 Solution

For point masses $m_1,m_2$ separated by distance $r$, Newton's law of universal gravitation gives an attractive central force of magnitudeThe vector force on body 2 is $-G{m}_1{m}_2({\mathbf{r}}_2-{\mathbf{r}}_1)/|{\mathbf{r}}_2-{\mathbf{r}}_1{|}^3$. For extended bodies one adds their elemental interactions. The shell theorem explains why a spherically symmetric body acts externally like its total mass placed at its centre.

The law has experimental and astronomical support, rather than a deductive proof from definitions. Kepler's laws provide one test. For a circular orbit of radius $r$ and period $T$, the required centripetal acceleration is $4{\pi }^{2}r/T^2$. With $T^2\propto r^3$, it varies as $r^{-2}$. The full elliptical-orbit analysis, with the law of areas identifying a central interaction, gives the same inverse-square dependence and predicts the orbit's form.

The Earth–Moon comparison connects celestial motion to terrestrial falling. Using illustrative mean values $R_{\oplus }\simeq 6.37\times 10^6\mathrm{m}$, $r_M\simeq 3.84\times 10^8\mathrm{m}$ and $g_0\simeq 9.81\mathrm{m}{\mathrm{s}}^{-2}$, the inverse-square prediction isThe circular estimate $4{\pi }^2{r}_M/T_M^2$, with $T_M\simeq 27.3$ days, is about $2.72\times 10^{-3}\mathrm{m}{\mathrm{s}}^{-2}$. Their agreement is the relevant check; the numbers are approximate mean-orbit values, not exact observations of a circular Moon.

The Cavendish experiment adds a laboratory test. Small suspended masses twist a fine fibre when larger source masses are moved nearby. Comparing the torque with the suspension's calibrated restoring response measures the weak attraction between laboratory bodies. Reversing source positions helps separate the gravitational signal from a persistent instrumental offset. Planetary perturbations and additional laboratory geometries provide further tests of the common law. Its excellent accuracy in ordinary weak-field conditions does not make it an exact theory in every regime.