Universal Law of Gravitation

Kepler's laws described planetary motion perfectly but gave no reason why planets move that way. Newton supplied the reason: a single, universal force law.

The law itself. A particle of mass $M_1$ attracts any other particle of mass $M_2$ in the universe with a force that is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them:

Here $\hat{r}$ is the unit vector from $M_1$ towards $M_2$, and $G = 6.67\times10^{-11}\ \text{N m}^2\text{kg}^{-2}$ is the universal gravitational constant. The negative sign is essential: it says the force on $M_2$ always points back toward $M_1$, i.e. gravity is always attractive. By Newton's third law, $\vec{F}_{12} = -\vec{F}_{21}$ -- the force $M_1$ feels from $M_2$ is equal and opposite to the force $M_2$ feels from $M_1$.

Worked check (Example 6.1 style). Two masses $m_1=1\ \text{kg}$ and $m_2=2\ \text{kg}$ are $r=10\ \text{m}$ apart. The force of attraction is

This is an extremely small force -- which is exactly why we never feel the gravitational attraction of ordinary nearby objects; only when one of the masses is planet-sized does the force become significant.

Important features.

• The force falls off as $1/r^2$: double the separation and the force drops to a quarter. This is why Uranus, much farther from the Sun than Earth, feels a far weaker solar pull than Earth does.
• Gravitational forces always occur as action-reaction pairs: the Sun pulls the Earth toward itself, and the Earth pulls the Sun toward itself with an equal and opposite force.
• The formula $F=GM_1M_2/r^2$ strictly treats both masses as point masses. This is an excellent approximation whenever the separation is much larger than the sizes of the bodies (true for Sun-Earth). One special extended-body case does work out exactly like a point mass, though: a uniform hollow sphere of mass $M$ attracts an external point mass $m$ exactly as if all of $M$ were concentrated at the sphere's centre. And remarkably, a mass placed inside a hollow shell feels zero net force from the shell, from any position inside it (Figure 6.5) -- a result whose full proof is left to higher classes, but which becomes important later when we calculate $g$ at a depth below Earth's surface.
• The famous insight that "the falling apple and the orbiting Moon obey the same force" is the whole triumph of the law: comparing the apple's measured acceleration ($9.8\ \text{m/s}^2$) against the Moon's centripetal acceleration (from its 27.3-day orbit at $60$ Earth radii) gives a ratio of almost exactly $3600 = 60^2$, precisely matching the inverse-square prediction $a_{apple}/a_{Moon} = (R_{Moon}/R_{Earth})^2$. …