Gravitation
One rule explains a falling apple, the Moon's orbit, your weight, and why a ship of steel floats. This is that rule, and everything that follows from it.
The big picture
Every object in the universe pulls on every other object. That is the whole idea. It sounds almost too simple to be useful, and yet it accounts for the apple leaving the branch, the Moon staying in the sky, and the number the bathroom scale shows you.
The second half of this chapter looks like a different subject — pressure, floating, sinking — but it is not. Things float because a fluid is heavy, and a fluid is heavy because of gravity. The two halves are one chapter for a reason.
The universal law
- F = G m₁m₂ / r²
- Inverse square
- G is tiny
- Forces come in equal pairs
Free fall
- g = GM / R²
- The mass cancels
- v = u + gt
- g varies with place
Mass and weight
- Mass is matter (kg)
- W = mg (newtons)
- Weight changes, mass does not
Pressure in fluids
- P = thrust / area
- Pascal (Pa)
- Pushes in every direction
- Grows with depth
Buoyancy
- Upthrust from a pressure difference
- Archimedes’ principle
- Density decides
Everything pulls on everything
Newton’s claim was that the force pulling an apple down and the force keeping the Moon in orbit are the same force. Before him, the sky and the ground were assumed to run on different rules. His universal law says they do not:
Every object attracts every other object with a force that is proportional to the product of their masses, and inversely proportional to the square of the distance between them.
Written out:
F = G × (m₁ × m₂) / r²
Read it in pieces, because each piece is a claim about the world.
The masses multiply. Double either mass and the force doubles. Note it is the product, not the sum — so a force needs two things with mass. One object alone feels nothing.
The distance is squared, and it divides. Move twice as far apart and the force does not halve — it drops to a quarter. This is called an inverse-square law, and it is the part almost everyone gets wrong on first meeting. Drag the slider and watch how fast the pull collapses.
The starting point. Step forward and watch the bar, not the number.
- Distance
- 1.00 r₀
- Force
- 100.0% of F₀
- Weaker by
- 1.00 ×
G is the constant that fixes the scale. Its measured value is
G = 6.673 × 10⁻¹¹ N m² kg⁻²
Look at how small that number is. Ten to the power of minus eleven. This is why gravity, despite being universal, is a fantastically weak force. Two people standing a metre apart do attract each other — the force is around a hundred-millionth of a newton, which is nothing. Gravity only becomes noticeable when one of the masses is enormous, like a planet. That is the honest reason things fall toward the Earth rather than toward each other.
Newton’s third law is hiding here
The equation is symmetric — swap m₁ and m₂ and nothing changes. So when the Earth pulls an apple down with some force, the apple pulls the Earth up with exactly the same force.
Why does the Earth not visibly leap toward the apple, then? Because acceleration is force divided by mass, and the Earth’s mass is about 6 × 10²⁴ kg. The same force produces an acceleration so small it could never be measured. The forces are equal; the effects are wildly unequal, because the masses are.
What the one law explains
The syllabus asks you to know why this law matters, and the answer is that four separate things you had no reason to connect turn out to be the same phenomenon:
- The force that binds us to the Earth — and holds the atmosphere down with us.
- The motion of the Moon around the Earth.
- The motion of planets around the Sun.
- The tides in the seas, caused by the Moon and the Sun pulling on the water.
Before Newton, each of these needed its own explanation. After him, they need one. That is what makes the law universal — not that it is important, but that it is the same rule everywhere.
Why the Moon does not fall down
This is the question that made the law famous, and the answer is better than it first sounds: the Moon is falling. It has been falling for four billion years. It simply keeps missing.
Imagine firing a cannonball horizontally from a very tall mountain. Fire it gently and it curves down and lands nearby. Fire it harder and it lands further away. All the while it is falling at the same rate — it just travels further sideways before it arrives.
Now turn the speed up and watch what happens to the landing point.
Lands further around than the step before. The fall takes just as long; the ball simply travels further sideways while it happens.
- Launch speed
- 0.40 × v_orbit
- Outcome
- lands
- Lowest point
- the ground
Nothing is holding the Moon up. Gravity is pulling it toward us the whole time, and it is accelerating toward us the whole time. Its sideways velocity is what turns a fall into a circle.
Free fall, and the value of g
An object moving under gravity alone is in free fall. From the universal law, the force on an object of mass m sitting on the Earth’s surface is
F = G × M × m / R² where M is the Earth’s mass and R its radius.
But Newton’s second law also says F = m × a. Both describe the same force, so they can be set equal — and something remarkable falls out. Step through it:
1The universal law
F = G M m / R²
The pull of the Earth on an object of mass m sitting on its surface. M is the Earth’s mass, R its radius.
2Newton’s second law
F = m a
The same force, written a different way — force is mass times the acceleration it produces.
3Set them equal
m a = G M m / R²
There is only one force here, so the two expressions for it must be the same thing.
4Cancel the m
a = G M / R²
The object’s mass appears on both sides, so it divides out — and takes with it any dependence on how heavy the object is. This is why a feather and a stone fall together in a vacuum.
5Give it a name
g = G M / R²
That acceleration belongs to the Earth, not to whatever is falling. We call it g.
The object’s own mass has vanished from the answer. How fast something accelerates under gravity does not depend on how heavy it is.
Put the Earth’s numbers in — M = 6 × 10²⁴ kg, R = 6.4 × 10⁶ m:
g = (6.673 × 10⁻¹¹ × 6 × 10²⁴) / (6.4 × 10⁶)² ≈ 9.8 m/s²
Because free fall is just motion with constant acceleration, the three equations of motion carry over unchanged. Replace a with g, and distance with height h:
- v = u + g t
- h = u t + ½ g t²
- v² = u² + 2 g h
The two graphs below are the same drop seen two ways. Watch how differently they grow: velocity climbs in a straight line, but distance curves upward, because each second the object is already moving faster than it was in the one before.
Sign convention matters more than memorising the equations. Pick a direction as positive and stay with it: if you take downward as positive, then g is +9.8 for a falling object, and an object thrown upward has a negative starting velocity.
Mass and weight are not the same thing
These two words are used interchangeably in ordinary speech and they mean genuinely different things in physics.
Mass is how much matter an object contains. It measures how hard the object is to accelerate. It is the same on Earth, on the Moon, and drifting in deep space. Kilograms.
Weight is the force gravity exerts on that mass: W = m × g. It is a force, so it is measured in newtons, and it changes with wherever you happen to be standing.
| Part | What it does | Why it's there |
|---|---|---|
| Mass (m) | Measures the amount of matter, in kilograms | Never changes — it is a property of the object itself |
| Weight (W = mg) | The force gravity pulls with, in newtons | Changes with location, because g changes |
| g | Acceleration due to gravity at a place, ≈ 9.8 m/s² on Earth | Depends on the planet you are on and how far you are from its centre |
| G | The universal gravitational constant, 6.673 × 10⁻¹¹ N m² kg⁻² | Identical everywhere in the universe — it sets the strength of gravity itself |
Watch what each of them does as the place changes. One number refuses to move.
Earth: g is 9.8 m/s², so the same 50 kg of matter weighs 490 N. Watch the top bar — it has not moved.
- Place
- Earth
- g there
- 9.8 m/s²
- Mass
- 50 kg
- Weight
- 490 N
On the Moon
The Moon has both less mass and a smaller radius than the Earth. Working through g = GM/R² with the Moon’s figures gives about 1.63 m/s², which is close to one-sixth of Earth’s 9.8.
So a student with a mass of 50 kg has:
- On Earth — mass 50 kg, weight 50 × 9.8 = 490 N
- On the Moon — mass still 50 kg, weight 50 × 1.63 ≈ 81.5 N
Same amount of matter, one-sixth the pull. This is why the Apollo astronauts could bounce around in heavy suits.
The second half: pressure in fluids
Everything from here follows from one new idea — that a force spread over a large area does something different from the same force concentrated on a small one.
Thrust is the force acting perpendicular to a surface. Pressure is that thrust divided by the area it acts over:
P = thrust / area
The SI unit is the pascal (Pa), equal to one newton per square metre.
This is why a sharp knife cuts and a blunt one does not — same push from your hand, but a sharp edge concentrates it onto a tiny area, so the pressure is enormous. It is why a camel has wide feet and a nail has a fine point. The force did not change; the area did.
Keep the force fixed and shrink the area it acts on. The pressure does not creep up — it runs away.
Spread over 0.02 m² — about a shoe — the same 500 N becomes 25,000 Pa. The arrow never changed; only the area under it did.
- Force
- 500 N
- Area
- 0.02 m²
- Pressure
- 25,000 Pa
- Roughly like
- a shoe
Fluids — liquids and gases — exert pressure on everything in them, and crucially they push in all directions, not just downward. That sideways-and-upward push is the key to everything below.
Buoyancy: why anything floats
Put an object in water. Water pressure increases with depth, so the water pushing up on the bottom of the object is at a greater depth, and therefore stronger, than the water pushing down on its top. Those two pushes do not cancel. What is left over is a net upward force called the buoyant force, or upthrust.
That is the entire mechanism. Buoyancy is not a special force — it is just the leftover of pressure being bigger lower down.
Push the block deeper and watch both arrows grow. Then watch the number that does not.
At 0.3 m both arrows are longer than at the last step — yet the gap between them is still 1.96 kPa. The upthrust has not changed at all.
- Push down on the top
- 2.94 kPa
- Push up on the bottom
- 4.90 kPa
- Difference
- 1.96 kPa
- Upthrust
- 196 N
Archimedes’ principle
Archimedes worked out exactly how big that upward force is, and the answer is remarkably clean:
When an object is immersed wholly or partly in a fluid, it experiences an upward force equal to the weight of the fluid it displaces.
So to find the buoyant force you do not measure the object at all. You ask how much fluid it pushed out of the way, and weigh that.
Change the block’s density below and watch two things at once: how deep it settles, and what happens to the two arrows.
It settles with 60% under — the same as its relative density, 0.60. That is not a coincidence.
- Density
- 600 kg/m³
- Relative density
- 0.60
- Result
- 60% under
This immediately explains floating and sinking:
- If the object weighs more than the fluid it displaces, weight wins and it sinks.
- If it weighs less, the upthrust wins, it rises, and it settles floating with just enough of itself submerged that the fluid displaced weighs exactly what the object weighs.
Compare like for like and it reduces to density — mass per unit volume. An object denser than the fluid sinks; less dense, it floats.
Where this gets used
Archimedes’ principle is on the syllabus because things are designed around it:
- Ships and submarines. A submarine dives by flooding its ballast tanks — taking on water raises its average density above the water’s. To surface it blows the tanks empty with compressed air.
- Hydrometers, for measuring the density of a liquid. The instrument floats deeper in a thinner liquid, so the depth it settles to is the reading.
- Lactometers, a hydrometer specialised for milk. Milk with water added is less dense, so the lactometer sinks lower — which is how adulteration is spotted.
Relative density
Relative density is the density of a substance divided by the density of water:
relative density = density of the substance / density of water
It is a ratio of two densities, so the units cancel and it has no unit at all. Water’s density is 1000 kg/m³, so a substance with relative density greater than 1 sinks in water and less than 1 floats. It is a quick way to answer “will this float?” without carrying units around — and in Fig. 11 it is also, exactly, the fraction of the block that ends up under water.
Go deeper
Beyond the syllabus. Nothing below is examinable in Class 9 — everything you are assessed on is above this line. These are here for when the chapter has clicked and you want to pull on a thread:
- Work the numericals both ways. Do not only calculate the buoyant force given the volume — practise going backwards, finding the volume submerged given that something floats. Backwards problems are where the exam questions live.
- Ask why the orbit is an ellipse, not a circle. Fig. 3 is drawing real ellipses: notice that below orbital speed the path is an ellipse that happens to intersect the ground. Circles are the special case. Kepler’s laws are the natural next step.
- Look up the Cavendish experiment. G is in every equation in this chapter, and it was measured by hanging lead balls from a wire in 1798. Finding out how anyone weighed the Earth is worth an evening.