Paper 4 Gravitational Fields Answers

Q1 Inverse-square data

1. $q$ values are approximately $0.0204$, $0.0156$, $0.0123$, $0.0100$ and $0.00694$. [2]
2. Axes and units [1]; suitable scale [1]; accurate plots [2]; best-fit straight line [1]. [5]

3. Since $q=10^{12}/r^2$,

$$g=\frac{GM}{10^{12}}q.$$

The fitted gradient is about $400\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}$, so

$$M=\frac{(400)(10^{12})}{6.67\times 10^{-11}}=6.00\times 10^{24}\mathrm{kg}.$$

Accept follow-through from the candidate’s best-fit line. [3] 4. Newton’s law uses centre-to-centre separation for point masses or spherical bodies. Height above the surface would underestimate the distance from the planet’s centre. [2] 5. Planet may not be perfectly spherical/uniform, measurements may be inside the atmosphere, local mass anomalies may exist, or other bodies may contribute fields. [1]

Q2 Orbit data

1. Plot $T^2$ against $r^3$. [1]
2. From $T^2=4{\pi }^2{r}^3/(GM)$, gradient $=4{\pi }^2/(GM)$, so $GM=4{\pi }^2/\text{gradient}$. [2]
3. The orbit radius is measured from the planet’s centre to the satellite. Height above the surface must be added to the planet radius. [2]
4. The orbit should be circular, the planet’s mass dominates, and non-gravitational forces should be negligible. [1]