Newton’s law of universal gravitation

Each mass attracts the other. For two positive point masses, $F_G=G\frac{m_1m_2}{r^2}$ gives the force magnitude. Each force points along the line joining the masses towards the other object. They have equal magnitudes but act on different bodies, so they must not be cancelled in the free-body diagram of either body.[1]

r is positive and measured between the mass locations in the model. The centres may be used exactly for two non-overlapping, spherically symmetric bodies. For irregular bodies this is a far-field approximation requiring their dimensions to be small compared with r.[2]

r is the distance between the two point masses; for two non-overlapping spherically symmetric bodies, it is the centre-to-centre distance. For two separate spheres, the visible gap is not the distance used in the equation. Add both radii to that gap. For an object above a planet, add the planet’s radius to altitude. This small geometry step often matters more than the next calculator operation because distance is squared in the denominator.[1][2]

Near an irregular body, centre-of-mass distance alone is insufficient: nearby and distant parts of the source pull with different strengths and directions. Far compared with the body’s dimensions, its total mass can be approximated as concentrated at one point. Inside an extended body, its mass distribution is needed.[2]

The law connects different observations: falling near Earth, celestial motion and weak attraction between laboratory masses. Those observations test its form. A calculation can show the consequences of an assumed law, but units alone cannot prove it. The constant G is measured too, rather than being assigned an exact SI value.[1][5]

An isotropic spherical field spreads its field lines across areas $4\pi r^2$. Constant total flux is therefore compatible with $g\propto\frac1{r^2}$. This illustrates the model’s geometry while already assuming its field structure; it is not an independent proof without experiments.[2]

As a consistency check, a circular orbit obeys $\frac{GMm}{r^2}=\frac{mv^2}{r}$ and $v=\frac{2\pi r}{T}$. Cancelling gives $T^2=\frac{4\pi^2r^3}{GM}$. The model thus reproduces the period–radius relationship for a low-mass companion. Reversing this calculation does not replace tests of other orbits or laboratory forces.[3]

Kepler’s third law extends the relation between orbital period and size to the relative two-body orbit. Its distance parameter is the semimajor axis; a circular orbit is a special case.[7]

Convert 40 cm to 0.40 m before squaring. Convert 6371 km to 6.371×10⁶ m; a factor of 1000 error in distance produces a factor of one million error in force. At fixed geometry, doubling one mass doubles force. Doubling distance reduces force to one quarter. Use these scalings as quick checks.

G is the universal gravitational constant, with SI unit N·m²/kg². By contrast, g is the local gravitational field strength, or free-fall acceleration in the Newtonian model without additional forces. Its unit N/kg is equivalent to m/s². Outside a spherically symmetric source, $g=\frac{GM}{r^2}$: G is constant, whereas g depends on source mass M and centre-to-centre distance r.[1][2]

Two equal source masses on opposite sides of a test mass produce opposing contributions that cancel at the midpoint. Each force magnitude is nevertheless positive. For an asymmetric arrangement, resolve and add components; simply adding magnitudes works only for forces pointing in the same direction.

An extended source can be treated as individual mass elements. Outside a spherical source its total mass determines the field. Inside a spherically symmetric distribution, only mass enclosed within the observation radius contributes to radial force. For a uniform sphere, field magnitude grows proportionally to r from the centre rather than as 1/r².[2]

A larger test body experiences greater gravitational force at the same position. More force is also required to change its velocity at the same rate. In Newton’s model its mass therefore cancels in $a=F_G/m$. This concerns acceleration due to gravity alone. Drag and other forces can still depend on size, shape and mass.[2]

The law supplies the force for a motion calculation, not a completed trajectory. Initial position, initial velocity and Newton’s equation of motion are also needed. For two mobile bodies, account for both motions; the larger body is not automatically stationary. A detailed many-body calculation sums all included mass contributions. For a circular orbit around a dominant central mass, circular-orbit speed follows by equating gravitational force with the required centripetal force.[3]

In a laboratory setup, tiny mutual attraction is often masked by other forces. A calculator result with many digits is not a measurement of matching precision. Practical uncertainty also depends on masses, separation, geometry and environmental disturbances; separation enters quadratically.[1]

The point-mass form is unsuitable when nearby irregular bodies are represented by just their centres, or when surface separation is used for r. Divergence as $r\to0$ does not imply infinite contact forces between real bodies: the assumption of separate point masses is inappropriate there.

Newtonian gravity approximates many weak-field, slow-motion situations well. Light deflection, relativistic precision and strong gravitational fields require general relativity. An orbital estimate using F_G alone also does not calculate a satellite clock’s time correction. In a Newtonian central field, gravitational potential energy describes the corresponding energy of the mass system; its zero level must also be specified.[4][6]

Use centre separation for point masses, separate spherically symmetric bodies or a justified far-field approximation. Identify directions and receiving bodies as well as magnitudes. G is a measured constant; g describes the local field. Inverse-square scaling supplies quick checks, but the force law alone does not specify a trajectory.