The law in simple words
Newton saw that the force pulling an apple down and the force holding the Moon in its path are the same kind of force. He said:
Every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
F = G m₁ m₂ / r²
- F = force of gravity (newton, N)
- m₁, m₂ = the two masses (kg)
- r = distance between their centres (m)
- G = universal gravitational constant
In vector form, the force on m₂ due to m₁ is F₂₁ = −G m₁ m₂ r̂₁₂ / r². The minus sign shows the force points back toward m₁: gravity always attracts.
Features of the gravitational force
- Always attractive. There is no push-apart gravity.
- Central force: it acts along the line joining the two centres.
- Equal and opposite pair: F₁₂ = −F₂₁ (Newton's third law). The Earth pulls you with 500 N; you pull the Earth with 500 N too. The Earth barely moves because its mass is huge.
- Does not need a medium. It works through empty space.
- Does not depend on what is in between and does not depend on the material of the bodies.
- Very weak for everyday objects, but huge for planets and stars because their masses are huge.
The constant G and how it was measured
G = 6.67 × 10⁻¹¹ N m² kg⁻². Its dimensional formula is [M⁻¹ L³ T⁻²].
G is called universal because it has the same value everywhere: on Earth, on the Moon, between galaxies. It does not change with the medium or the bodies.
Henry Cavendish first measured it (1798) with a torsion balance: two small lead balls on a light rod hung from a thin wire. Two big lead spheres were brought near them. The tiny gravity pull twisted the wire by a small angle. Knowing how stiff the wire is, he found the force and then G.
Do not mix up G and g. G is a fixed constant of nature. g is the acceleration due to gravity at a place (about 9.8 m/s² on Earth) and it changes from place to place.
Superposition: many masses at once
If several masses pull one body, find each pull separately using F = G m₁ m₂ / r², as if the others were not there. Then add them as vectors:
F_net = F₁ + F₂ + F₃ + …
If two pulls are at right angles, use Pythagoras: F_net = √(F₁² + F₂²). If they are opposite, subtract.
Big spheres and shells
A uniform sphere pulls an outside mass as if all its mass were at its centre. That is why we use r = distance between centres for the Earth and the Moon. A uniform hollow shell pulls nothing on a mass kept inside it: the pulls from all sides cancel.
Try it: feel the inverse square
Hold a torch 1 m from a wall and mark the bright patch. Move it to 2 m. The patch becomes 2 times wider and 2 times taller, so 4 times the area. The same light is spread over 4 times the area, so each part gets one quarter. Gravity spreads out in space the same way, which is why it goes as 1/r². In the 3D, drag r from 1 m to 2 m and check that F becomes one quarter.
Key formulas and definitions
- F = G m₁ m₂ / r²
- G = 6.67 × 10⁻¹¹ N m² kg⁻²
- [G] = M⁻¹ L³ T⁻²
- F₁₂ = −F₂₁ (equal and opposite)
- F_net = F₁ + F₂ + … (vector sum)
- r doubled → F becomes F/4
Worked examples
1. Find the gravitational force between two 50 kg children sitting 1 m apart.
F = 6.67 × 10⁻¹¹ × 50 × 50 / 1² = 1.67 × 10⁻⁷ N. Far too small to feel.
2. The force between two masses is 12 N. What is it if the distance is made 2 times?
F ∝ 1/r². New F = 12 / 2² = 3 N.
3. The force between two masses is F. One mass is doubled and the distance is halved. Find the new force.
F' = G(2m₁)m₂/(r/2)² = 2 × 4 × F = 8F.
4. Find the force between the Earth (6 × 10²⁴ kg) and the Moon (7.4 × 10²² kg), 3.84 × 10⁸ m apart.
F = 6.67 × 10⁻¹¹ × 6 × 10²⁴ × 7.4 × 10²² / (3.84 × 10⁸)². Top = 2.96 × 10³⁷; bottom = 1.47 × 10¹⁷. F ≈ 2.0 × 10²⁰ N.
5. Find the Earth's pull on a 60 kg student on its surface (M = 6 × 10²⁴ kg, R = 6.4 × 10⁶ m).
F = 6.67 × 10⁻¹¹ × 6 × 10²⁴ × 60 / (6.4 × 10⁶)² = 2.40 × 10¹⁶ / 4.1 × 10¹³ ≈ 586 N. This is the student's weight (≈ 60 × 9.8).
6. Three equal masses m sit at three corners of a square of side a. Find the net force on the mass at the corner between the other two (the right-angle corner).
Each neighbour pulls with F = Gm²/a², at 90° to each other. F_net = √(F² + F²) = √2 Gm²/a², along the diagonal.
7. Where between the Earth and the Moon (distance d, Earth 81 times heavier) is the net pull on a body zero?
At distance x from Earth: G(81M)/x² = GM/(d − x)². So 9/x = 1/(d − x), 9d − 9x = x, x = 0.9 d from the Earth.
Common mistakes
- Thinking a heavier body pulls harder on a lighter one than the lighter pulls back. The two forces are always equal and opposite.
- Using r as the distance between surfaces of two spheres. Use the distance between centres.
- Writing F ∝ 1/r. Gravity is inverse square: halve r and F becomes 4 times.
- Mixing up G and g. G = 6.67 × 10⁻¹¹ N m² kg⁻² is universal; g ≈ 9.8 m/s² changes with place.