₁ m₂ / r²	Used in weight formula W = m g

Mass and Weight

Mass:

Amount of matter in a body.
It remains the same everywhere.
Unit of mass is kilogram (kg).

Weight:

Force with which a planet (like Earth) attracts the body.
It can change with location.
Unit of weight is newton (N).

Formula: Weight (W) = mass (m) × g

Mass vs. Weight: Key Differences

No.	Mass (How much "stuff" is in an object)	Weight (How hard gravity pulls on that object)
1	It's a scalar quantity (has just magnitude, no direction).	It's a vector quantity (has both magnitude and downward direction).
2	Mass is always the same no matter where you are (Earth, Moon, space).	Weight changes depending on gravity, and it's zero in space or at the Earth's center.
3	Measured with a traditional balance (like a beam balance).	Measured with a spring balance.
4	The main unit is the kilogram (kg).	The unit is the Newton (N), because it's a force.
5	Mass can never be zero.	Weight can be zero if there is no gravity pulling on it (like in deep space).

Relation Between Weight on Earth and Weight on Moon

Weight on the Moon

According to the universal law of gravitation, the weight of an object on the moon is given by:

W_m = (G · M_m · m) / R_m²

W_m = weight on the moon,
G = gravitational constant,
M_m = mass of the moon,
m = mass of the object,
R_m = radius of the moon.

Weight on the Earth

According to the universal law of gravitation, the weight of an object on the Earth is given by:

W_e = (G · M_e · m) / R_e²

W_e = weight on Earth,
M_e = mass of the Earth,
R_e = radius of the Earth.

Comparing Weights

Now, by dividing the weight on the moon by the weight on Earth, we get:

W_m / W_e = (M_m / R_m²) / (M_e / R_e²)

Substituting the values, we get

W_m / W_e = (7.36 x 10²² / (1.74 x 10⁶)²) / (5.98 x 10²⁴ / (6.37 x 10⁶)²) ≈ 0.165

So:

Weight on Moon = (1 / 6) × Weight on Earth

The Moon's gravity is weaker than Earth's — roughly 1/6th of Earth's gravity.

Example: If a person weighs 60 kgf (approx) on Earth, then on the Moon they would weigh about 10 kgf (i.e., 1/6th).

Understanding Free Fall

When you throw something up, it goes a little way and then falls back down. This happens because the Earth's gravity is pulling on it.
This action—where an object falls only because of the pull of Earth's gravity—is called free fall of an object.

Motion of Objects Under Gravity (Vertical Motion)

When an object moves vertically under gravity (near Earth's surface) we use these equations (assuming g constant):

v = u + g t
s = u t + ½ g t²
v² = u² + 2 g s

Where:

u = initial velocity
v = final velocity
s = displacement
t = time
g = acceleration due to gravity

NOTE: Take g = -9.8 m/s² if the motion is upwards and take g = +9.8 m/s² if the motion is downwards.

Derivation: Mass of the Earth

From the formula g = G × (M / R²), rearrange to find Earth's mass:

M = g × R² / G

Substitute the values:

g = 9.8 m/s²
R = 6.4 × 10⁶ m
G = 6.67 × 10⁻¹¹ N·m²/kg²

On calculation, M ≈ 6 × 10²⁴ kg (mass of Earth).

Variations in g (Why g changes)

1. Variation with Altitude

g decreases as you go higher above the Earth’s surface because the distance from Earth's center increases. (g ∝ 1 / R²)

2. Variation with Depth

Inside the Earth (below surface), g decreases with depth because the mass pulling you effectively reduces as you go deeper (only the mass inside the radius matters).

3. Variation with Latitude

g is slightly larger at the poles and smaller at the equator. Reason:

Earth is not a perfect sphere — it bulges at the equator, so the equatorial radius is larger (distance from center increases → lower g).
Also, Earth's rotation produces a small centrifugal effect that slightly reduces effective g at the equator.

Kepler's Laws of Planetary Motion

Kepler's Law of Orbits (The Law of Ellipses):

What it means: Planets don't orbit in perfect circles; they orbit in ovals (ellipses) called elliptical orbits.
Key detail: The Sun isn't in the center; it's slightly off to the side, at one of the two foci.

Kepler's Law of Areas (The Law of Equal Areas):

What it means: A planet speeds up when it gets closer to the Sun and slows down as it moves farther away.
Key detail: This happens because a line drawn from the planet to the Sun sweeps out the equal areas in equal time intervals, no matter where the planet is in its orbit.

Kepler's Law of Periods (The Harmonic Law):

What it means: There's a mathematical relationship between how long a planet takes to orbit the Sun (its period) and how far away it is from the Sun (its average distance).
Key detail: The square of the orbital period is directly proportional to the cube of the average distance. Basically, the farther away a planet is, the much longer its year will be.

Thrust and Pressure

Thrust: Force acting on an object perpendicular to the surface is called thrust. S.I. Unit is Newton (N).
Pressure: It is defined as thrust acting per unit area.

Formula: Pressure = Thrust / Area

SI unit of pressure = Pascal (Pa) = N/m²
Example: A box of weight 200 N on base area 0.5 m² exerts pressure 400 Pa.

Liquid Pressure

Pressure exerted by liquids is caused by the weight of the liquid itself.

Key Characteristics of Liquid Pressure

Depth: Pressure increases as you go deeper.
Uniformity: At any specific depth, the pressure is the same in all directions.
Container: The pressure doesn't depend on the size or shape of the container holding the liquid.

Practical example: A diver feels greater pressure as depth increases.

Buoyant Force (Upthrust)

Have you ever felt an object seem lighter when you put it in water? That's the buoyant force at work!

It's an upward push that a liquid (or any fluid) exerts on anything that is placed in it, whether it's floating or fully sunk. This upward push is also called Upthrust.

Because of this force, an object submerged in a fluid appears to lose some of its weight—it feels lighter than it does in the air.

If buoyant force equals the weight of the body → the body floats.
If buoyant force is less than the weight → the body sinks.

What Affects the Buoyant Force?

Two main things determine how strong the buoyant force will be:

Density of the Fluid:

The denser the liquid, the stronger the buoyant force.
Think of it this way: Saltwater is denser than plain water, so it pushes up harder. That's why it's easier to float in the ocean (or the Dead Sea) than in a swimming pool!

Volume of the Object:
- The more space (volume) an object takes up underwater, the stronger the buoyant force.
- The object pushes (or displaces) the fluid out of the way. The more fluid it displaces, the bigger the upward push it gets. This is why a big, empty ship can float, while a small pebble sinks.

Mathematical form: F_b = ρ × V_sub × g

F_b = buoyant force
ρ = density of the fluid
V_sub = volume of fluid displaced
g = acceleration due to gravity

Law of Floatation Explained

The Law of Floatation tells us whether an object will float, sink, or hover in a liquid. It all comes down to a comparison between the object's density and the liquid's density.

What Happens to an Object? (Density Comparison)

1. FLOAT (Lighter than the liquid): An object will float on the surface if its density is less than the density of the liquid.

2. HOVER / SUSPENDED (Same density as the liquid): An object will hover (or be in equilibrium) at any depth if its density is equal to the density of the liquid.

3. SINK (Heavier than the liquid): An object will sink to the bottom if its density is greater than the density of the liquid.

Floatation in Terms of Force

We can also look at this by comparing the upward Buoyant Force and the object's downward Weight:

Floating: Buoyant Force = Weight of the object.

Sinking: Buoyant Force < Weight of the object.

Suspended (Hovering): Buoyant Force = Weight AND Object Density = Liquid Density.

Relative Density (Specific Gravity)

Relative density of a substance is the ratio of its density to the density of water (at 4°C):

Relative density = density of substance / density of water

This is a unitless number (no units). It tells whether a substance will float or sink in water (values < 1 float, > 1 sink).

Archimedes' Principle Explained

Imagine putting a toy boat into a tub of water. When you put the boat in, the water level rises—or water spills out. That spilled water is the "displaced fluid."

The upward push (Buoyant Force) that an object feels when it's in a fluid is exactly equal to the weight of the fluid that it pushes out of the way (the displaced fluid).

In short: Buoyancy equals the weight of the water pushed aside!

Real-World Applications

Object / Example	What it does	Principle at work
Submarine (sinking / floating)	Sinking: The submarine fills its ballast tanks with more water, so it becomes heavier. Floating: It pumps water out of the tanks, making itself lighter.	When the submarine's total weight becomes greater than the buoyant force (weight of water displaced), it sinks. When the total weight is less than the buoyant force, it rises and floats.
Hot-air balloon (rises or falls)	The air inside the balloon is heated, so it becomes less dense (lighter) than the surrounding air.	The buoyant force from the displaced cooler air becomes greater than the weight of the balloon + hot air, so the balloon rises. If the air cools, it loses lift and descends.
Hydrometer / Lactometer (measures density)	A hydrometer floats at different levels in different liquids: it sinks more in less dense liquids and floats higher in denser liquids.	In a denser liquid, the device displaces a smaller volume to balance its weight, so it floats higher. In a less dense liquid it must displace more volume and sinks lower.
Ship (large hollow hull) (even though made of steel)	A ship is built with a large hollow volume so that the overall average density of the ship + air inside is less than the density of water.	The buoyant force (weight of the water displaced by the hull) is greater than the ship's total weight, so it stays afloat.

Key phrase: Buoyant force = weight of fluid displaced. An object floats when buoyant force ≥ weight; it sinks when buoyant force < weight.

Key Formulas Summary

F = G × (m₁ m₂) / r² — Universal law
g = G × (M / R²) — acceleration due to Earth
W = m g — weight
Pressure = Thrust / Area
F_b = ρ × V_sub × g — buoyant force

Short Examples for Practice

Calculate the weight of a 5 kg mass on Earth. (Use g = 9.8 m/s².)
Find the gravitational force between two 1 kg masses 1 m apart. (Use G = 6.67×10⁻¹¹.)
A block displaces 0.02 m³ of water. If density of water = 1000 kg/m³, find buoyant force. (Use g = 9.8 m/s².)

NAMAN GARG

Sound – Class 9 Science Complete Notes