Gravitation - Universal law of gravitation, acceleration due to gravity, mass versus weight, and principles of buoyancy

Hello students! Welcome to one of the most exciting chapters in Class 9 Physics: Gravitation.

Have you ever wondered why an apple falls down from a tree instead of floating up into the sky? Why doesn't the Moon drift away into deep space? Or why huge steel ships float effortlessly on water while a small iron nail sinks right to the bottom?

In this comprehensive tutorial, we will break down all these fascinating phenomena step-by-step!

Visualizing Core Gravitation Concepts

To help you build a clear mental model, let's look at how universal attraction, gravity, weight, and buoyancy interact in our physical world:

1. The Universal Law of Gravitation

In 1687, Sir Isaac Newton proposed a revolutionary idea: The force that makes an apple fall to the Earth is the exact same force that keeps the planets orbiting around the Sun.

What does the Law State?

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.

Let's break this down algebraically:

1. Consider two masses, $M$ and $m$, separated by a distance $d$.
2. The gravitational force $F$ is directly proportional to the product of masses: $F \propto M \times m$
3. The gravitational force $F$ is inversely proportional to the square of the distance between them: $F \propto \frac{1}{d^2}$

Combining both statements: $F \propto \frac{M \times m}{d^2}$

To turn this proportion into an equation, we insert the Universal Gravitational Constant ($G$):

$F = G \frac{M \times m}{d^2}$

Key Properties of $G$ (Universal Gravitational Constant)

• SI Unit: $\text{N}\cdot\text{m}^2/\text{kg}^2$
• Accepted Value: $G = 6.673 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ (calculated by Henry Cavendish).
• Why "Universal"? Because its value remains constant everywhere in the universe—whether you are on Earth, Mars, or in deep space!

💡 Teacher's Analogy: Imagine invisible elastic bands connecting every particle in the cosmos. Heavy objects have extra-thick bands pulling strongly, but if you pull objects further apart, the tension weakens rapidly because of the inverse-square rule ($d^2$).

2. Acceleration Due to Gravity ($g$)

When an object falls towards the Earth solely under the influence of gravitational force, it is said to be in Free Fall.

During free fall, the direction of motion remains unchanged, but the speed increases every second. This change in velocity produces an acceleration called Acceleration due to Gravity, denoted by $g$.

Derivation of $g$

According to Newton's Second Law of Motion: $\text{Force } (F) = \text{mass } (m) \times \text{acceleration } (g)$

From the Universal Law of Gravitation: $F = \frac{G M m}{R^2}$ (where $M$ is the mass of Earth, $m$ is the mass of the object, and $R$ is the radius of Earth)

Equating both expressions for force: $m \cdot g = \frac{G M m}{R^2}$

Canceling $m$ from both sides:

$g = \frac{G M}{R^2}$

Key Observations about $g$:

1. Independent of Object's Mass: The value of $g$ does not depend on the mass ($m$) of the falling body. A heavy stone and a light feather dropped in a vacuum will hit the ground at the exact same instant!
2. Value on Earth Surface: Substituting $G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$, $M = 6 \times 10^{24} \text{ kg}$, and $R = 6.4 \times 10^6 \text{ m}$: $g \approx 9.8 \text{ m/s}^2$

Distinguishing $G$ vs $g$

3. Mass versus Weight

In everyday language, we often mix up mass and weight. But in Physics, they are completely different quantities!

                  MASS                                     WEIGHT
       [ Total matter contained ]              [ Force of gravitational pull ]
       • Measured in kilograms (kg)            • Measured in Newtons (N)
       • Scalar quantity (constant)            • Vector quantity (changes with g)

Detailed Comparison

1. Mass ($m$):

   • The measure of inertia and quantity of matter contained in an object.
   • SI Unit: Kilogram ($\text{kg}$).
   • Constant everywhere (remains $50\text{ kg}$ on Earth, Moon, or Space).

2. Weight ($W$):

   • The force with which an object is pulled towards Earth's center.
   • Formula: $W = m \times g$
   • SI Unit: Newton ($\text{N}$).
   • Variable because $g$ varies.

Weight of an Object on the Moon

The Moon's mass is much smaller than Earth's. As a result, its gravitational attraction is weaker.

$W_{\text{Moon}} = \frac{1}{6} \times W_{\text{Earth}}$

If you weigh $600\text{ N}$ on Earth, you will weigh only $100\text{ N}$ on the Moon!

4. Thrust, Pressure, and Principles of Buoyancy

Have you noticed why a sharp knife cuts vegetables effortlessly while a blunt knife fails? Or why camel feet are broad so they don't sink in desert sand?

Thrust and Pressure

• Thrust: The net force acting perpendicular (at $90^\circ$) to a surface. Unit: Newton ($\text{N}$).
• Pressure: The thrust per unit area.

$\text{Pressure } (P) = \frac{\text{Thrust}}{\text{Area}} = \frac{F}{A}$

• SI Unit: $\text{N/m}^2$ or Pascal ($\text{Pa}$).
• Rule: For a fixed force, smaller surface area creates higher pressure!

Buoyancy and Upthrust

When an object is immersed in a liquid (water, oil, etc.), it experiences an upward force exerted by the fluid. This upward force is called Buoyant Force or Upthrust.

Why do objects float or sink?

• Sinks: If the object's density is greater than the liquid's density (Weight of object > Upthrust).
• Floats: If the object's density is less than or equal to the liquid's density (Upthrust $\ge$ Weight of object).

Archimedes' Principle

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.

$\text{Buoyant Force} = \text{Weight of Fluid Displaced}$

    [ Immersed Object ] ---> Displaces Fluid 
                                  │
                                  ▼
    Upward Buoyant Force = Weight of that Displaced Fluid

Applications of Archimedes' Principle:

1. Designing ships and submarines.
2. Lactometers (used to determine the purity of milk).
3. Hydrometers (used to measure the density of liquids).

Guided Practice Questions with Detailed Solutions

Let's test our understanding with three classic numerical and conceptual problems step-by-step.

Question 1: Gravitational Force Calculation

Mass of Earth is $6 \times 10^{24}\text{ kg}$ and mass of the Moon is $7.4 \times 10^{22}\text{ kg}$. If the distance between Earth and Moon is $3.84 \times 10^5\text{ km}$, calculate the force exerted by Earth on the Moon. (Take $G = 6.7 \times 10^{-11}\text{ N}\cdot\text{m}^2/\text{kg}^2$)

Solution:

Step 1: Write down the given values in standard SI units.

• Mass of Earth ($M$) = $6 \times 10^{24}\text{ kg}$
• Mass of Moon ($m$) = $7.4 \times 10^{22}\text{ kg}$
• Distance ($d$) = $3.84 \times 10^5\text{ km} = 3.84 \times 10^8\text{ m}$
• Gravitational Constant ($G$) = $6.7 \times 10^{-11}\text{ N}\cdot\text{m}^2/\text{kg}^2$

Step 2: Apply Universal Law Formula. $F = \frac{G \cdot M \cdot m}{d^2}$

Step 3: Substitute values and calculate. $F = \frac{6.7 \times 10^{-11} \times 6 \times 10^{24} \times 7.4 \times 10^{22}}{(3.84 \times 10^8)^2}$

$F = \frac{297.48 \times 10^{35}}{14.7456 \times 10^{16}}$

$F \approx 2.017 \times 10^{20}\text{ N}$

Answer: The force exerted by the Earth on the Moon is $2.02 \times 10^{20}\text{ N}$.

Question 2: Free-Fall Motion Equations

A ball is thrown vertically upwards and rises to a height of $20\text{ m}$. Calculate:

1. The velocity with which the object was thrown upwards.
2. The total time taken by the object to reach the highest point. (Take $g = 9.8\text{ m/s}^2$)

Solution:

Step 1: Identify given conditions.

• Final velocity ($v$) at highest point = $0\text{ m/s}$
• Distance/Height ($s$) = $20\text{ m}$
• Acceleration ($a$) = $-g = -9.8\text{ m/s}^2$ (Negative because moving upwards against gravity)

Step 2: Solve part (1) using the third equation of motion ($v^2 = u^2 + 2as$). $0^2 = u^2 + 2(-9.8)(20)$ $0 = u^2 - 392$ $u^2 = 392$ $u = \sqrt{392} \approx 19.8\text{ m/s}$

Step 3: Solve part (2) using the first equation of motion ($v = u + at$). $0 = 19.8 + (-9.8)t$ $9.8t = 19.8$ $t = \frac{19.8}{9.8} \approx 2.02\text{ seconds}$

Answer:

1. Initial throw velocity $u = \mathbf{19.8\text{ m/s}}$
2. Time taken to reach peak $t = \mathbf{2.02\text{ s}}$

Question 3: Density, Buoyancy & Floating Condition

A sealed block of volume $500\text{ cm}^3$ has a mass of $600\text{ g}$. Will the block float or sink in water? (Density of water = $1\text{ g/cm}^3$)

Solution:

Step 1: Formula for Density. $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

Step 2: Compute density of the block. $\text{Density of block} = \frac{600\text{ g}}{500\text{ cm}^3} = 1.2\text{ g/cm}^3$

Step 3: Compare with liquid density.

• Density of Block = $1.2\text{ g/cm}^3$
• Density of Water = $1.0\text{ g/cm}^3$

Since the density of the block ($1.2\text{ g/cm}^3$) is greater than the density of water ($1.0\text{ g/cm}^3$), the gravitational force pulling it down exceeds the maximum upthrust force of the water.

Answer: The block will sink in water.

Chapter Summary Checklist

• Universal Law: $F = G\frac{Mm}{d^2}$ connects mass, distance, and gravity everywhere.
• Gravity Acceleration: $g = 9.8\text{ m/s}^2$ at Earth's surface; independent of mass $m$.
• Mass vs Weight: Mass ($\text{kg}$) stays fixed; Weight ($\text{N}$) changes with $g$.
• Buoyancy: Objects float if their density is lower than the fluid's density due to net upward upthrust force.

Keep practicing your NCERT numericals, stay inquisitive, and keep reaching for the stars! Happy learning!