= Cavendish experiment
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The <Cavendish experiment> makes the attraction predicted by <Newton's law of universal gravitation> measurable between laboratory <masses>. Two small test <masses> hang at the ends of a horizontal rod in a <Cavendish torsion balance>; nearby larger source <masses> exert a tiny <torque>. A thin suspension fiber supplies an opposing <torque> as it twists. Measuring its twist and calibrating its <torsion constant> lets an experimenter infer the gravitational <force> and, in a modern analysis, the <gravitational constant>. https://www.phys.uconn.edu/~hamilton/phys258/N/cav.pdf[University of Connecticut, Physics 258/259, The Cavendish Experiment: apparatus and equations of motion, pp. 1–3]{ref}
The beautiful feature is that the two horizontal <forces> can cancel as vectors while their <torques> add. The instrument converts weak attraction into a rotation, with sensitivity set by the fiber's <torsion constant>. Earth's much larger downward <gravitational weight> is supported by the suspension rather than supplying the intended <torque> about its vertical axis. A laboratory measurement can therefore probe the same <Newtonian gravitational field> that governs the <two-body problem>.
Henry Cavendish's original paper was \i[Experiments to determine the Density of the Earth], published in \i[Philosophical Transactions], volume 88 (1798), pp. 469–526, and read on 21 June 1798. On p. 469 he credits John Michell with the apparatus idea and explains that Michell died before using it; the instrument reached Cavendish through Francis John Hyde Wollaston. Cavendish's stated aim was Earth's mean <mass density>. Calling the experiment a determination of the <gravitational constant> is a modern interpretation, not Cavendish's own formulation of his result. https://www.famaf.unc.edu.ar/~wolf/FisicaI/Phil.%20Trans.%20R.%20Soc.%20Lond.-1798-Cavendish-rstl.1798.0022.pdf[Cavendish (1798), original paper, opening p. 469; DOI 10.1098/rstl.1798.0022]{ref}
The connection between Earth's mean <mass density> and the <gravitational constant> follows in a spherical, nonrotating model of <Earth>. With Earth <mass> $M_E$, radius $R_E$, and surface <gravitational acceleration> $g$, the <spherical shell theorem> and <Newton's law of universal gravitation> give
$$
g=\frac{GM_E}{R_E^2},\qquad \overline\rho_E=\frac{3M_E}{4\pi R_E^3},\qquad G=\frac{3g}{4\pi R_E\overline\rho_E}.
$$
Thus a local comparison of gravitational attractions, together with knowledge of Earth's size and surface <gravitational acceleration>, reaches a property of the entire planet. These are modern equations explaining the connection, not equations quoted from Cavendish's paper.
For a worked example, use an explicitly simplified two-near-pair model. Each test <point mass> is $m$, each source <point mass> is $M$, and each <lever arm> is $L$, so the rod's full <length> is $2L$. Neglect the rod's <mass>. At zero twist put the test <masses> at $(-L,0)$ and $(L,0)$, and the sources at $(-L,-r)$ and $(L,r)$. Here $r$ is the near-pair center separation. Ignore each source's attraction on the test <mass> at the opposite end. Evaluate the two near-pair <forces> at zero twist and treat their <torque> as constant for a small twist, requiring $L|\theta|\ll r$. This is a teaching model, not a precision model of Cavendish's actual apparatus.
By <Newton's law of universal gravitation>, each retained <force> has magnitude $F=GmM/r^2$. The left <force> points down and the right <force> points up. Using <torque> about the suspension axis, both contributions are counterclockwise when viewed from above:
$$
\tau_{z,\mathrm{left}}=(-L)(-F)=LF,\qquad \tau_{z,\mathrm{right}}=LF,\qquad |\tau_g|=2L\frac{GmM}{r^2}.
$$
The net retained horizontal <force> is zero, yet the net <torque> is positive. Moving the sources to $(-L,r)$ and $(L,-r)$ reverses both <forces> and hence the <torque>. In a real apparatus, the opposite source also attracts each test <mass>; that correction is explicitly included in the university derivation. https://www.phys.uconn.edu/~hamilton/phys258/N/cav.pdf[University of Connecticut, The Cavendish Experiment, p. 2, equation (1): opposite-source correction]{ref}
\Image[/cavendish-torsion-balance-schematic.png]
{title=Two near-pair forces give the same counterclockwise torque about a vertical suspension fiber}
For illustrative parameters, take $m=0.010\,\mathrm{kg}$, $M=1.0\,\mathrm{kg}$, $r=0.050\,\mathrm m$, $L=0.10\,\mathrm m$, and <torsion constant> $\kappa=1.0\times10^{-8}\,\mathrm{N\,m/rad}$. Use $G=6.67430\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$, the central value in NIST's 2022 CODATA table; that table gives standard uncertainty $0.00015\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$. This is an input for the example, not a measurement made with these assumed parameters. <Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others>{ref}
$$
F=\frac{(6.67430\times10^{-11})(0.010)(1.0)}{(0.050)^2}=2.66972\times10^{-10}\,\mathrm N,\qquad |\tau_g|=2(0.10)F=5.33944\times10^{-11}\,\mathrm{N\,m}.
$$
This sub-nanonewton <force> produces a measurable twist in the <Cavendish torsion balance>. The displayed digits make the toy calculation reproducible; they do not describe experimental precision.
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