How this instrument works
Newton set out universal gravitation in the Principia in 1687, and the load-bearing word is universal: whatever drops an apple also keeps the Moon on its arc, under one rule with one constant. He sat on the result for years while proving the awkward part — that a sphere of uniform density pulls on anything outside it exactly as though the whole of it were squeezed into a point at the middle. That shell theorem is why the separation here runs centre to centre, and why an astronomer may treat a planet as a dot without apology.
Complete though it was, the law could not produce a number until somebody measured G. Henry Cavendish closed that gap in 1798 with a torsion balance: a 1.8-metre wooden rod slung from a fine wire, small lead weights at its ends, and two 158 kg lead spheres wheeled up beside them. He watched the deflection through a telescope from an adjoining room so his own body heat would not stir the air, and reported Earth's mean density as 5.48 times that of water; the constant was teased out of his figures by later hands. It is still the least well known of the fundamental constants — 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², good to 22 parts per million, whereas the electron's mass is pinned some seventy thousand times more sharply.
Two conditions bound the quotient. The bodies must be point-like, or spherical and far enough apart not to deform one another, so a ring, a dumbbell or a pair of touching spheres wants an integral instead. And the field must be weak. Across most of the solar system the Newtonian figure holds to roughly one part in a hundred million, yet it left Mercury's perihelion creeping forward by an extra 43 arcseconds each century with no explanation until general relativity supplied the missing term in 1915. Close to a neutron star, the arithmetic on this page stops describing anything real.
- Type your figure into First mass, choosing g, kg or t on its unit menu rather than writing out exponents by hand.
- Fill Second mass the same way. Order makes no difference — the expression is symmetric, so swapping the two entries returns an identical answer.
- Set Distance between centres in mm, cm, m or km. Measure from the middle of one body to the middle of the other; zero is rejected.
- Read Gravitational attraction in N, mN or kN. Ten decimal places are carried, because ordinary bench-scale pairs land far below a millinewton.
Worked example — the constant, weighed out
Put 1 kg into First mass, 1 kg into Second mass, and 1 m into Distance between centres. Gravitational attraction reads 6.6743 × 10⁻¹¹ N. That is G itself, digit for digit, because with every other factor set to unity there is nothing else left in the expression — which makes this trio the standard sanity check on any implementation of the law.
Seven hundredths of a nanonewton corresponds to the weight of about seven nanograms on Earth, which is why no one has ever felt it. Cavendish worked with roughly 1.5 × 10⁻⁷ N, some two thousand times more, and even that demanded a wire so fine a draught would spoil the reading. Stand two 70 kg adults a metre apart and this sheet returns 3.3 × 10⁻⁷ N — well under the weight of a grain of sand, and hopelessly swamped by the friction beneath their shoes.
Questions
Does the lighter body feel a weaker pull?
No. Both feel the same magnitude in opposite directions, which is the third law surfacing inside Newton's own gravity formula. Earth tugs a 1 kg book with about 9.8 N, and the book tugs Earth back with about 9.8 N. What differs is the response: divide by our planet's 5.972 × 10²⁴ kg and it accelerates at some 1.6 × 10⁻²⁴ m/s², a figure no instrument will ever resolve. Nothing in the expression distinguishes the heavy partner from the light one.
Should the separation be measured between surfaces or between centres?
Between centres, always — the shell theorem is what licenses treating each body as a point there. Two billiard balls of 3 cm radius resting in contact sit 6 cm apart for this purpose, not zero, so a surface measurement would predict an infinite pull instead of a real one. Orbital work springs the same trap: altitude above ground has to be added to the planet's radius before it goes in. Since the separation is squared, halving it by mistake multiplies the answer by four.
Why is G known so much less precisely than other constants?
Because gravitation cannot be shielded, amplified or switched off, so every determination is a delicate null experiment fighting seismic tremor, thermal drift and the tug of whoever walks past the bench. CODATA 2018 gives 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻², a relative uncertainty of 2.2 × 10⁻⁵, and rival laboratories still disagree by more than their quoted error bars. Orbit work sidesteps the whole problem: satellite ranging fixes the product GM for Earth at 3.986004418 × 10¹⁴ m³/s², nine solid digits, because a product is what an orbit actually senses.
What happens as the separation approaches zero?
For true point masses the quotient runs away to infinity, which is why this sheet refuses a separation of zero. Real bodies are spared that fate. Once you are inside a sphere only the material closer to the middle than you keeps pulling, the outer shells contributing nothing at all, so the attraction drops off in a straight line and reaches zero dead centre. Bore a hole to the middle of Earth and you would hang weightless. What appears here describes bodies from the surface outward.
How do I get surface gravity or weight out of this?
Put the planet into one mass field and its radius into the separation field. Earth's 5.972 × 10²⁴ kg at 6.371 × 10⁶ m yields 9.82 N for every kilogram of the other body — a shade above the conventional 9.80665 m/s², since that standard also allows for spin and for a globe flattened at the poles. Swap in the Moon, 7.346 × 10²² kg at 1.737 × 10⁶ m, and the same kilogram is drawn with 1.62 N.
Why are the answers so tiny next to everyday weights?
Because G is a number beginning eleven decimal places down, and nothing on a bench carries enough matter to make up the deficit. Gravitation turns conspicuous only when one partner reaches planetary scale: it takes Earth's 5.972 × 10²⁴ kg to convert that minuscule constant into the 9.8 newtons per kilogram holding you to the floor. Run the other way and the totals turn absurd — the Moon is held on station by 1.98 × 10²⁰ N, and the Sun grips Earth with 3.54 × 10²² N.