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

Introduction to Gravitation

Have you ever wondered why an apple falls downwards from a tree? 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, fundamental force of nature: Gravitation. This chapter from the NCERT Class 9 Science syllabus delves into this fascinating force that governs the motion of everything, from a tiny pebble to the largest galaxies.

The concept of gravitation was famously articulated by Sir Isaac Newton. The story goes that he was inspired by a falling apple, which led him to ponder that the same force responsible for the apple's fall might also be the force keeping the Moon in its orbit around the Earth. Gravitation is not just a force that pulls things down; it is a universal force of attraction that exists between any two objects with mass in the universe. In this comprehensive guide, we will explore the Universal Law of Gravitation, understand the concept of free fall and acceleration due to gravity, differentiate between mass and weight, and investigate the principles of thrust, pressure, and buoyancy in fluids.

The Universal Law of Gravitation

Isaac Newton proposed that every object in the universe attracts every other object with a force. This force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This is known as the Universal Law of Gravitation.

Let's 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) is directly proportional to the product of their masses: F ∝ M × m
• The force of attraction (F) is inversely proportional to the square of the distance between them: F ∝ 1/d²

Combining these two relationships, 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.

F = G * (M * m) / d²

This equation is a powerful tool that allows us to calculate the gravitational force between any two objects if we know their masses and the distance between them.

The Gravitational Constant (G)

The constant 'G' in the formula is called the Universal Gravitational Constant. It is 'universal' because its value is believed to be the same throughout the entire universe, for all objects and at all times. The value of G was first experimentally determined by Henry Cavendish in 1798 using a highly sensitive torsion balance. The currently accepted value is:

G = 6.673 × 10⁻¹¹ N m²/kg²

The unit N m²/kg² can be derived directly from the gravitation formula. Since G = Fd² / (Mm), its unit is (Unit of Force) × (Unit of distance)² / (Unit of mass)². This gives us N × m² / (kg × kg), which simplifies to N m²/kg².

Importance of the Universal Law of Gravitation

The law of gravitation is not just an academic formula; it explains many phenomena we observe in our daily lives and in the cosmos:

• Binding us to the Earth: It is the gravitational force between the Earth and our bodies that keeps us firmly on the ground.
• Planetary Motion: The Sun's immense gravitational pull keeps all the planets, including Earth, revolving in their respective orbits.
• Lunar Motion: The Earth's gravity keeps the Moon in its orbit, preventing it from drifting away.
• Tides: The gravitational pull of the Moon and, to a lesser \textent, the Sun on the Earth's oceans causes the rhythmic rise and fall of sea levels, known as tides.
• Formation of Stars and Galaxies: Gravitation is the force responsible for pulling together dust and gas to form stars, planets, and entire galaxies.

Free Fall

When you drop a stone from a height, it accelerates downwards. This motion is a classic example of free fall. An object is said to be in free fall when it is moving under the influence of the Earth's gravitational force alone. In a true free fall, no other forces like air resistance are acting on the object. While this is only perfectly achievable in a vacuum, the motion of a dense object falling a short distance through the air is a very close approximation.

Acceleration Due to Gravity (g)

Since a freely falling body experiences a continuous force, its velocity changes continuously. This change in velocity implies that the object is accelerating. This acceleration is called the acceleration due to gravity, denoted by the symbol 'g'.

We can derive an expression for 'g' using Newton's second law of motion (F = ma) and the Universal Law of Gravitation. Consider an object of mass 'm' near the surface of the Earth. The force acting on it is the gravitational force exerted by the Earth.

F = G * (M * m) / R²

Here, M is the mass of the Earth, and R is the radius of the Earth (approximating the distance to the center).

From Newton's second law, this force produces an acceleration 'g' in the object: F = m × g

Equating the two expressions for F:

m × g = G * (M * m) / R²

Canceling 'm' from both sides, we get:

g = G * M / R²

This important equation shows that the acceleration due to gravity depends only on the mass of the Earth (M) and its radius (R), and not on the mass of the falling object ('m'). This is why a feather and a bowling ball fall at the same rate in a vacuum.

By substituting the values for G, M (approx. 6 × 10²⁴ kg), and R (approx. 6.4 × 10⁶ m), the value of 'g' is calculated to be approximately 9.8 m/s². This value is not perfectly constant; it varies slightly with altitude and is greater at the poles than at the equator because the Earth is not a perfect sphere.

Equations of Motion for Freely Falling Bodies

For objects in free fall, the standard equations of motion can be adapted by replacing the acceleration 'a' with 'g'.

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

Here, 'u' is the initial velocity, 'v' is the final velocity, 's' is the distance (height), and 't' is the time. It's crucial to use a consistent sign convention. Typically, motion downwards is taken as positive (g is +9.8 m/s²), and motion upwards is taken as negative (g is -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 concepts.

Mass (m)

Mass is the measure of the amount of matter contained within an object. It 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 it.

• Constant Quantity: The mass of an object is constant. It does not change whether the object is on Earth, on the Moon, or in deep space.
• Scalar Quantity: Mass has only magnitude and no direction.
• SI Unit: The SI unit of mass is the kilogram (kg).

Weight (W)

Weight is the gravitational force exerted on an object by a celestial body, like the Earth. It is the force with which the Earth pulls an object towards its center. Since weight is a force, we can calculate it using Newton's second law, F = ma.

Here, the acceleration 'a' is the acceleration due to gravity 'g'. Therefore, the weight (W) of an object is:

W = m × g

• Variable Quantity: The weight of an object is not constant. It depends on the local value of 'g'. Since 'g' changes with location (e.g., Earth vs. Moon), the weight of the object also changes. An object's weight is slightly different at the poles than at the equator.
• Vector Quantity: Weight is a force and has both magnitude and direction. Its direction is always towards the center of the Earth (or the celestial body it is on).
• SI Unit: The SI unit of weight is the same as the unit of force, which is the Newton (N).

Weight of an Object on the Moon

The Moon has significantly less mass and a smaller radius than the Earth. Therefore, its acceleration due to gravity (g_moon) is much weaker. In fact, the Moon's gravity is about one-sixth that of Earth's (g_moon ≈ 1.63 m/s²). Consequently, the weight of an object on the Moon is about 1/6th of its weight on Earth. If an astronaut weighs 600 N on Earth, they would weigh only about 100 N on the Moon. However, their mass would remain the same.

Difference between Mass and Weight

Thrust and Pressure

The concepts of thrust and pressure are crucial for understanding how forces act on surfaces, especially in fluids.

Thrust

Thrust is defined as the force acting on an object perpendicular to its surface. For example, when you stand on the ground, your weight acts as a thrust on the ground. When you push a pin into a board, the force you apply is the thrust. Since thrust is simply a perpendicular force, its SI unit is the Newton (N).

Pressure (P)

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

Pressure = Thrust / Area

The SI unit of pressure is N/m², which is also called the Pascal (Pa), in honor of the scientist Blaise Pascal. 1 Pa = 1 N/m².

The relationship shows that for a given force (thrust), the pressure is inversely proportional to the area. This has many practical applications:

• Sharp Objects: A knife or a nail has a very sharp edge or point. This tiny area of contact concentrates the applied force, creating a very high pressure that allows it to cut or pierce easily.
• Wide Straps: School bags and backpacks have wide straps to distribute the bag's weight (thrust) over a larger area of your shoulders. This reduces the pressure, making the bag more comfortable to carry.
• Building Foundations: Buildings have wide foundations to spread their immense weight over a large area of the ground, reducing the pressure and preventing the building from sinking.
• Tractor Tires: Tractors have broad tires to reduce the pressure on soft soil, preventing them from getting stuck.

Pressure in Fluids

Liquids and gases are collectively known as fluids. A key property of fluids is that they exert pressure on the base and walls of their container. Unlike a solid, which exerts pressure only downwards, a fluid in a container exerts pressure in all directions. This pressure increases with depth.

Buoyancy and Archimedes' Principle

Buoyancy

When you try to push a cork into water, you feel an upward push. When you lift a bucket of water from a well, it feels lighter when it's inside the water compared to when it's out of the water. This upward force exerted by a fluid on an object partially or fully immersed in it is called the buoyant force or upthrust.

The magnitude of this buoyant force depends on two factors: the density of the fluid and the volume of the object submerged in the fluid. The reason for buoyancy lies in the pressure difference in the fluid. The pressure at the bottom of an immersed object is greater than the pressure at its top, resulting in a net upward force.

Why Objects Float or Sink

The fate of an object in a fluid—whether it floats or sinks—is determined by a contest between its weight (pulling it down) and the buoyant force (pushing it up). This can be more easily understood by comparing the density of the object with the density of the fluid.

• If the density of the object is greater than the density of the fluid, the object's weight will be greater than the buoyant force. The net force will be downwards, and the object will sink (e.g., an iron nail in water).
• If the density of the object is less than the density of the fluid, the buoyant force will be greater than the object's weight when fully submerged. The object will rise to the surface and float, partially submerged (e.g., a piece of cork or wood in water).
• If the density of the object is equal to the density of the fluid, the object's weight is exactly balanced by the buoyant force. The object will be in neutral buoyancy and will float fully submerged wherever it is placed within the fluid.

Archimedes' Principle

The Greek scientist Archimedes discovered a fundamental principle that quantifies the buoyant force. 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 other words, Buoyant Force = Weight of the displaced fluid.

This principle has numerous applications in science and engineering:

• Designing Ships and Submarines: A ship, despite being made of iron (which is much denser than water), floats because its hollow shape displaces a huge volume of water. The weight of this displaced water is equal to the total weight of the ship, providing the necessary buoyant force. Submarines use ballast tanks to control their density, allowing them to sink or surface.
• Lactometers: These are instruments used to check the purity of a milk sample. They work on the principle that the density of milk changes with its water content.
• Hydrometers: These are used for determining the density of various liquids.

Relative Density

It is often convenient to express the density of a substance in comparison to a standard substance, which is usually water. The 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 is a pure number and has no units. For example, the density of silver is 10800 kg/m³, and the density of water is 1000 kg/m³. So, the relative density of silver is 10800 / 1000 = 10.8. This tells us that silver is 10.8 times denser than 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.

Important Questions and Answers

Question 1: State the universal law of gravitation. Why is it called 'universal'?

Answer: The universal law of gravitation, proposed 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. The mathematical expression is F = G(M*m)/d².
It is called 'universal' because it is applicable to all objects in the universe, irrespective of their size, shape, or chemical composition. It applies to celestial bodies like planets, stars, and galaxies as well as to terrestrial objects on Earth.

Question 2: What is the difference between the mass of an object and its weight?

Answer: The main differences between mass and weight are:

• Definition: Mass is the amount of matter in an object, while weight is the gravitational force acting on that object (W = mg).
• Constancy: Mass is a constant property of an object and does not change with location. Weight is variable and changes depending on the acceleration due to gravity (g) at that location.
• SI Unit: The SI unit of mass is the kilogram (kg), whereas the SI unit of weight is the Newton (N).
• Nature: Mass is a scalar quantity (has only magnitude), while weight is a vector quantity (has both magnitude and direction, which is towards the center of the gravitating body).

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

Answer: This can be explained by the concept of pressure. Pressure is defined as force (or thrust) per unit area (P = F/A). The weight of the school bag acts as the force. When the strap is made of a thin string, the area over which this force acts on the shoulder is very small. According to the formula, if the area (A) is very small, the pressure (P) becomes very large. This large pressure on the shoulder causes pain and makes it difficult to hold the bag. A wider strap increases the area of contact, thereby reducing the pressure and making it more comfortable.

Question 4: An object is thrown vertically upwards and rises to a height of 10 m. Calculate (i) the velocity with which the object was thrown upwards and (ii) the time taken by the object to reach the highest point. (Take g = 9.8 m/s²)

Answer: Given: Distance (height), s = 10 m Final velocity at the highest point, v = 0 m/s Acceleration due to gravity, g = -9.8 m/s² (negative because the object is moving against gravity) (i) To find the initial velocity (u): We use the third equation of motion: v² = u² + 2gs 0² = u² + 2 × (-9.8) × 10 0 = u² - 196 u² = 196 u = √196 u = 14 m/s So, the velocity with which the object was thrown upwards is 14 m/s. (ii) To find the time taken (t): We use the first equation of motion: v = u + gt 0 = 14 + (-9.8) × t 0 = 14 - 9.8t 9.8t = 14 t = 14 / 9.8 t ≈ 1.43 s So, the time taken to reach the highest point is approximately 1.43 seconds.

Chapter Summary

• Universal Law of Gravitation: Every object attracts every other object with a force F = G(Mm/d²), where G is the universal gravitational constant.
• Free Fall: The motion of an object solely under the influence of Earth's gravity is called free fall.
• Acceleration Due to Gravity (g): The acceleration produced in a freely falling body is 'g', approximately 9.8 m/s² near the Earth's surface. It is independent of the object's mass.
• Mass vs. Weight: Mass is the constant amount of matter (in kg), while weight is the variable gravitational force (W = mg, in N).
• Thrust and Pressure: Thrust is the perpendicular force on a surface. Pressure is thrust per unit area (P = F/A), measured in Pascals (Pa).
• Buoyancy: It is 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. This principle explains why objects float or sink.
• Relative Density: It is the ratio of the density of a substance to the density of water. It is a unitless quantity that helps determine if a substance will float or sink in water.