Quick questions on Newton's law of universal gravitation: WACE Year 12 Physics Unit 3

Putting one mass as a planet $M$ and the other as a test mass $m$ at the surface (radius $r$), the weight is $mg=GMm/r^2$, so

The law as written applies strictly to point masses, yet WACE happily uses it for planets, moons and satellites. The justification is Newton's shell theorem: a body with a spherically symmetric mass distribution produces, at any external point, exactly the field it would if all its mass were concentrated at its centre. This is why $r$ is always the centre-to-centre distance and why we can treat Earth as a point of mass $M$ at its core when computing surface gravity or orbital forces. The approximation fails only when the separation is comparable to the bodies' own sizes or when the mass distribution is markedly non-spherical, neither of which arises in standard WACE problems.

Many gravitation marks come from ratio reasoning rather than full substitution. The trick is always to write the target quantity as a formula, form the ratio of the two cases, and let every unchanged quantity cancel. For force, $\dfrac{F_2}{F_1}=\dfrac{m_{1,2}m_{2,2}}{m_{1,1}m_{2,1}}\left(\dfrac{r_1}{r_2}\right)^2$; for surface gravity, $\dfrac{g_2}{g_1}=\dfrac{M_2}{M_1}\left(\dfrac{r_1}{r_2}\right)^2$. Doing this avoids ever needing $G$ and slashes arithmetic errors.