Why do things fall? Long before Newton, people knew apples drop and rivers flow downhill, but nobody could explain why the Moon does not fall to Earth the way an apple does. Newton's brilliant leap was to realise that the same force pulls the apple down and keeps the Moon in orbit. He called it gravitation, and he stated it as a single rule that works for any two objects in the universe.
Newton's Universal Law of Gravitation: Every object in the universe attracts every other object with a force that is (a) directly proportional to the product of their masses, and (b) inversely proportional to the square of the distance between their centres. In symbols:
- $F \propto m_1 m_2$ — double either mass and the force doubles.
- $F \propto \dfrac{1}{r^2}$ — double the distance and the force falls to one-quarter (the inverse-square law).
Combining these gives the famous equation $F = \dfrac{G m_1 m_2}{r^2}$, where $G$ is the universal gravitational constant. Its value, measured by Henry Cavendish, is $G = 6.674 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$. Because $G$ is so tiny, the pull between everyday objects is far too weak to feel — gravitation only becomes obvious when at least one body (like a planet) is enormous.
Acceleration due to gravity ($g$): When Earth pulls a freely falling object, the object accelerates. This acceleration is given the special symbol $g$, and near Earth's surface $g \approx 9.8\ \text{m/s}^2$ (often rounded to $10\ \text{m/s}^2$ for quick calculations). Crucially, $g$ does not depend on the falling object's mass — a feather and a coin fall with the same $g$ in a vacuum.
How $g$ is linked to the law: Put the falling object of mass $m$ at Earth's surface. The gravitational pull is $F = \dfrac{G M m}{R^2}$, where $M$ is Earth's mass and $R$ its radius. But this force also equals $mg$ (force = mass × acceleration). Setting them equal, the $m$ cancels and we get $g = \dfrac{G M}{R^2}$. This single result explains a lot: $g$ depends only on the planet's mass and radius, not on the object dropped.
Why $g$ varies:
- Altitude: higher up, $r$ increases, so $g$ decreases.
- Depth: deep inside Earth only the mass below pulls, so $g$ decreases, reaching zero at the centre.
- Shape: Earth is flattened at the poles, so $g$ is largest at the poles ($\approx 9.83$) and smallest at the equator ($\approx 9.78\ \text{m/s}^2$).
The same law explains the orbits of planets and why a satellite stays up: it keeps ‘falling’ toward Earth but moves sideways fast enough to miss it.