Newton's law of gravitation

Universal gravitation

Newton's law of universal gravitation states that two masses attract each other with a force

F=Grac{m_1m_2}{r^2}

where $m_1$ and $m_2$ are the masses, $r$ is the distance between their centers, and $G$ is the gravitational constant.

The force is attractive and acts along the line joining the masses. In vector form, the force on mass $m$ due to a central mass $M$ is often written

ec{F}=-Grac{Mm}{r^2}hat{r}

where $hat{r}$ points outward from the central mass. The negative sign shows the force points inward.

Inverse-square behavior

Gravity weakens with the square of distance. If distance doubles, gravitational force becomes one fourth as strong. If distance triples, it becomes one ninth as strong.

This inverse-square behavior comes from geometry: the influence of a point source spreads over the surface area of spheres, and sphere area grows as $4pi r^2$.

Gravitational field

The gravitational field is force per unit mass:

ec{g}=rac{ec{F}}{m}

For a spherical mass $M$,

ec{g}=-Grac{M}{r^2}hat{r}

Near Earth's surface, this field has magnitude approximately

$gapprox 9.8 m/s^2$

The familiar weight formula $W=mg$ is a near-surface approximation of universal gravitation.

Connection to near-Earth gravity

At Earth's surface,

g=Grac{M_E}{R_E^2}

where $M_E$ is Earth's mass and $R_E$ is Earth's radius. For heights small compared with Earth's radius, $g$ changes only slightly, so treating it as constant is usually accurate.

For satellites and planets, however, the variation of gravity with $r$ is essential.

Gravitational potential energy

For two masses separated by distance $r$, gravitational potential energy is

U(r)=-Grac{Mm}{r}

The zero of potential energy is chosen at infinite separation. The negative sign means the masses are gravitationally bound; energy must be added to separate them to infinity.

The force comes from the potential:

F_r=-rac{dU}{dr}

Taking the derivative of $U(r)$ gives the inverse-square force.

Escape speed

Escape speed is the minimum speed needed to reach infinity with zero final speed, ignoring air resistance and additional propulsion. Set initial mechanical energy equal to zero:

rac{1}{2}mv_{esc}^2-Grac{Mm}{r}=0

Solving gives

v_{esc}=sqrt{rac{2GM}{r}}

The object's mass cancels, so escape speed from a given planet at a given radius does not depend on the escaping object's mass.

Shell theorem

For spherically symmetric masses, Newton's shell theorem says that outside the sphere, gravity acts as if all mass were concentrated at the center. This is why Earth can often be treated as a point mass for external gravitational calculations.

Inside a uniform spherical shell, the net gravitational force is zero. This result is less intuitive but follows from symmetry and inverse-square geometry.

The big idea

Newton's law of gravitation unifies falling objects, planetary motion, tides, satellites, and escape speed. Gravity is an inverse-square attractive force with associated field and potential energy. Near Earth's surface, it reduces to the simpler constant-$g$ model, but on orbital scales the full $1/r^2$ law is essential.