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A small-disturbance, two-dimensional potential flow model for a thin airfoil, using surface tangency and smooth flow departure at its trailing edge to determine circulation. Its lift coefficient slope is $2\pi$ per radian, with camber shifting the zero-lift angle of attack. See airfoil for assumptions, examples and the aerodynamic stall limit.

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.

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.

A triangle is the convex hull of three noncollinear points: the three vertices must not lie on one line. Its boundary consists of the three joining line segments; depending on context, "triangle" can mean this boundary or the filled region it encloses. Here a triangle's area refers to the filled region. The noncollinearity assumption excludes a collapsed shape with zero area, for which ordinary interior angles are not defined.

For example, the positions $(0,0)$, $(4,0)$ and $(0,3)$ form a right triangle with a right angle at the origin. Taking the side of length $4$ as its base gives perpendicular height $3$, so its area is $4\cdot 3/2=6$. The height is measured at a right angle to the chosen base; it is not generally the length of another side. In this example it happens to be a side because the triangle is a right triangle.

By contrast, side lengths $3,4,6$ define a valid triangle: each side is shorter than the sum of the other two, as required by the strict triangle inequality. But $3^2+4^2=25\ne 36=6^2$, so this is not a right triangle and the Pythagorean theorem cannot calculate its longest side from the other two. The law of cosines instead gives the opposite interior angle $C$ throughThis example separates the definition of a triangle from the additional assumption of a right angle.

Every triangle in the Euclidean plane has interior angles summing to $\pi$ radians, a result proved in Euclid's Elements I.32.^[ref] It lets two measured angles determine the third, and underlies the angle-angle criterion for similar triangles. The qualification "Euclidean plane" matters: the earlier spherical triangle demonstrates a different sum.

Trigonometric functions turn an angle into a coordinate or a ratio of lengths. Start at $(1,0)$ on the unit circle and rotate counterclockwise through $\theta$ radians: the resulting point isThe cosine records horizontal position and the sine records vertical position. This definition continues around the whole circle, including negative angles, so the signs encode direction rather than requiring a separate rule for each right triangle.

The unit circle picture explains why these functions are useful beyond measuring triangles. The two coordinates of a point rotating at constant angular speed are periodic functions. Scaling the radius to $A$ and using $\theta =\omega t+\varphi$ gives $x=A\cos (\omega t+\varphi )$ and $y=A\sin (\omega t+\varphi )$. A circular motion becomes two simple repeating traces, connecting Euclidean geometry, simple harmonic motion, and Fourier series.

For an acute angle in a right triangle, rescaling its hypotenuse to one recovers the familiar ratios: sine is opposite side divided by hypotenuse, and cosine is adjacent side divided by hypotenuse. The tangent function is their ratio, $\tan \theta =\sin \theta /\cos \theta$, wherever the denominator is nonzero; the secant function, cosecant, and cotangent give the other reciprocal ratios.

The same names extend to complex numbers through the complex exponential:These definitions agree with the circle interpretation for real angles. For complex arguments, the values need not lie between $-1$ and $1$; the literal circle-coordinate picture is restricted to real arguments. ^[ref]

Choose two point-like astronomical sources with $({\theta }_1,I_1)=(-0.00625,1)$ and $({\theta }_2,I_2)=(0.009375,0.65)$, where the angles are in radians and strengths use arbitrary common units. Their separation is $\Delta \theta =0.015625$ radians. These are invented model parameters, not historical measurements. Their complex visibility follows immediately from the preceding derivation, and its squared magnitude isAt $u=0$, $V=1.65$; at $u=400$, $V=-1+0.65i$; at $u=800$, $V=0.35$. The unequal strengths prevent the complex visibility magnitude from reaching zero. Its modulation contains the source separation, while a common displacement of both sources changes wave phase without changing this magnitude. Amplitude alone loses that location information.

The sparse case samples $u=\pm 12.5k$ for $k\in \{2,5,9,14,20,28,36,44,52,60,64\}$; the denser case samples every $k=1,\dots ,64$, also with both signs. All weights are one. Both reach $|u|=800$, so their shortest fringe period is the same $0.00125$ radians. Their dirty beams and false subsidiary features differ because the intermediate spatial frequencies differ. Neither case includes $u=0$ or frequencies above that maximum. The figure uses a periodic angular interval of length $0.08$ radians; this artificial boundary makes a discrete Fourier transform calculation exact for the chosen grid, rather than modelling an all-sky observation.

The plotted dirty image is the sum of two shifted dirty beams, not a fitted or deconvolved estimate of the source. Adding samples in this example reduces prominent spurious peaks; it does not establish a universal monotonic improvement for every weighting or sky. A missing zero baseline leaves the periodic dirty image with zero mean, so it cannot by itself recover the total positive brightness. Missing short spacings also weaken constraints on extended smooth emission.

Reconstruction is an inverse problem. A perturbation whose Fourier transform vanishes at every sampled coordinate can be added without changing the measurements: incomplete coverage leaves a null space. Nonnegative brightness, a restricted source region, or a model with a small number of components can reduce ambiguity, but each is additional information. Removing dirty beam artefacts means estimating a sky consistent with both data and such assumptions; a narrow central peak alone does not certify a unique image. Measurement error, uncertain calibration and source variability add further limitations.

Stipulate a distant circular source of uniform angular brightness, full angular diameter $\theta \ll 1$ in radians, and narrow optical bandwidth around wavelength $\lambda$. Distinct source patches emit independently. Assume each collector is small enough not to resolve the disk individually. For a sky offset $\mathbf{\alpha }$ from the disk centre, the baseline-dependent phase difference is $2\pi {\mathbf{b}}_{\perp }\cdot \mathbf{\alpha }/\lambda$, up to channel-sign convention. Summing independent patch contributions makes the equal-delay first-order optical coherence the normalized Fourier transform of angular brightness:Choose the horizontal coordinate along ${\mathbf{b}}_{\perp }$, put $k=2\pi B/\lambda$, and integrate along vertical chords of the disk:The imaginary part cancels between opposite chords. Here $J_1$ is the order-one Bessel function of the first kind. One way to obtain the last equality is to integrate the exponential's power series term by term on the finite disk. Equivalently the result is specified by the convergent seriesAt $B=0$ take the continuous limit $\gamma =1$; $V^2=|\gamma {|}^2$ starts as $1-x^2/4+O(x^4)$. This is why very short interferometric baselines give little leverage on angular diameter. Larger disks lose spatial optical coherence at shorter baselines.

The first positive zero of $J_1$ is $x_1\approx 3.831705970$, soThis is a full diameter convention. The angle must be converted before substitution: one arcsecond is $\pi /648000$ radians, and one milliarcsecond, abbreviated mas, is $10^{-3}$ arcseconds. For the stipulated $\theta =5\mathrm{mas}$ and $\lambda =500\mathrm{nm}$, $\theta =2.4240684\times 10^{-8}\mathrm{rad}$ and $B_0\approx 25.1575\mathrm{m}$. At $B=10\mathrm{m}$, the model gives $x\approx 1.52309$ and $V^2\approx 0.542783$.

Beyond $B_0$, $\gamma$ changes sign but $V^2$ stays nonnegative and develops weak further lobes. A single measured squared visibility can therefore match several angular diameters; sampling several interferometric baselines constrains the fit. Limb darkening, spectral averaging, or resolved collector apertures change the uniform-disk prediction. A uniform-disk diameter is a parameter of that brightness model; converting it to a physical stellar radius also requires distance and an appropriate brightness profile.

A wavenumber measures how rapidly a wave phase changes with distance. Two conventions coexist: angular wavenumber counts radians per unit length, while ordinary spatial frequency counts cycles per unit length. For a monochromatic wave with wavelength $\lambda >0$,Here $K\ge 0$ is the magnitude of the angular wavenumber. The symbol $k$ can instead denote a signed one-dimensional angular wavenumber: a factor $e^{ikx}$ has wavelength $2\pi /|k|$ for real $k\ne 0$. Both conventions have dimensions of inverse length; writing $\mathrm{rad}{\mathrm{m}}^{-1}$ or “cycles per metre” exposes the convention. A quoted value in ${\mathrm{m}}^{-1}$ alone does not resolve the factor of $2\pi$. Spectroscopic wavenumber usually uses the cycles-per-length convention.^[ref]

For a real monochromatic plane wave, fix the conventionThe wavevector $\mathbf{k}$ supplies both the spatial wave phase slope and the normal to the wavefronts; its dot product with position makes the wave phase constant on parallel planes. The angular frequency is $\omega =2\pi f$, with $f>0$ the ordinary frequency. Moving a distance $\lambda$ along $\mathbf{k}$ changes the wave phase by $2\pi$. Holding a wave crest's wave phase fixed gives its phase velocity $\omega \mathbf{k}/K^2$ and phase speedIn one dimension the signed phase velocity is $\omega /k$. This identifies a moving pattern; it does not identify the motion of the medium's particles or a wave packet's group velocity.

For the synthetic plane wave below, choose $\lambda =0.40\mathrm{m}$ and $f=5\mathrm{Hz}$. Then $K=5\pi \mathrm{rad}{\mathrm{m}}^{-1}$, ${\nu }_x=2.5$ cycles per metre, $\omega =10\pi \mathrm{rad}{\mathrm{s}}^{-1}$, and $v_p=2\mathrm{m}{\mathrm{s}}^{-1}$. The wave crests are $0.40\mathrm{m}$ apart at one instant; after a quarter oscillation period, each has advanced $0.10\mathrm{m}$. The lower panel retains whole turns of wave phase, making its slope visible instead of reducing it modulo $2\pi$.

The zenith is the point on the celestial sphere in the direction of the observer's local upward vertical. In the horizontal coordinate system, its unit vector is ${\mathbf{e}}_U$ and its altitude is $90^{\circ }$. The zenith belongs to the observer's site; it is not generally the north celestial pole, whose altitude depends on geographical latitude. A direction's zenith distance is its nonnegative angle from the zenith, so $z=90^{\circ }-\gamma$. A direction of altitude $30^{\circ }$ therefore has zenith distance $60^{\circ }$. ^[ref].

At the zenith, every value of azimuth produces $(E,N,U)=(0,0,1)$. There is no horizontal component to identify a bearing: $atan2(0,0)$ cannot provide a geometrically meaningful azimuth, even if software returns a number. This coordinate singularity is a defect of the angular description; the viewing direction itself remains regular. The same ambiguity occurs at the opposite downward vertical. Near the zenith, two directions at fixed altitude separated by a small azimuth increment $d\eta$ have separation $\cos \gamma |d\eta |$ to first order, with angles in radians. At altitude $89^{\circ }$, a $1^{\circ }$ azimuth increment spans only about $0.01745^{\circ }$ on that small arc. An optical telescope can consequently require a large bearing change for a small change of pointing near the zenith; working with unit vectors avoids inventing a preferred bearing at the singular point.