Reading F–r, g–r, U–r and φ–r graph shapes
Force-type curves (F, g) fall as 1/r²; energy-type curves (U, φ) fall as 1/r. Plotted with sign, U and φ are negative and rise towards zero, and the 1/r curve approaches zero more slowly than the 1/r² curve. All four tend to zero as r → ∞.
What this actually means
The examinable content is SHAPE, not numbers. Learn four sketches: g and F as inverse-square decays, φ and U as negative curves climbing towards the axis from below. Every graph MCQ in this topic is one of those four or a combination.
The discriminator between the two families is how fast they die. Because 1/r² shrinks faster than 1/r, the force curve collapses towards the axis while the potential curve is still noticeably below it. In the standard MCQ the potential-type curve is the one that approaches zero more gradually.
One honesty note the deck glosses over: you cannot literally compare a force in newtons with an energy in joules on a shared vertical axis. The diagram is about relative rate of decay and sign, not about which line is numerically higher, so answer on shape.
Sign discipline. If the question plots F with sign (attractive taken as negative) then both F and U sit below the axis. If it plots the MAGNITUDE of g, it sits above. Read the axis label before you pick.
For the two-body case along the line joining Earth and Moon, the g graph dips to zero at the neutral point and then reverses sign, while the φ graph has a MAXIMUM there and stays negative throughout. Those two shapes are a favourite paired MCQ, and confusing them is the single commonest slip.
Inside a uniform planet the picture changes: g rises linearly from zero at the centre out to the surface, then switches to the 1/r² decay. The kink is at r = R, which is where every 'sketch g from the centre outwards' question hides its marks.
Drawing φ crossing the axis, or giving the potential curve the same steepness as the field curve.
Prove it — watch it be true
- Put the φ–r and g–r graphs on screen together and compare how fast each approaches zero.
- Note φ is negative throughout while the g curve decays faster towards the axis.
- Cross-check the inside-the-planet behaviour on the /gravity-field g vs r graph, where g rises linearly to the surface then turns over.