= Torsion constant
{title2=$\kappa$}
= Torsional spring constant
{synonym}
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:
$$
\tau_{\mathrm{fiber}}=-\kappa\theta,\qquad \kappa=-\frac{d\tau_{\mathrm{fiber}}}{d\theta}>0.
$$
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 $C=G_sJ_p$, with polar <second moment of area> $J_p=\pi r^4/2$. Uniform twist gives $\theta/\ell$ as the twist per unit <length>, so
$$
\kappa=\frac C\ell=\frac{\pi G_s r^4}{2\ell},\qquad U=\frac12\kappa\theta^2.
$$
Here $U$ 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 $\langle\theta^2\rangle=k_B\mathcal T/\kappa$, 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 use
$$
\tau_{\mathrm{cal}}=aF,\qquad \kappa=\frac{\tau_{\mathrm{cal}}}{\Delta\theta}.
$$
Several 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>. https://cdn.pasco.com/product_document/Coulombs-Law-Apparatus-Manual-ES-9070.pdf[Bruce Lee, PASCO ES-9070 manual, pp. 9–10: mass calibration and force-per-degree convention]{ref}
\Image[/torsion-balance-calibration.png]
{title=Toy torsion calibrations: static slope and a period corrected for viscous damping}
The figure's ideal static calibration uses $\kappa=1.0\times10^{-8}\,\mathrm{N\,m/rad}$: an applied <torque> of $2.0\times10^{-10}\,\mathrm{N\,m}$ 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>:
$$
I\ddot\theta+\kappa\theta=0,\qquad T_0=2\pi\sqrt{\frac I\kappa},\qquad \kappa=\frac{4\pi^2I}{T_0^2}.
$$
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.
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