Gravitation — Class 9 (CBSE)

When Newton saw the apple fall, he didn't ask "why does it fall?" Falling was obvious. He asked, "Does the same force that pulls the apple also reach the moon?" The answer turned out to be yes — and that single insight unified terrestrial and celestial physics for the first time. This chapter is about how gravity works on Earth, how it relates to weight, and how the same physics explains why heavy ships float.

1. The story — from an apple to a universal law

In 1666, with the plague closing Cambridge University, 23-year-old Isaac Newton retreated to his family farm. There he watched an apple fall from a tree (or so the famous story goes), and made the conceptual leap:

"If gravity reaches the top of a tree, why not the top of a mountain? And if the top of a mountain, why not the moon?"

He went on to formulate Newton's universal law of gravitation, published in 1687: every particle of matter in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

This was the FIRST time a single law explained both terrestrial physics (falling apples) and celestial physics (orbiting planets). Before Newton, these were thought to obey different rules. After Newton, the universe became unified.

2. Newton's universal law of gravitation

For two point masses $m_1$ and $m_2$ separated by distance $r$:

$\boxed{F=G\frac{m_1{m}_2}{r^2}}$

Where:

• $F$ = gravitational force between them (newtons, N).
• $G$ = gravitational constant = $6.674\times 10^{-11}{\text{N⋅m}}^2/{\text{kg}}^2$. Universal and unchanging.
• $m_1,m_2$ = the two masses (kg).
• $r$ = distance between centres (m).

Three important properties of gravity

1. Always attractive — never repulsive (unlike electricity).
2. Acts at a distance — no contact needed; works across the vacuum of space.
3. Universal — the same law applies to atoms, balls, planets and galaxies.

Why we don't feel gravity from objects around us

Because $G$ is incredibly small. Two 70 kg people standing 1 m apart attract each other with force:

$F=(6.674\times 10^{-11})\cdot \frac{70\times 70}{1^2}\approx 3.3\times 10^{-7}\text{N}$

That's 0.00000033 newtons — totally imperceptible. It takes a massive body (like the Earth, $6\times 10^{24}\text{kg}$) to produce a noticeable force.

3. Free fall and the acceleration due to gravity (g)

Drop an object near Earth's surface. It falls toward the Earth — that's free fall. The acceleration it experiences is $g$.

From Newton's universal law applied to an object near Earth's surface:

$F=G\frac{Mm}{R^2}$

Where $M$ is Earth's mass ($\approx 6\times 10^{24}$ kg) and $R$ is Earth's radius ($\approx 6.4\times 10^6$ m).

But by Newton's 2nd law, $F=mg$ (weight). So:

$g=\frac{GM}{R^2}=\frac{(6.674\times 10^{-11})(6\times 10^{24})}{(6.4\times 10^6{)}^2}$

Calculating: $g\approx 9.8{\text{m/s}}^2$.

That's why a free-falling object accelerates at $9.8{\text{m/s}}^2$ regardless of its mass — the mass cancels in $F=mg=GMm/R^2$.

What varies the value of g?

• Altitude: $g$ decreases as you go higher. Top of Mount Everest: $g\approx 9.77{\text{m/s}}^2$.
• Depth: $g$ also decreases as you go deeper into Earth (mass below shrinks). Centre of Earth: $g=0$.
• Latitude: Earth bulges at the equator due to spin, so $R$ is larger at the equator → $g$ is slightly smaller. Equator: $g\approx 9.78$. Poles: $g\approx 9.83$.

For Class 9 problems, $g$ is taken as either $9.8{\text{m/s}}^2$ or $10{\text{m/s}}^2$ (whichever simplifies calculations).

Galileo's experiment revisited

Aristotle: heavy objects fall faster. Galileo (~1590) dropped a heavy iron ball and a light wooden ball from the Leaning Tower of Pisa — both hit the ground at the same time.

Modern proof: on the Moon (no air resistance), an astronaut dropped a hammer and a feather. They fell side by side and hit the lunar surface together. Beautifully confirmed by Apollo 15 astronaut David Scott in 1971.

The reason: $g$ is mass-independent. All objects experience the same acceleration in free fall (assuming negligible air resistance).

4. Mass vs weight

The most-confused distinction in physics. Let's be clear.

Mass ($m$) — the amount of matter in a body. Scalar. SI unit: kilogram (kg). Universal — doesn't change with location.

Weight ($W$) — the force exerted by gravity on a body. Vector (directed down). SI unit: newton (N).

$W=mg$

How they differ

"I weigh 60 kg"

In everyday language, you say "I weigh 60 kg," but you're actually stating your mass. Your weight on Earth is 60 × 9.8 = 588 N. On the Moon: 60 × 1.6 = 96 N. Your mass is the same in both places.

On the Moon, you'd feel "lighter" because the gravitational force on you is smaller. Your inertia (resistance to motion) is the same.

5. Thrust and pressure

Thrust = force acting perpendicular to a surface. SI unit: newton (N).

Pressure = thrust per unit area.

$P=\frac{F}{A}$

Where $F$ is force (in N) and $A$ is area (in m²). SI unit of pressure: pascal (Pa) = N/m².

Why pressure matters

Two examples with the same force but different pressure:

• A blunt knife (large contact area) doesn't cut bread easily.
• A sharp knife (small contact area) cuts through the same bread — same force, same arm, same cutting motion. The difference is higher pressure due to smaller area.

This is why:

• Knives are sharpened (reduce area → increase pressure).
• Camels have broad feet (large area → low pressure → don't sink in sand).
• A drawing pin's tip is sharp (high pressure on the wall) while its head is broad (low pressure on your finger).
• Trucks have wider tyres than cars (distribute weight over more area → less damage to roads).

Pressure in fluids

Liquids and gases exert pressure on the walls of their container AND on any object placed in them.

Pascal's principle: pressure applied to an enclosed fluid is transmitted equally in all directions. Basis of hydraulic systems (car brakes, lifts, hydraulic presses).

Pressure due to a column of fluid:

$P=\rho gh$

Where $\rho$ is fluid density (kg/m³), $g$ is gravity, $h$ is height. So pressure increases linearly with depth.

6. Buoyancy and Archimedes' principle

When an object is partially or fully immersed in a fluid, the fluid exerts an upward force on it called buoyant force or upthrust. This is buoyancy.

Why buoyancy exists

A submerged object experiences pressure from all sides. The pressure at the bottom is higher than at the top (deeper fluid → more pressure). The pressure difference produces a net upward force = buoyant force.

Archimedes' principle

The buoyant force on an object equals the weight of the fluid the object displaces.

$F_b=\text{weight of displaced fluid}={\rho }_{\text{fluid}}\cdot V_{\text{submerged}}\cdot g$

Three cases when an object is placed in a fluid:

1. Object sinks: object's density > fluid's density. Buoyant force < object's weight. Net force downward → sinks.
2. Object floats: object's density ≤ fluid's density. Buoyant force = object's weight. Net force zero → floats in equilibrium.
3. Object floats partly submerged: object adjusts how much it submerges until displaced fluid's weight equals its own weight.

Why a ship floats and a coin sinks

The coin: small volume, high density (much higher than water). It displaces only a tiny volume of water; the displaced water's weight is much less than the coin's weight. Coin sinks.

The ship: hollow shape means LARGE volume despite the steel's high density. The ship displaces a huge volume of water; the displaced water's weight equals the ship's total weight. Ship floats.

Relative density

Relative density (specific gravity) = ratio of an object's density to water's density.

$\text{Relative density}=\frac{\text{Density of substance}}{\text{Density of water}}=\frac{{\rho }_{\text{substance}}}{1000{\text{kg/m}}^3}$

If relative density < 1: floats in water. If > 1: sinks.

Common values:

• Cork: ≈ 0.2 (floats).
• Wood: ≈ 0.7 (floats).
• Ice: ≈ 0.92 (floats — but only just; 92% submerged).
• Water: 1 (reference).
• Aluminium: 2.7.
• Iron: 7.8.
• Lead: 11.3.
• Gold: 19.3.

This is also why icebergs are dangerous to ships — only ~8% of the ice is above water; ~92% is hidden below.

7. Worked example — buoyancy in water

A solid block of density $800{\text{kg/m}}^3$ and volume $0.5{\text{m}}^3$ is placed in water. Will it sink or float? If it floats, what fraction is submerged?

Step 1 — Density check.

• Block's density: 800 kg/m³.
• Water's density: 1000 kg/m³.
• Block's density < water's density → floats.

Step 2 — Find fraction submerged.

The block floats in equilibrium → buoyant force = weight of block.

${\rho }_{\text{water}}\cdot V_{\text{submerged}}\cdot g={\rho }_{\text{block}}\cdot V_{\text{total}}\cdot g$

Cancel $g$: ${\rho }_{\text{water}}\cdot V_{\text{submerged}}={\rho }_{\text{block}}\cdot V_{\text{total}}$ $\frac{V_{\text{submerged}}}{V_{\text{total}}}=\frac{{\rho }_{\text{block}}}{{\rho }_{\text{water}}}=\frac{800}{1000}=0.8$

Answer: 80% of the block is submerged. The remaining 20% is above water.

This is a general result: fraction submerged = ratio of densities.

8. Closing thought

The universal law of gravitation is one of those rare equations that connects everything:

• Why apples fall.
• Why the moon doesn't fall but orbits.
• Why ocean tides exist (moon's gravity pulling on the oceans).
• Why galaxies form (gravity over cosmic time pulling matter together).
• Why GPS satellites need clock corrections (gravity slows time, slightly).

The same equation gives you everyday weight and buoyancy via the simple chain $F=GMm/R^2=mg$ → $W=mg$ → buoyant force = weight of displaced fluid.

Three centuries on, this law is so well-tested that NASA still uses Newton's gravity (not Einstein's general relativity) to compute spacecraft trajectories. Only at extreme conditions (very strong fields or very high speeds) do Einstein's corrections matter. Newton built the physics that put humans on the moon.