Before you can read U=−rGMm, you must recognise every squiggle in it and every idea that hides behind them. We build them in order — each one uses only the ones before it.
The distinction is only for bookkeeping — both masses actually pull each other equally. We just find it easier to picture the little one moving in the field of the big one.
Gravity's force is always a pull inward, toward the big mass. In the figure below the red arrow on the ball points back toward the planet — never away.
For now, hold on to just this: gravity is a vector pointing inward, and its strength depends on the distance r. We are not yet ready to write the full formula — two of its symbols (r^ and the meaning of r2 on the bottom) still have no picture. We build those in §5 and §6, and only then assemble the complete law.
That r2 in the denominator is why gravity has a special, non-flat shape — and it's precisely why the potential energy will turn out to be 1/r and not something simpler. Hold that thought for §9.
Now every symbol has a picture, so we can finally read the complete force law from Newton's Law of Universal Gravitation:
F=−r2GMmr^.
Decode it piece by piece: the strength is GMm/r2 (constant G × the two masses, weakened by the inverse-square 1/r2), and the direction is −r^ (inward, from §5). Every symbol here was earned above.
The pull's strength changes as r changes (that's the 1/r2). To total the work over a whole journey we can't just multiply once — we slice the path into tiny steps dr, on each of which the force is nearly constant, and add up all the slivers. That "add up infinitely many tiny pieces" machine is the integral∫.
You don't need to do integrals to read this topic — you need to see that ∫∞r means "sum the tiny bits of work all the way from infinitely far in to distance r," and that summing 1/r2 produces −1/r.
Recall Self-test: can you answer each before reading on?
What does r measure, and from where? ::: The straight-line distance between the two masses' centres (not height off the surface).
Why can r never be zero, and what happens to U as r→0? ::: The domain is r>0; r=0 would put both centres at one point. As r→0, U=−GMm/r→−∞ — a singularity (infinitely deep well).
What is r^ and how long is it? ::: A pure-direction arrow of length exactly 1, pointing outward from the centre.
Why does gravity's force carry a minus sign in F=−r2GMmr^? ::: The pull is inward (−r^) while r^ points outward, so the direction flips.
Difference between G and g? ::: G is the universal constant (same everywhere); g is a local field strength computed from G, a planet's mass and radius.
What is dr? ::: A tiny vector step along the path — a microscopic footstep with a small length and a direction.
Full dot-product rule and what it extracts? ::: A⋅B=∣A∣∣B∣cosθ; it extracts the part of one arrow lying along the other, with sign from cosθ.
Why must gravity be conservative for U to exist? ::: Only then does the work depend on endpoints alone, so a single position-number U(r) can describe it.
Why do we need an integral rather than force × distance? ::: The force changes with r; the integral sums infinitely many tiny constant-force slivers.
What is ∫r−2dr and why? ::: −1/r, by the power rule (raise −2 to −1, divide by −1); checked because drd(−1/r)=1/r2.
Why is −GMm/r always negative but mgh can be positive? ::: Different chosen zero points — r=∞ for the former (everything below it is negative), an arbitrary floor for the latter.
What does r=R+h mean? ::: Centre-to-surface distance R plus height above surface h = full centre-to-centre distance r.
