Draw the graph of the Earth's gravitational field strength…

Draw the graph of the Earth's gravitational field strength against the distance $r$ of a point from the Earth's centre, $G = G(r)$.

Step 1: Idea

The field strength is the force per kilogram, so this is Problem 13 divided by the test mass. We treat the Earth as a uniform sphere and use the shell theorems: outside, the whole mass acts from the centre; inside, only the mass below the point counts.

Step 2: Outside and inside

Outside ($r \ge R$) the field falls as the inverse square; inside ($r \le R$) the attracting mass $M r^3/R^3$ grows as $r^3$, which divided by $r^2$ gives a linear increase. With $G_0 = \gamma M/R^2 = g$, the surface value:

\[ G(r) = G_0\frac{r}{R}\quad (r \le R), \]

\[ G(r) = G_0\frac{R^2}{r^2}\quad (r \ge R). \]

\[G_{\max} = G_0 = g \approx 9.8\ \mathrm{N/kg}\ \text{at}\ r = R\]

Step 3: Shape of the graph

From the centre the graph is a straight line rising from zero to its peak $G_0$ at the surface. There it has a corner (the slope changes abruptly), and then it falls off along the curve $1/r^2$: at $2R$ it is a quarter of $G_0$, at $3R$ a ninth. The field is strongest on the surface itself.

Check

At $r = 2R$: $G = g/4 = 2.45\ \mathrm{N/kg}$; at $r = R/2$: $G $ $= g/2 $ $= 4.9\ \mathrm{N/kg}\ \checkmark$

The graph is the slope of the potential graph in Problem 32 $\checkmark$