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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 torqueThe 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. ^[ref]

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. ThusFor 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 $a=0.050\mathrm{m}$, $F=2.0\times 10^{-9}\mathrm{N}$, and $\kappa =1.0\times 10^{-8}\mathrm{N}m/rad$, the applied torque is $1.0\times 10^{-10}\mathrm{N}\mathrm{m}$ and the equilibrium twist is $0.010\mathrm{rad}\simeq 0.573^{\circ }$. 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. ^[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. ^[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. ^[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 $\pm 0.01\mathrm{K}$, 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. ^[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.

Torsion balance zero drift is slow motion of the unloaded reference angle ${\beta }_0(t)$ during a measurement. If the torsion constant remains fixed, a constant applied torque gives ${\beta }_{\mathrm{eq}}={\beta }_0(t)+{\tau }_{\mathrm{ext}}/\kappa$. For two equal and opposite applied torques measured at the same reference zero, their equilibrium angles differ by $2{\tau }_{\mathrm{ext}}/\kappa$. A stable offset cancels; a changing zero contributes ${\beta }_0(t_{+})-{\beta }_0(t_{-})$ and does not cancel automatically.

A simple symmetric sequence makes the limitation explicit. Let $b_1,b_2,b_3$ be settled equilibrium angles for applied torques $+\tau ,-\tau ,+\tau$ at equally spaced times $t_0-h,t_0,t_0+h$. If the unloaded zero is linear in time, ${\beta }_0(t)=u+vt$, and $\kappa$ and $\tau$ remain fixed, thenThe arithmetic average of the outer zero values equals the middle zero, so both the constant offset and linear drift cancel. Curved zero motion, unequal time spacing, changing torsion constant, changing signal strength, or unsettled oscillations violate these assumptions. This is an algebraic control example, not a claim that every torsion balance follows a linear drift law.

The torsion constant, or torsional spring constant, is the positive restoring torque per unit twist angle of a complete suspension in its linear elasticity range:Its SI unit is $\mathrm{N}m/rad$. The minus sign describes restoration toward the untwisted orientation. The derivative specifies a local stiffness if the response is not globally linear; using one constant over a finite range additionally requires a straight calibration curve. Return to the same zero after unloading tests recoverable elastic deformation; repeat loading in both directions to check whether the response depends on its previous loading. Staying below permanent deformation is necessary, but does not alone prove proportionality between torque and twist.

For an ideal uniform circular wire of length $\ell$, radius $r$, and shear modulus $G_s$, the filament torsional stiffness is $C=G_s{J}_p$, with polar second moment of area $J_p=\pi r^4/2$. Uniform twist gives $\theta /\ell$ as the twist per unit length, soHere $U$ is stored elastic energy. Doubling the wire's radius multiplies the ideal torsion constant by $16$; doubling its length halves it. This explains why a long thin fiber is useful for weak torque measurement. The formula assumes linear elasticity with uniform material and circular section; it does not calibrate arbitrary ribbons, clamps, or a compound suspension. Within this ideal model, the equipartition theorem predicts an equilibrium thermal mean-square angular displacement $⟨{\theta }^2⟩=k_B\mathcal{T}/\kappa$, where $k_B$ is the Boltzmann constant and $\mathcal{T}$ the uniform absolute temperature. A softer suspension magnifies both the signal and its thermal fluctuations.

Static calibration supplies known torques without assuming the unknown interaction being measured. Apply a known perpendicular force $F$ at lever arm $a$, record the change from the unloaded twist, and useSeveral positive and negative settings test proportionality; fit the slope of applied torque against twist and retain a possible intercept. A nonzero intercept must not be silently absorbed into the slope. Convert degrees to radians before reporting $\mathrm{N}m/rad$. PASCO's procedure turns the Coulomb balance on its side and adds small known masses to generate a gravitational weight at the sphere's lever arm, returning the rod to its index mark by turning the dial. Its tabulated coefficient is force per degree: multiply by the lever arm and $180/\pi$ to obtain our torsion constant. ^[ref]

The figure's ideal static calibration uses $\kappa =1.0\times 10^{-8}\mathrm{N}m/rad$: an applied torque of $2.0\times 10^{-10}\mathrm{N}\mathrm{m}$ gives $0.020\mathrm{rad}$. Both signs lie on the same straight line. The plotted points are calculated examples, not experimental observations; they cannot demonstrate a real fiber's linearity.

Dynamic calibration instead measures the oscillation period $T_0$ with known total suspended moment of inertia $I$. For negligible damping, small free oscillations about a fixed reference obey simple harmonic motion:Include the rod, mirrors, and any attachments in $I$ unless their contributions are demonstrably negligible. Keep the applied torque absent or independent of angle. If an external torque has derivative $g_{\tau }=d{\tau }_{\mathrm{ext}}/d\theta$ at equilibrium, linearization gives effective stiffness $\kappa -g_{\tau }$: timing alone then measures that combination, and stability requires $\kappa -g_{\tau }>0$. A constant applied torque merely shifts equilibrium. Non-negligible damping changes the measured period; damped torsional oscillation gives a correction under a specified damping model. Agreement between static and dynamic calibrations is a useful check on the assumed moment of inertia, reference stability, and restoring law, rather than a reason to ignore their assumptions.