= Torsion balance
A <torsion balance> measures a weak <torque> by suspending a body from a thin fiber and reading its twist <angle>. The fiber supports the body's <gravitational weight> while permitting rotation about the suspension axis. In the range of <linear elasticity>, a twist $\theta$ in <radians>, measured from the untwisted reference orientation, produces a restoring <torque>
$$
\tau_{\mathrm{fiber}}=-\kappa\theta,\qquad \kappa>0.
$$
The <torsion constant> $\kappa$ sets the sensitivity. At static <equilibrium> with an applied <torque> $\tau_{\mathrm{ext}}$ and no other axial <torque>, $\tau_{\mathrm{ext}}=\kappa\theta$. A perpendicular <force> $F$ at <lever arm> $a$ therefore gives $\theta=aF/\kappa$. The <torsion balance> measures a turning effect: a <force> whose line of action passes through the suspension axis gives no axial <torque>. The <Cavendish torsion balance> is the gravitational application; <Coulomb's law> supplies an electrical one. https://cdn.pasco.com/product_document/Coulombs-Law-Apparatus-Manual-ES-9070.pdf[Bruce Lee, PASCO ES-9070 Coulomb Balance manual, pp. 1 and 7: twist measurement of electric force]{ref}
\Image[/torsion-balance-readout.png]
{title=Torsion balance readout: fiber twist, force geometry, and returning the body to a fixed angle}
The original schematic separates two ways of reading the same <torque>. In deflection operation the top attachment stays fixed and the body rotates. In null operation the experimenter rotates the top attachment by $\alpha$ while returning the body's laboratory <angle> $\beta$ to its original value. If the initially untwisted configuration has $\alpha=\beta=0$, the fiber twist is $\theta=\beta-\alpha$, not $\beta$ alone. Thus
$$
\tau_{\mathrm{fiber}}=-\kappa(\beta-\alpha),\qquad \tau_{\mathrm{ext}}=\kappa(\beta-\alpha).
$$
For a positive applied <torque>, holding $\beta=0$ requires $\alpha=-\tau_{\mathrm{ext}}/\kappa$. Keeping the body at the same <angle> keeps the <lever arm> and source geometry fixed, making comparison of different applied <forces> easier. Either readout still requires a calibrated <torsion constant> and a stable reference orientation.
An illustrative calculation shows the amplification: with $a=0.050\,\mathrm m$, $F=2.0\times10^{-9}\,\mathrm N$, and $\kappa=1.0\times10^{-8}\,\mathrm{N\,m/rad}$, the applied <torque> is $1.0\times10^{-10}\,\mathrm{N\,m}$ and the <equilibrium> twist is $0.010\,\mathrm{rad}\simeq0.573^\circ$. These are stipulated values, not measured data. Reducing the <torsion constant> tenfold increases this twist tenfold, provided the fiber remains in its calibrated <linear elasticity> range. The cost is a slower response, explained by the <oscillation period> in <torsion constant>.
A concrete laboratory experiment is PASCO's Coulomb balance: level its counterbalanced rod, discharge both conducting spheres, and set the unloaded zero. Charge the spheres at maximum separation; switch off the charging supply. Move the second sphere to a known center separation $R$, then turn the suspension dial until the rod returns to its index mark. Record the dial change and $R$, recharge, and repeat at several separations. The manual recommends a draft-free room, clean insulating supports, and recharging before each reading: leakage, especially in humid conditions, otherwise changes the <electric charges> during the sequence. At close separations the conducting spheres redistribute charge, so the <point charge> approximation needs correction. https://cdn.pasco.com/product_document/Coulombs-Law-Apparatus-Manual-ES-9070.pdf[Bruce Lee, PASCO ES-9070 manual, pp. 4–8: setup, force-versus-distance procedure, and corrections]{ref}
For constant <electric charges> and a fixed perpendicular <lever arm>, the <Coulomb's law> prediction is $|\theta|\propto R^{-2}$. Plotting the <logarithm> of twist magnitude against the <logarithm> of separation gives $\log|\theta|=\text{constant}-2\log R$, so the predicted slope is $-2$. Absolute <force> measurement additionally needs the <torsion constant>; comparing ratios can eliminate it if it remains stable. Repeating a reference separation between other settings distinguishes a separation effect from a changing signal. A declining reading at the reference separation would invalidate the assumption of unchanged <electric charges>, even if a distance plot looks smooth.
The historical electrical example is Coulomb's first memoir of 1785. His apparatus enclosed a suspended needle with a small ball and a second ball; an adjustable head twisted the suspension. After charging the balls by contact, he reported angular separations $36^\circ$, $18^\circ$, and $8.5^\circ$ with head settings $0^\circ$, $126^\circ$, and $567^\circ$. The total twists were approximately $36^\circ$, $144^\circ$, and $576^\circ$: fourfold restoring <torque> accompanied approximately halved separation. These are his reported observations, not data from our diagrams. https://cnum.cnam.fr/pgi/redir.php?ident=8CA121-1&onglet=c[Coulomb, Premier mémoire (1785), reprinted in Mémoires de Coulomb, pp. 104–109, CNAM digitization]{ref} A precise reconstruction must use center distances and the changing <lever arm>, rather than identify angular separation exactly with distance; Alberto A. Martínez's replication explicitly analyzes that geometry. https://martinezwritings.com/m/Coulomb_files/CoulombExperiment-1.pdf[Martínez, Replication of Coulomb's Torsion Balance Experiment (2006), pp. 522–525]{ref}
Sensitivity also exposes changes of the apparatus itself. Fleischer and colleagues' cryogenic <torsion balance> used a temperature-controlled enclosure to reduce vacuum-vessel top-plate <temperature> fluctuations to approximately $\pm0.01\,\mathrm K$, limiting changes in suspension geometry from <thermal expansion>. They used a mild bake-out near $40^\circ\mathrm C$ to speed initial fiber relaxation and reduce slow zero motion. Their optical readout compared the pendulum mirror with an internal reference mirror to reject shared optical motion. Room-temperature <thermal radiation> entering through the optical opening heated the pendulum; a cold filter reduced that load. The authors could not directly measure the pendulum's <temperature> without disturbing it, and explicitly distinguished the cold shield's <temperature> from the body's. Cooling therefore requires checking the actual noise and thermal environment, rather than assuming that the fiber and body share the shield's <temperature>. https://arxiv.org/pdf/2206.02890[Fleischer et al., A Cryogenic Torsion Balance Using a Liquid-Cryogen Free, Ultra-Low Vibration Cryostat (2022), §§II B–C and III A]{ref}
A useful control strategy is to record unloaded readings and <temperature> before and after each signal sequence, recalibrate the <torsion constant> under the same conditions, and compare forward and reverse settings. The algebra of <torsion balance zero drift> shows precisely which offsets such comparisons can cancel. A fine angular scale alone does not establish a small <measurement error>.
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