The Garden Shed Torsion Wire

Henry Cavendish, the Michell Balance, and Weighing the Earth

Mounted in the brick wall of a darkened outbuilding at Clapham Common, South London, two brass vernier telescopes pointed through narrow glass keyholes into a sealed mahogany box six feet wide. Inside that inner case hung a slender wooden deal rod supporting two lead balls two inches in diameter, illuminated by weak beams of candlelight directed through glass windows from outside the room. Throughout the autumn of 1797 and spring of 1798, the natural philosopher Henry Cavendish stood outside in the cold weather, pulling a rope-and-pulley mechanism to swing a pair of massive 348-pound lead cannonballs to within nine inches of the small suspended spheres and logging the sub-millimeter shift of the wooden arm across an ivory scale engraved to twentieths of an inch.

Isaac Newton's Principia of 1687 had established that every pair of masses $M$ and $m$ attracts each other with a force $F = G M m / r^2$, yet for more than a century neither Newton nor his successors knew the numerical value of the universal gravitational constant $G$ or the total mass of the Earth $M_{\oplus}$. Astronomical observations of the Moon's orbit or a seconds pendulum yielded only the combined product $G M_{\oplus}$. Separating the two variables—"weighing the world," as Cavendish called it—required measuring the gravitational pull between two known laboratory masses of lead and comparing that horizontal force against the downward pull of the planet.

The experimental barrier lay in the extreme weakness of gravity. Two lead spheres weighing 348 pounds ($158\text{ kg}$) and 1.61 pounds ($0.73\text{ kg}$) whose centers are separated by 8.85 inches ($22.5\text{ cm}$) attract each other with a force of roughly $1.74 \times 10^{-7}\text{ Newtons}$—equivalent to the weight of a single grain of fine sand divided by three hundred, or one fifty-millionth of the small ball's downward weight. No vertical beam scale could ever detect such a feather-touch because the friction of a steel pivot supporting the balls is tens of thousands of times larger than their mutual gravitational pull.

Around 1783, the Yorkshire geologist and clergyman John Michell conceived a mechanical way to decouple horizontal measurement from vertical weight. By hanging the six-foot wooden rod horizontally from its midpoint on a thin, vertical thirty-nine-inch silvered copper wire, the entire downward weight of the lead balls was borne by the tensile strength of the vertical wire, while horizontal motion was resisted only by the minuscule elastic restoring torque of the wire twisting around its axis. Michell died in 1793 before conducting the experiment, and his apparatus passed to Cavendish, who rebuilt the instrument from scratch to defeat a hidden adversary Michell had underestimated: thermal air convection.

Cavendish discovered that if a human observer stood near the balance case, the radiant heat of the observer's body warmed one side of the wooden box by a fraction of a degree Celsius. That slight temperature gradient set up rising plumes of warm air inside the chamber whose aerodynamic drag pushed the suspended lead balls far harder than the gravity of the 348-pound weights. To silence thermal drafts, Cavendish enclosed the torsion rod inside a tight mahogany box painted black, locked the apparatus inside a garden room, and turned the iron frame carrying the twin 348-pound weights from outside via a worm gear.

When the heavy lead weights $M$ were swung from one side of the small balls $m$ (on a rod of half-length $L$) to the other, the gravitational torque rotated the deal rod through an angle $\theta$ balanced by the wire's torsional stiffness $\kappa$, which was calibrated independently from the natural oscillation period $T$ and rotational inertia $I$ of the suspended rod:

With his original thin copper wire, $\kappa$ was so delicate that the six-foot rod took nearly fifteen minutes ($T \approx 900\text{ s}$) to complete a single slow swing, and the small balls occasionally drifted against the case walls. Cavendish substituted a stiffer wire for later runs that shortened the oscillation period to seven minutes ($T \approx 427\text{ s}$). Because air damping inside the box was so slight that the balance never came to rest, Cavendish timed three consecutive extreme turning points of the oscillating ivory vernier through his telescope to calculate the dynamic midpoint of the swing, moving the 348-pound weights from the positive to the negative position to double the measured deflection to roughly four millimeters.

Correcting for the gravitational pull of the iron suspension rods, the copper wire, and the mahogany box, Cavendish completed seventeen experimental runs comprising twenty-nine measurements. In June 1798, he reported to the Royal Society that Earth's mean density was $5.448 \pm 0.033$ times the density of water—within one percent of the modern geophysical value of $5.515\text{ g/cm}^3$ and corresponding to $G = 6.74 \times 10^{-11}\text{ m}^3\text{kg}^{-1}\text{s}^{-2}$. For nearly a century after his June 1798 paper in the Philosophical Transactions, no physicist on Earth improved significantly upon the gravitational constant implied by the twenty-nine telescope readings Cavendish took outside his Clapham shed.