Universal gravitation

01Two masses, one mutual attraction.

In Newtonian mechanics, every mass produces a gravitational field, even a small metal sphere. Both bodies matter: double either mass while keeping everything else fixed and the force doubles. Double the distance between their centres and the force becomes one quarter. This mass dependence and the inverse-square force law are supported by experiment and astronomical observations.

An isolated pair · initially at restPolished spheres · centre-to-centre separation

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Force on mass 1, towards mass 2

F_{2\to1}=1.07\times10^{-9}\,\mathrm{N}

$a_1=5.34\times10^{-11}\,\mathrm{m\,s^{-2}}$

Force on mass 2, towards mass 1

F_{1\to2}=1.07\times10^{-9}\,\mathrm{N}

$a_2=2.14\times10^{-10}\,\mathrm{m\,s^{-2}}$

The equal-length arrows compare the two forces at this instant; numerical readouts give their absolute size. Release both: the lighter body gains speed faster, while the shared centre of mass stays fixed. Motion stops before contact.

F\propto\frac{m_1m_2}{r^2},\qquad \frac{F}{F_0}=\frac{m_1}{20\,\mathrm{kg}}\frac{m_2}{5\,\mathrm{kg}}\left(\frac{2.5\,\mathrm m}{r}\right)^2=1.000

F₀ is the force in the restored example: 20 kg and 5 kg, 2.5 m apart. The live multiplier shows what changed. The distance r is between centres; it is not the gap between surfaces.

Mass 1 makes a field

g₁ means force per kilogram due to mass 1. More source mass makes a stronger field. Distance spreads the field geometrically.

g_1=\frac{Gm_1}{r^2}

Mass 2 responds to it

Multiplying that force per kilogram by mass 2 gives the force on mass 2. Reversing the roles gives exactly the same force magnitude on mass 1.

F_{1\to2}=m_2g_1=\frac{Gm_1m_2}{r^2}

02The constant G carries units.

The mass-and-distance factor gives the right pattern, but is not yet a force in newtons. Introduce a proportionality constant G. First determine the units it must have; its numerical value then has to be measured.

Force

A dimension tells us what kind of quantity we have. M means mass, L length, and T time. From F = ma, force is mass multiplied by length divided by time squared.

[F]=[M][L][T]^{-2},\qquad 1\,\mathrm{N}=1\,\mathrm{kg\,m\,s^{-2}}

[M]^1

mass

[L]^1

length

[T]^{-2}

time

F=G\frac{m_1m_2}{r^2},\qquad G\approx6.67430\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}

\mathrm{N\,m^2\,kg^{-2}}=\mathrm{m^3\,kg^{-1}\,s^{-2}}

$F$ Force magnitude: The attractive force on either body, in newtons (N).
$m_1,\ m_2$ Two masses: Both masses enter the product, measured in kilograms (kg).
$r$ Centre distance: The separation of the two point masses or spherical centres, in metres (m).
$G$ Gravitational constant: A measured constant setting the strength of Newtonian gravity. It is not g.

Matching dimensions is a consistency check. It cannot prove the inverse-square law, the product of masses, or the numerical value of G. Those require physical evidence. G is measured, not an exact defined number; its small value explains why ordinary objects attract so weakly.

03Earth pulls you. You pull Earth. The forces are equal.

Newton’s third law says that the two forces in an interaction are equal in magnitude and opposite in direction. Crucially, they act on different bodies. Select one body below and inspect only the force acting on it.

Only this pair’s gravitational forces are shownIllustration not to scale

Each arrow belongs to the object at its tail. These are two different forces on two different objects. They do not add to 2F on either one.

Your acceleration due to Earth

a_{\rm you}=\frac{F}{m}= 9.820\,\mathrm{m\,s^{-2}}

Earth’s acceleration due to you

a_{\rm Earth}=\frac{F}{M}= 9.87\times10^{-23}\,\mathrm{m\,s^{-2}}

Equal force does not mean equal acceleration: a = F/m. Earth’s enormous mass makes its acceleration due to you extraordinarily small. If you stand on the ground, the upward support force also acts on you; this illustration isolates gravity and does not claim you accelerate through the floor.

\vec F_{\rm Earth\to you}=-\vec F_{\rm you\to Earth},\qquad |F|=\frac{GMm}{r^2}

Do not count the interaction twice

Your force on Earth acts on Earth. Earth’s force on you acts on you. Each body feels one gravitational force of size GMm/r² from the other body, not “Earth’s gravity + your gravity”.

Why do they not cancel your fall?

Only forces on the same body are added when finding its net force. If we instead study Earth and you together as one isolated system, the two internal forces cancel in the system’s total force, so its centre of mass does not accelerate.

Check: Earth pulls you with 600 N. How strongly do you pull Earth?

600 N, in the opposite direction, acting on Earth. Your own gravitational force from Earth is still 600 N, not 1,200 N. The accelerations differ because the masses differ.

04Where this model applies.

Point masses and spheres

The simple formula is exact in Newtonian gravity for point masses and for two non-overlapping spherically symmetric bodies. Extended, irregular bodies require adding the contributions of their parts.

Not a relativistic description

General relativity describes gravity through spacetime geometry. Newton’s model is an excellent approximation for weak gravity and speeds small compared with light, but needs correction in stronger or faster regimes.

G and g are different

G is the universal constant in this law. g is local gravitational field strength, in N/kg (equivalently m/s²), and changes with location. Near Earth’s surface g is about 9.8 m/s².

Later: a different description in general relativity

Read furtherOpenStax · Universal gravitation NIST · Measuring G OpenStax · Newton’s third law NASA · Earth data