Gravitational Potential Graphs

• The vertical axis is potential φ, in J kg⁻¹. The horizontal axis is distance r from the centre.
• Outside the planet, potential follows the relation below. The curve stays below zero and rises towards zero as r increases.
  $$\phi=-\dfrac{GM}{r}$$
• At $2R$, potential is half as negative. At $4R$, it is one quarter as negative (9702/42/F/M/25 Q2(a)).
• A satellite in a circular orbit stays at one radius. Its potential does not change with time, so a φ–time graph is a horizontal line (9702/42/F/M/25 Q2(b)).
• Do not copy the g–r shape. Outside the planet, g changes as $1/r^2$, but φ changes as $-1/r$.

• Rewrite the equation in straight-line form.
  $$\phi=(-GM)\left(\dfrac{1}{r}\right)$$
• This is a straight-line form. Plot φ on the vertical axis and $1/r$ on the horizontal axis.
• The gradient is $-GM$. Its negative sign means the line slopes down.

A φ–r graph is a negative curve. A φ against 1/r graph is a straight line through the origin with gradient −GM.