Newton's Law of Universal Gravitation

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Newton's Law

Every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of their separation

Universal Gravitation

Gravitation Detailed

Newton's Law of Universal Gravitation states that every particle of matter in the universe 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 their centres:

where:

• $F$ = gravitational force (N) — always attractive
• $G$ = universal gravitational constant = $6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}$
• $M$, $m$ = masses of the two objects (kg)
• $r$ = centre-to-centre distance between the masses (m)

The force acts along the line joining the centres of the two masses. It is a mutual force — if mass $M$ exerts a force on mass $m$, then mass $m$ exerts an equal and opposite force on mass $M$ (Newton's third law).

The Inverse Square Relationship

The force is inversely proportional to the square of the distance:

$F \propto \dfrac{1}{r^2}$

This means:

• If $r$ doubles, $F$ becomes $\dfrac{1}{4}$ (one-quarter)
• If $r$ triples, $F$ becomes $\dfrac{1}{9}$ (one-ninth)
• If $r$ is halved, $F$ becomes $4$ times larger

The gravitational field strength $g$ at a point is defined as the gravitational force per unit mass experienced by a small test mass placed at that point:

where $r$ is the centre-to-centre distance from the centre of the massive object (e.g. planet) to the point where the field strength is being calculated. For a point at height $h$ above a planet's surface:

$r = R + h$

where $R$ is the radius of the planet and $h$ is the altitude above the surface.

At Earth's Surface

At Earth's surface, $r = R_{\text{Earth}} = 6.37 \times 10^6 \text{ m}$:

$g = \dfrac{GM}{R^2} = \dfrac{(6.67 \times 10^{-11}) \times (5.97 \times 10^{24})}{(6.37 \times 10^6)^2} = 9.83 \text{ m/s}^2$

This value ($9.83 \text{ m/s}^2$) is close to the commonly quoted $9.8 \text{ m/s}^2$. The small difference arises from Earth's rotation and the fact that Earth is not a perfect sphere.

At Height $h$ Above the Surface

For a point at altitude $h$ above a planet's surface, the field strength $g'$ can be found in two equivalent ways:

The inverse square law is one of the most important patterns in physics. If a quantity spreads out uniformly in three-dimensional space from a point source, its intensity falls off as the inverse square of the distance.

Visualising the Inverse Square Law

Imagine gravitational force spreading out uniformly over the surface of an expanding sphere centred on the source mass. The surface area of a sphere is $4\pi r^2$, so as the sphere expands, the same total force is spread over an area that increases as $r^2$. The force per unit area therefore decreases as $1/r^2$.

Graphical Representation

When we plot gravitational force $F$ against distance $r$, we get a curve that decreases rapidly:

• Graph of $F$ vs $r$: a hyperbolic curve that approaches zero asymptotically
• Graph of $F$ vs $\dfrac{1}{r^2}$: a straight line through the origin with gradient $GMm$

The linear relationship between $F$ and $1/r^2$ is powerful because it allows us to verify the inverse square law experimentally — plotting measured force against $1/r^2$ should yield a straight line.

Other Inverse Square Laws in Physics

The inverse square law applies to any phenomenon where a quantity radiates uniformly from a point source in three dimensions:

• Gravitational force: $F = \dfrac{GMm}{r^2}$
• Electrostatic force (Coulomb's law): $F = \dfrac{kq_1q_2}{r^2}$
• Light intensity: $I = \dfrac{P}{4\pi r^2}$ where $P$ is the power of the source