D.1.2—Universal gravitation

Core idea

Any two point masses attract along the line joining them. In the equation below, $m_1$ and $m_2$ are the masses, $r$ is their centre-to-centre separation, and $G=6.67\times10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}$.

Worked example — Earth and a book

For $m=1.0\,\mathrm{kg}$, $M=6.0\times10^{24}\,\mathrm{kg}$ and $r=6.4\times10^6\,\mathrm{m}$, $F=(6.67\times10^{-11})(1.0)(6.0\times10^{24})/(6.4\times10^6)^2=9.8\,\mathrm{N}$. This is the book’s weight; the book attracts Earth with the same force magnitude.

Read the scaling

Doubling either mass doubles the force. Doubling the separation reduces the force to one quarter. The two masses exert equal-magnitude forces on each other in opposite directions; the equation gives the interaction force, not two independent forces on the same object.

Choose the model

Use the point-mass equation when the bodies can be treated as point masses, or when a spherically symmetric body is outside its surface and the centre-to-centre separation is used. For extended irregular bodies, the simple centre-to-centre model may not be valid.

Common trap

Do not use the radius of one body as $r$ unless the other mass is effectively at its centre. Convert kilometres to metres before substituting, and remember that gravitational force is attractive rather than repulsive.