Gravitational Field Strength Graphs

• At a point inside the planet, matter farther from the centre forms complete spherical shells. The pulls from each complete shell cancel.
• Only the mass inside radius r produces a net field at that point. For uniform density, this mass is proportional to the volume. The volume is proportional to $r^3$.
• Substitute the enclosed mass into the field-strength equation. The result is directly proportional to r inside the planet.
  $$g=\dfrac{GM_{\text{inside}}}{r^2}\propto\dfrac{r^3}{r^2}=r$$

1. At the centre, pulls from all directions cancel. The field strength is zero.
2. Inside a uniform-density planet, field strength is directly proportional to radius. Draw the straight-line section from the centre to the surface.
3. At the surface, g has its largest value, $g_0$.
4. Outside the planet, field strength follows an inverse-square curve. The two check values are shown below.
   $$g(2R)=\dfrac{g_0}{4}, \qquad g(3R)=\dfrac{g_0}{9}$$

• If the horizontal axis is signed displacement x through the whole planet, the field has opposite signs on the two sides. At equal distances on the left and right, the field sizes are equal, but every field arrow still points towards the centre (9702/41/O/N/23 Q1(b)(iii)).

Centre: zero. Inside a uniform planet: straight line. Surface: maximum. Outside: inverse-square curve towards zero.