NCERT Class 9 Science Chapter 10: Gravitation - A Comprehensive Guide

Introduction to Gravitation

Welcome, students! Have you ever wondered why a ball thrown upwards always comes back down? Or why the planets revolve around the Sun in fixed orbits? What keeps the Moon from flying off into space or crashing into the Earth? The answer to all these profound questions lies in a single, universal force: Gravitation. In this comprehensive guide to Chapter 10 of the NCERT Class 9 Science textbook, we will embark on a journey to understand this fundamental force that governs the entire cosmos.

Gravitation is not just about apples falling from trees; it's the invisible cosmic glue that holds galaxies together, dictates the motion of celestial bodies, and keeps our feet firmly on the ground. This chapter will introduce you to Sir Isaac Newton's revolutionary Universal Law of Gravitation, the concept of free fall, the crucial difference between mass and weight, and the principles of thrust, pressure, and buoyancy that explain why a massive ship floats while a tiny nail sinks. Understanding these concepts is crucial as they form the bedrock of classical physics and help us make sense of the universe we live in.

The Universal Law of Gravitation

The story of gravitation is famously linked to Sir Isaac Newton and a falling apple. While the story might be an oversimplification, it captures the essence of his brilliant insight: the force that pulls an apple to the ground is the very same force that keeps the Moon in its orbit around the Earth. This unified view was a revolutionary idea that changed our understanding of the heavens and the Earth.

Newton's Insight: The Apple and the Moon

Newton reasoned that the Moon is constantly 'falling' towards the Earth. Imagine throwing a stone horizontally. It travels some distance before gravity pulls it to the ground. If you throw it faster, it travels further. Newton imagined throwing it so fast that as it falls, the Earth's surface curves away beneath it at the same rate. The stone would then be in a continuous state of free fall, never hitting the ground – it would be in orbit. He concluded that the Moon's orbit is a result of this same principle. The Moon is constantly being pulled towards the Earth by gravity, but its tangential velocity (its tendency to move in a straight line) keeps it from crashing. This balance between its forward motion and the Earth's gravitational pull results in its nearly circular orbit.

Stating the Universal Law of Gravitation

Based on his observations and the work of astronomers like Johannes Kepler, Newton formulated the Universal Law of Gravitation. This law is 'universal' because it applies to all objects in the universe, regardless of their size or composition, from the smallest subatomic particles to the largest galaxy clusters.

The law states that: Every object in the universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The Mathematical Formulation of the Law

Let's break down the law into a mathematical formula, which is the cornerstone of this chapter.

Consider two objects, A and B, with masses M and m, respectively. Let the distance between their centers be 'd'.

• According to the law, the force of attraction (F) between them is directly proportional to the product of their masses. This means if you increase the mass of either object, the gravitational force increases. Mathematically, this is written as: F ∝ M × m
• The law also states that the force is inversely proportional to the square of the distance between them. This is known as the 'inverse-square law'. It means that if you double the distance between the objects, the force becomes four times weaker (1/2² = 1/4). If you triple the distance, the force becomes nine times weaker (1/3² = 1/9). Mathematically: F ∝ 1/d²

Combining these two proportionalities, we get:

F ∝ (M × m) / d²

To turn this proportionality into an equation, we introduce a constant of proportionality, G, known as the Universal Gravitational Constant.

The final equation is:

F = G * (M × m) / d²

Understanding the Gravitational Constant (G)

G is a fundamental constant of nature. Its value is the same everywhere in the universe. It was first experimentally measured by Henry Cavendish in 1798. The accepted value of G is:

G = 6.673 × 10⁻¹¹ N m²/kg²

The units (Newton meter squared per kilogram squared) are derived from the equation itself to ensure the force F is in Newtons. The incredibly small value of G explains why we don't feel the gravitational pull from everyday objects like tables or books. The force is there, but it is \textremely weak unless at least one of the objects has a massive mass, like a planet or a star.

Importance of the Universal Law of Gravitation

This single law successfully explains several phenomena that were previously mysteries:

• The force that binds us to the Earth: It is the gravitational pull between the Earth and our bodies that gives us weight and keeps us on the ground.
• The motion of the Moon around the Earth: As Newton reasoned, the Moon's orbit is a perfect example of gravitational force providing the necessary centripetal force.
• The motion of planets around the Sun: The entire solar system is held together by the Sun's immense gravitational pull on the planets.
• The tides in the seas and oceans: Tides are primarily caused by the gravitational pull of the Moon and, to a lesser \textent, the Sun on the Earth's water.

Free Fall

We've established that the Earth attracts all objects towards its center. When an object falls towards the Earth under the influence of this gravitational force alone, we say that the object is in a state of free fall.

What is Free Fall?

In a true free fall, no other forces, such as air resistance, are acting on the object. In reality, on Earth, air resistance is always present. However, for dense, heavy objects falling over short distances, its effect is often negligible, and we can approximate the motion as free fall. A key and somewhat counter-intuitive feature of free fall is that the acceleration of the falling object does not depend on its mass. This means a heavy stone and a light feather, if dropped in a vacuum (where there is no air resistance), would fall at the same rate and hit the ground simultaneously.

Acceleration Due to Gravity (g)

Since a falling object is acted upon by a net force (gravity), it must accelerate according to Newton's Second Law of Motion (F=ma). This acceleration is called the acceleration due to gravity and is denoted by the symbol 'g'.

Calculating the Value of g

We can derive an expression for 'g' using the Universal Law of Gravitation and Newton's Second Law of Motion.

• Let M be the mass of the Earth and m be the mass of an object on or near its surface.
• Let R be the radius of the Earth (which is the distance 'd' between the object and the Earth's center).
• The force of gravity on the object is given by: F = G * (M × m) / R²
• From Newton's second law, this force also produces an acceleration 'g', so: F = m × g

By equating these two expressions for F, we get:

m × g = G * (M × m) / R²

Notice that the mass of the object, 'm', appears on both sides of the equation and can be cancelled out. This mathematically proves that the acceleration due to gravity does not depend on the mass of the falling object!

We are left with the formula for 'g':

g = G × M / R²

Now, let's plug in the known values for the Earth:

• G = 6.67 × 10⁻¹¹ N m²/kg²
• Mass of Earth (M) = 6 × 10²⁴ kg
• Radius of Earth (R) = 6.4 × 10⁶ m

Calculating this gives a value of g ≈ 9.8 m/s². This is the standard value we use for calculations near the Earth's surface.

Variation in the Value of g

The value of 'g' is not constant everywhere on Earth. It varies slightly due to two main reasons:

1. Altitude: As we go higher above the Earth's surface, our distance 'R' from the center increases. Since 'g' is inversely proportional to R², its value decreases with increasing altitude.
2. Shape of the Earth: The Earth is not a perfect sphere; it is slightly flattened at the poles and bulges at the equator. This means the radius at the poles is less than the radius at the equator. Since 'g' is inversely proportional to R², the value of 'g' is slightly greater at the poles and slightly lesser at the equator.

Motion of Objects Under the Influence of Gravity

Since free fall is a motion with uniform acceleration (g), we can use the three familiar equations of motion by replacing the acceleration 'a' with 'g':

• v = u + gt
• s = ut + ½gt²
• v² = u² + 2gs

A small sign convention is important here. When an object is falling downwards, its velocity is increasing, so 'g' is taken as positive (+9.8 m/s²). When an object is thrown upwards, its velocity is decreasing as it fights against gravity, so 'g' is taken as negative (-9.8 m/s²).

Mass and Weight

In everyday language, we often use the terms 'mass' and 'weight' interchangeably. However, in physics, they are two distinct and different concepts. Understanding this difference is crucial.

Defining Mass: The Measure of Inertia

Mass is the amount of matter contained in an object. It is a fundamental property of that object. Mass is also a measure of an object's inertia – its resistance to a change in its state of motion. The more mass an object has, the harder it is to accelerate or decelerate it.

• The SI unit of mass is the kilogram (kg).
• Mass is a scalar quantity, meaning it only has magnitude and no direction.
• Crucially, the mass of an object is constant. It does not change no matter where the object is in the universe – whether on Earth, on the Moon, or in deep space.

Defining Weight: The Force of Gravity

Weight is the force with which an object is attracted towards the center of a planet or other celestial body. It is simply the force of gravity acting on an object's mass.

Using Newton's second law (F=ma), we can write the formula for weight (W) as:

W = m × g

Where 'm' is the mass and 'g' is the acceleration due to gravity at that location.

• Since weight is a force, its SI unit is the Newton (N).
• Weight is a vector quantity because it has both magnitude and a direction (always acting downwards towards the center of the planet).
• Unlike mass, the weight of an object is not constant. It changes depending on the value of 'g'. An astronaut has the same mass on Earth and on the Moon, but their weight on the Moon is much less because the Moon's 'g' is weaker.

The Key Differences Between Mass and Weight

Weight of an Object on the Moon

Let's compare the weight of an object on Earth and the Moon. The Moon's mass is less than Earth's, and its radius is also smaller. This results in the acceleration due to gravity on the Moon (gₘ) being about 1/6th of the acceleration due to gravity on Earth (gₑ).

If an object has a weight Wₑ on Earth, then Wₑ = m × gₑ.

Its weight on the Moon, Wₘ, will be Wₘ = m × gₘ = m × (gₑ/6) = (m × gₑ)/6.

Therefore, Wₘ = Wₑ / 6.

An object's weight on the Moon is one-sixth of its weight on Earth. This is why astronauts can take giant leaps on the lunar surface.

Thrust and Pressure

The effects of forces can be very different depending on how they are applied. The concepts of thrust and pressure help us understand this difference, especially in the context of fluids (liquids and gases).

Understanding Thrust

Thrust is defined as the force acting on an object perpendicular (at a 90° angle) to its surface. It's a simple concept: if you push a book resting on a table straight down, the force you apply is thrust. The weight of the book itself is also a thrust acting on the table.

• Thrust is simply a perpendicular force.
• Its SI unit is the same as force: the Newton (N).

Defining Pressure

Pressure is defined as the thrust per unit area. It tells us how concentrated a force is on a surface.

The formula for pressure (P) is:

P = Thrust / Area = F / A

From this formula, we can see two important things:

1. Pressure is directly proportional to the force applied. More force means more pressure.
2. Pressure is inversely proportional to the area over which the force is applied. A smaller area results in greater pressure for the same force.

The SI unit of pressure is the Pascal (Pa), which is defined as one Newton of force applied over an area of one square meter (1 Pa = 1 N/m²).

Everyday Examples and Applications of Pressure

The inverse relationship between pressure and area has many practical applications:

• Sharp Knives and Needles: A knife has a very sharp edge, which means it has a very small surface area. This allows even a small force from your hand to create a very high pressure, enabling it to cut through vegetables easily. Similarly, a needle has a sharp tip to create high pressure to pierce cloth.
• School Bag Straps: School bags have broad straps, not thin strings. The broad straps increase the area over which the bag's weight is distributed on your shoulders. This reduces the pressure, making the bag more comfortable to carry.
• Foundations of Buildings: Buildings and dams have wide foundations. This large area distributes the massive weight of the structure over a large patch of ground, reducing the pressure and preventing the building from sinking.
• Camel's Feet: Camels, the 'ships of the desert', have large, flat feet. This large area reduces the pressure they exert on the sand, preventing them from sinking in.

Pressure in Fluids

Liquids and gases (fluids) also exert pressure. A fluid exerts pressure on the bottom and walls of its container. A key property is that pressure exerted by a fluid is transmitted equally in all directions.

Buoyancy and Archimedes' Principle

Have you ever felt lighter when you are in a swimming pool? Or noticed how a plastic bottle pushed into water springs back up to the surface? This upward push experienced in a fluid is due to a force called buoyancy.

What is Buoyancy? The Upward Force

When an object is immersed partially or wholly in a fluid (a liquid or a gas), the fluid exerts an upward force on it. This upward force is called the buoyant force or upthrust. This force acts in the opposite direction to the force of gravity (weight) acting on the object.

The magnitude of this buoyant force determines whether an object will float or sink.

Why Do Objects Float or Sink?

An object placed in a fluid is acted upon by two main forces: its weight (gravitational force) acting downwards, and the buoyant force acting upwards.

• If the buoyant force is greater than the object's weight, the net force is upwards, and the object will rise to the surface and float.
• If the buoyant force is less than the object's weight, the net force is downwards, and the object will sink.
• If the buoyant force is equal to the object's weight, the object will be suspended and float just below the surface of the fluid.

This principle can also be understood in terms of density. An object will float if its average density is less than the density of the fluid it is placed in. It will sink if its density is greater than the fluid's density.

Archimedes' Principle: The 'Eureka' Moment

The Greek scientist Archimedes discovered a fundamental principle that allows us to calculate the magnitude of the buoyant force. Legend has it that he discovered this while stepping into a bath and observing the water level rise, after which he ran through the streets shouting 'Eureka!' ('I have found it!').

Archimedes' Principle states that: When a body is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it.

In simple terms, the buoyant force on an object is equal to how much the fluid it pushed out of the way weighs. This is a powerful principle with many applications.

Applications of Archimedes' Principle

• Designing Ships and Submarines: A ship is made of iron and steel, which are much denser than water. So why does it float? A ship has a hollow shape. This shape displaces a very large volume of water. According to Archimedes' principle, this creates a massive buoyant force. The ship is designed so that this buoyant force is equal to the total weight of the ship (including its cargo), allowing it to float. Submarines use ballast tanks; they take in water to increase their weight and sink, and pump out water to decrease their weight and rise.
• Lactometers: These are instruments used to check the purity of a milk sample. They work on the principle that purer milk has a specific density. The lactometer will sink to a certain level in pure milk. If the milk is adulterated with water, its density changes, and the lactometer will sink to a different level.
• Hydrometers: These instruments are used to determine the density of various liquids based on how deep they sink.

Relative Density

It is often convenient to compare the density of a substance with the density of a standard substance, which is usually water.

Defining Relative Density

Relative density of a substance is the ratio of its density to the density of water.

Relative Density = Density of Substance / Density of Water

Since relative density is a ratio of two similar quantities (densities), it has no units. For example, the density of gold is 19300 kg/m³ and the density of water is 1000 kg/m³. The relative density of gold is 19300 / 1000 = 19.3. This tells us that gold is 19.3 times as dense as water.

If the relative density of a substance is greater than 1, it will sink in water. If it is less than 1, it will float on water.

Important Questions and Answers

Here are some solved questions from the NCERT exercise to help you solidify your understanding.

Question 1: State the universal law of gravitation.

Answer: The universal law of gravitation, formulated by Sir Isaac Newton, states that 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 their centers. Mathematically, it is expressed as F = G * (M × m) / d², where F is the gravitational force, M and m are the masses of the two objects, d is the distance between their centers, and G is the universal gravitational constant (6.673 × 10⁻¹¹ N m²/kg²).

Question 2: What are the differences between the mass of an object and its weight?

Answer: The differences are as follows:

• Definition: Mass is the quantity of matter in an object, while weight is the gravitational force acting on that mass.
• Constancy: Mass is a constant property of an object and does not change with location. Weight varies depending on the local acceleration due to gravity (g).
• Units: The SI unit of mass is the kilogram (kg). The SI unit of weight is the Newton (N).
• Nature: Mass is a scalar quantity. Weight is a vector quantity, as it has a direction (towards the center of the Earth).
• Zero Value: An object's mass can never be zero. An object's weight can be zero in a place where gravity is zero, such as in deep space or at the center of the Earth.

Question 3: Why is it difficult to hold a school bag having a strap made of a thin and strong string?

Answer: This is a practical application of the concept of pressure. Pressure is defined as force per unit area (P = F/A). The weight of the school bag is the force acting downwards. When the strap is made of a thin string, the area (A) over which this force acts on the shoulder is very small. According to the formula, if the area is very small, the pressure exerted on the shoulder will be very large. This high pressure causes pain and discomfort, making it difficult to hold the bag. Broad straps increase the area of contact, which reduces the pressure for the same amount of force (weight), making the bag more comfortable to carry.

Question 4: A ball is thrown vertically upwards with a velocity of 49 m/s. Calculate (i) the maximum height to which it rises, (ii) the total time it takes to return to the surface of the earth.

Answer:

Given:

• Initial velocity (u) = 49 m/s
• Final velocity at maximum height (v) = 0 m/s (The ball momentarily stops at its highest point)
• Acceleration due to gravity (g) = -9.8 m/s² (Negative because the ball is moving upwards against gravity)

(i) To find the maximum height (s):

We use the third equation of motion: v² = u² + 2gs

0² = (49)² + 2 × (-9.8) × s

0 = 2401 - 19.6s

19.6s = 2401

s = 2401 / 19.6

s = 122.5 m

So, the maximum height the ball rises to is 122.5 meters.

(ii) To find the total time taken:

First, let's find the time taken to reach the maximum height (t₁). We use the first equation of motion: v = u + gt

0 = 49 + (-9.8) × t₁

9.8t₁ = 49

t₁ = 49 / 9.8

t₁ = 5 s

The time taken for the ball to fall back to the ground from the maximum height will be the same as the time it took to rise. So, time of descent (t₂) = 5 s.

Total time = time of ascent + time of descent = t₁ + t₂

Total time = 5 s + 5 s = 10 s

Therefore, the total time it takes to return to the surface of the Earth is 10 seconds.

Chapter Summary

Here is a quick recap of the key concepts and formulas from the chapter on Gravitation:

• Universal Law of Gravitation: Every object attracts every other object. The force is F = G * (M × m) / d².
• G (Universal Gravitational Constant): G = 6.673 × 10⁻¹¹ N m²/kg². It is constant throughout the universe.
• Free Fall: The motion of an object under the influence of Earth's gravity alone. The acceleration is 'g'.
• Acceleration Due to Gravity (g): g = G × M / R². Its standard value on Earth is approximately 9.8 m/s². It varies with altitude and the shape of the Earth.
• Equations of Motion for Free Fall: v = u + gt, s = ut + ½gt², and v² = u² + 2gs.
• Mass (m): The amount of matter in an object. It is constant and its SI unit is the kilogram (kg).
• Weight (W): The force of gravity on an object (W = m × g). It varies with location and its SI unit is the Newton (N).
• Thrust: Force acting perpendicular to a surface.
• Pressure (P): Thrust per unit area (P = F/A). Its SI unit is the Pascal (Pa).
• Buoyancy: The upward force exerted by a fluid on an immersed object.
• Archimedes' Principle: The buoyant force is equal to the weight of the fluid displaced by the object.
• Relative Density: The ratio of the density of a substance to the density of water. It has no units.