The universal law of gravitation
From a falling apple to the Moon
An apple falls from a tree towards the Earth. The Moon goes round the Earth. Newton asked: could the same pull be doing both jobs?
The Moon is about 60 Earth radii away. If the pull weakens as one over distance squared, the Moon’s acceleration should be g divided by 60 squared, which is 3600.
That gives 9.8 divided by 3600, about 2.7 × 10⁻³ metres per second squared. Watch the tiny arrow at the Moon, pointing at the Earth.
Now use circular motion. The Moon goes once round in 27.3 days, on a circle of radius 3.84 × 10⁸ metres. Its centripetal acceleration, 4π²R/T², also comes out to about 2.7 × 10⁻³.
The two numbers match. The force that makes an apple fall is the same force that holds the Moon in its orbit.
Newton went further. Every body in the universe attracts every other body. The force is proportional to the product of their masses and inversely proportional to the square of the distance between them.
In symbols, F equals G m₁ m₂ over r squared. G is the universal gravitational constant, the same everywhere: on Earth, on the Moon, in distant galaxies.
Double one mass and the force doubles. Double the distance and the force drops to a quarter. Watch the arrows shrink as the two balls move apart.
Vector form and the shell theorem
Force is a vector, so let us give it a direction. Put mass m₁ at the origin and mass m₂ at position r. The unit vector r̂ points from m₁ to m₂.
The force on m₂ is F equals minus G m₁ m₂ over r squared, times r̂. The minus sign says the force points back towards m₁. Gravity always attracts.
The force on m₁ is equal in size and opposite in direction. The Earth pulls an apple, and the apple pulls the Earth just as hard: Newton’s third law.
If several masses pull on one body, add their forces as vectors. Watch three arrows join head to tail to give the net force.
The law is for point masses. What about the Earth, a big ball? Think of it as thin hollow shells, one inside another, like the layers of an onion.
Shell theorem, part one: a uniform shell pulls a point outside it as if all its mass were packed at its centre. So for things outside the Earth, the whole Earth acts like a point at its centre.
Part two: for a point inside a uniform shell, the pulls from all sides cancel. The net force is exactly zero, wherever you are inside.
Keep this in mind. It will tell us how g changes when we go deep down a mine.
Cavendish measures G
Newton knew the form of the law, but not the value of G. Gravity between everyday objects is so weak that measuring it is very hard.
In 1798, Henry Cavendish built a torsion balance. A light rod carries two small lead spheres at its ends and hangs from a very thin wire.
Two large lead spheres are brought close to the small ones, on opposite sides. Each small sphere is pulled towards its big neighbour.
The two pulls form a couple, and the rod turns through a tiny angle θ, twisting the wire. The wire twists back until the two torques balance.
A beam of light reflected from a small mirror on the wire makes the tiny twist easy to see on a far scale.
From the angle, the stiffness of the wire, the masses and the distances, Cavendish found G. The modern value is 6.67 × 10⁻¹¹ newton metre squared per kilogram squared.
That is tiny. Two 50 kg students one metre apart pull each other with less than a millionth of a newton. Gravity only becomes strong when one of the masses is huge, like a planet.
Try it: two masses
- Pick a pair, then change the masses and the distance.
- Watch the “compared with the start” readout: it follows m₁m₂/r².
- Try Earth & apple: the apple accelerates at about 9.8 m/s², the Earth hardly at all.
- The two green arrows are always the same length: equal and opposite forces.
Quick check
The distance between two masses is made three times larger. What happens to the gravitational force between them?
- It becomes one-third
- It becomes one-ninth
- It becomes three times larger
- It does not change, because the masses are the same
Answer
B. F ∝ 1/r². If r becomes 3r, F becomes F/3² = F/9.
Inside a shell
A small mass is placed inside a uniform hollow spherical shell, away from its centre. What is the net gravitational force of the shell on it?
- Towards the centre of the shell
- Towards the nearest part of the shell
- Zero
- GMm/r², as if the shell’s mass were at its centre
Answer
C. By the shell theorem, the pulls from different parts of a uniform shell cancel exactly at every point inside it. The nearer part is closer but has less mass in that direction; the farther part is more massive but farther away. The net force is zero.
Summary
- Universal law of gravitation:
F = G m₁m₂ / r² - Vector form (force on m₂):
F = −(G m₁m₂ / r²) r̂ - Gravitational constant:
G = 6.67 × 10⁻¹¹ N m² kg⁻²
- Every body attracts every other body with a force ∝ m₁m₂ and ∝ 1/r². Gravity always attracts.
- The two bodies pull each other with equal and opposite forces (Newton’s third law).
- Shell theorem: outside a uniform shell or sphere, treat its mass as a point at the centre. Inside a uniform shell, the net force is zero.
- Cavendish measured G with a torsion balance. G is very small, so gravity is noticeable only when a huge mass is involved.