That is the whole law. Newton's claim was not that things fall — everyone knew that — but that the apple and the Moon are doing the same thing, governed by one equation with one number in it. This page takes that equation seriously: every figure below is computed from it live, and each one is shown next to the published measurement so you can see exactly where a point-mass model nails reality and where it quietly stops working.
G is a staggeringly small number, which is why gravity feels like a strong force only when one of the masses is a planet. Two people standing a metre apart attract each other with 3.3 × 10⁻⁷ N — about five ten-billionths of one person's weight. Slide the distance and watch the line: on these logarithmic axes the inverse square is a perfectly straight slope of −2, and the two markers show what that means in practice. Double the distance and you keep a quarter of the force, every time, at every scale.
Put one of the masses at the centre of the Earth and divide out the other, and the law stops being about two objects and becomes a property of a place: g = GM/r². Feed in Earth's mass and mean radius and you get 9.820 m/s². The textbook value is 9.80665. That 0.13% gap is not rounding — it is the two things a point sphere cannot know about, and both are recoverable. Build your own world with the sliders, and note how much harder radius works than mass: it is squared, and it is on the bottom.
| world | mass (kg) | radius (km) | GM/r² says | measured | off by |
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Fire horizontally from 300 km up and vary only the speed. Slowly and the shell arcs into the ground; fast enough and the ground curves away underneath it at exactly the rate it falls, and it never lands. Nothing changes at that speed except the number — the same equation is running the whole time. Watch the dot: it visibly races through the low point of an ellipse and dawdles at the far end, which is Kepler's second law falling out of Newton's law without being asked for. Push past 10.93 km/s and the path stops closing altogether.
Set gravity equal to what a circle needs and the mass of the satellite cancels completely: v = √(GM/r), and the period follows as T = 2π√(r³/GM). The ISS is only 6% further from Earth's centre than you are, so it must move at 7.67 km/s, and 92.6 minutes is simply how long that takes. The same formula, unchanged, puts a satellite that circles once per day at 35,786 km — which is where every television dish in your street is pointing — and predicts the Moon's month to within 0.13%. The table checks Kepler's third law against all eight planets.
| planet | semi-major axis (AU) | T = a3/2 predicts | actual period | error |
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The Sun pulls on you 180 times harder than the Moon does. The Moon still wins the tides, because a tide is not caused by the pull — it is caused by the difference in the pull across the width of the Earth, and differencing an inverse square leaves you with an inverse cube. Being 390 times closer beats being 27 million times less massive. That difference points away from the centre on both the near and far sides, which is why there are two bulges and why you get two high tides a day, 12 h 25 min apart. Swing the Moon around and watch the two bulges add and cancel.
Everything on this page comes from F = Gm₁m₂/r² and nothing else, and it was all checked against published measurements before it was drawn. Kepler's third law is reproduced for all eight planets to within 0.06% from their semi-major axes alone. Escape velocity lands on 11.186 km/s, geostationary altitude on 35,793 km against a measured 35,786, and the Moon's sidereal month on 27.29 days against 27.32. The orbits in section 3 are integrated with velocity Verlet, not sketched: over a full circular orbit its energy drifts by 4 × 10⁻¹¹ %, and the period it measures agrees with the analytic 2π√(a³/GM) to five figures — which is the point, because if the picture and the formula disagreed, one of them would be wrong.
The 9.8 you are taught is where the honest trouble starts. GM/r² at Earth's mean radius gives 9.820, and the real thing ranges from 9.780 at the equator to 9.832 at the poles. Two effects a point mass cannot represent account for it: you are on a spinning ball, which subtracts up to ω²R = 0.034 m/s² of centrifugal effect at the equator, and Earth is not a sphere, so the poles sit 21 km closer to the centre. Put both back and the sphere formula lands within 0.08% of 9.80665 — but they had to be put back by hand. The worlds table shows the same failure on a larger scale: for the rocky bodies the sphere model is within a few tenths of a percent, while for Saturn it is 7% out, because Saturn is visibly squashed and its equator is being flung outward once every 10.6 hours.
The tides here are the equilibrium theory, and it predicts a spring range of about 0.78 m everywhere on Earth. Almost nowhere gets that. Real tides are the ocean sloshing in basins that have their own resonant periods, which is why the Bay of Fundy sees 16 m and the Mediterranean barely notices, why high tide arrives hours after the Moon is overhead, and why tide tables are built from decades of local measurements rather than from this equation. The model gives you the driving force correctly and the response not at all.
Left out: every orbit here is two bodies alone, with no perturbation from anything else, no atmospheric drag (which is why the real ISS needs reboosting several times a year, while the one in section 3 would circle forever), and no relativity. Newton's law is not the final word — Mercury's orbit turns 43 arcseconds per century more than this page can explain, and that discrepancy is exactly what general relativity was needed for. The Special Relativity lab is next door. What is remarkable is not that Newton's law eventually fails, but that a single equation with one constant gets you to five decimal places on almost everything first.