From 'action at a distance' to a field
Newton's law tells us how strong the pull between two masses is, but it is silent on how the Earth reaches out and tugs the Moon across 380,000 km of vacuum. Newton himself was uneasy about this 'action at a distance.' The modern fix is the idea of a field: the Earth fills the space around it with a gravitational field, and any mass placed in that field feels a force locally, right where it sits.
The gravitational field \vec{g} at a point is defined as the force per unit mass that a small test mass would feel there. It is a vector — it has a strength and a direction (toward the source) at every point in space.
The gravitational field is the force per unit mass. Its units are N/kg, which are exactly the same as m/s² — the field strength IS the acceleration a free mass would have.
The field of a planet: g = GM/r²
Put the two ideas together. A planet of mass M exerts a force F = GMm/r^{2} on a test mass m. Divide by m to get the field, and the test mass cancels out — the field belongs to the planet alone.
The gravitational field strength a distance r from the center of a mass M. At Earth's surface, r = R⊕ ≈ 6.37×10⁶ m and M⊕ ≈ 5.97×10²⁴ kg give g ≈ 9.8 m/s².
Mass versus weight — a stubborn confusion
Weight equals mass times the local gravitational field. On the Moon g ≈ 1.6 m/s², so your weight is about one-sixth of its Earth value — but your mass is unchanged.
A 60 kg astronaut has a mass of 60 kg on Earth, on the Moon, and drifting between them. On Earth she weighs 60 \times 9.8 \approx 590\ \mathrm{N}; on the Moon only about 60 \times 1.6 \approx 96\ \mathrm{N}. When people say something 'weighs 60 kilograms,' they are loosely quoting mass; the true weight is a force in newtons.
Apparent weight and the elevator
What a bathroom scale actually reads is not your weight but the support force pushing up on you — your apparent weight. Stand still and the two are equal. But ride an elevator that accelerates upward and the floor must push harder; the scale reads more, and you feel heavier. Accelerate downward and the scale reads less.
Now the punchline. If the elevator cable snapped and the whole thing fell freely, the floor would fall away just as fast as you do — it can no longer push on you at all, and the scale reads zero. You are not out of gravity's reach; you are in weightlessness, because you and your support are falling together. Hold onto that idea: it is the exact secret of how astronauts float in orbit, which we unpack in the next guide.