For a normalized electron speed distribution $f_T(v)$, the atomic transition rate coefficient is $q_{ij}(T)=\int {\sigma }_{ij}(v)v{f}_T(v)dv$. Its dimensions are volume per time, so the rate per target atom is $n_e{q}_{ij}$. Maxwellian detailed balance gives $q_{ji}=(g_i/g_j)e^{(E_j-E_i)/kT}{q}_{ij}$ for $E_j>E_i$.

Online text: openstax.org/books/college-physics/pages/27-1-the-wave-aspect-of-light-interference

Cited passages: section 27.1, wavelength in a medium: unchanged frequency and wavelength divided by refractive index.

A small organic helper involved in an enzyme's catalysis. It can be transformed in one chemical reaction and regenerated in another.

Time in physics orders physical events and measures durations between them. A clock realizes a time coordinate by counting a reproducible process, such as the oscillations of a simple pendulum. Describing a falling object's motion then means assigning a position to successive clock readings, making its velocity and acceleration measurable.

In Newtonian mechanics, a single common time parameter can be used throughout an experiment. In special relativity, the proper time accumulated by a particular clock depends on its world line, and assigning times to separated events requires a specified reference frame and clock synchronization. Two distant clocks therefore involve more physical structure than simply reading one local dial. The choice of a time origin is conventional; predictions concern durations and relations among measured events.

A mechanical constraint restricts the configurations or velocities a mechanical system can attain. A bead threaded on a rigid circular wire cannot move freely in three dimensions: the wire restricts its position to a circle, leaving one configuration degree of freedom. The supporting wire supplies a normal reaction that enforces this restriction.

A holonomic constraint can be expressed as an equation among configuration coordinates and possibly time, such as $x^2+y^2=a^2$ for a bead on a fixed circle. A nonholonomic constraint restricts allowed velocities without generally reducing to such a position equation; rolling without slipping can produce this situation. Using coordinates adapted to a mechanical constraint can remove unknown reaction forces from the equations of motion, while retaining the physical restriction they impose.

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 relational shape space of a particle configuration is obtained by removing collective translations and rotations; here scale is retained. For generic collision-free, noncollinear configurations of $N\ge 3$ labelled particles in three dimensions, the resulting configuration dimension is $3N-6$. Collinear configurations have nontrivial rotational stabilizers and require singular strata. Removing position and orientation does not automatically remove conserved angular momentum or its centrifugal effect; a dynamically faithful reduction must retain the appropriate momentum-level data.

The configuration space of a mechanical system is the space of its allowed positions or generalized coordinates. Its tangent bundle describes configurations and velocities, and its cotangent bundle describes configurations and conjugate momenta. For smooth holonomic constraints it is a manifold on which geometric Lagrangian and Hamiltonian descriptions can be built. Singular constraints or symmetry quotients can require strata rather than one global smooth manifold.

Inertia is a body's persistence in its state of rest or constant velocity when the net external force vanishes in an inertial frame. It is not an extra force that pushes a moving body forward. Newton's first law describes this persistence; for a point mass, Newton's second law quantifies how a nonzero net external force changes its momentum. At constant mass, ${\mathbf{F}}_{\mathrm{net}}=m\mathbf{a}$, so a larger mass gives a smaller acceleration for the same net force.

A moving cart does not need a forward net force to maintain its velocity. Ordinary carts slow because friction and drag force produce an opposing net external force; reducing those interactions reveals Newton's first law more clearly. The air-track test of inertial motion separates this persistence from the initial push. In overdamped particle dynamics, the inertial acceleration term is small compared with drag force over the time scales considered; that approximation does not mean that inertia has disappeared.

A body exchanging mass with material of inertial velocity $\mathbf{u}$ satisfies the displayed momentum conservation law for one stream, with signed $\dot{m}$ positive for entry and negative for exit. The force $\mathbf{F}$ excludes the incoming or outgoing momentum flux. A short time interval gives $d(m\mathbf{v})/dt=\mathbf{F}+\mathbf{u}\dot{m}$, proving the formula. For stationary incoming matter, $d(m\mathbf{v})/dt=\mathbf{F}$; for matter leaving at the body's own velocity, $m\dot{\mathbf{v}}=\mathbf{F}$. The rocket equation and accreting-drop motion are applications.

Online text: farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node53.html

Cited passages: Localized Current Distribution, equations 661–677: distant vector-potential expansion and dipole field; Localized Current Distribution, equations 678–679: closed-loop integral and planar area normal; Richard Fitzpatrick, Localized Current Distribution, equations 664–675: vanishing boundary terms, current identities and moment integral.

Online text: farside.ph.utexas.edu/teaching/em/lectures/node74.html

Cited passages: Richard Fitzpatrick, Magnetic susceptibility and permeability, equations (866)–(868): isotropic linear response and permeability.

Online text: farside.ph.utexas.edu/teaching/em/lectures/node69.html

Cited passages: Richard Fitzpatrick, Polarization, equations (807)–(812): displacement, polarization and susceptibility; this source uses epsilon for relative permittivity.

A chromatid is one of the two copies of a replicated chromosome. Each homologous chromosome contributes two sister chromatids before separation in meiosis; chromosomal crossovers can exchange segments between nonsister chromatids.

// Target: biology.bigb

Homologous chromosomes carry corresponding genetic loci, although their alleles may differ. In the diploid $AB/ab$ example, the two homologous chromosomes carry the parental haplotypes $AB$ and $ab$.