Physics (source code)
= Physics
{wiki}
Physics investigates matter, motion, and interactions by building <mathematical models> and testing their consequences against observations. A <physical law> supplies a relation; a model specifies the objects, conditions, and approximations used to apply it. <Classical mechanics>, <electromagnetism>, <thermodynamics>, and <quantum mechanics> organize different aspects of these investigations. Their usefulness comes from turning measurements into predictions that can be checked and refined.
Consider a <simple pendulum>: how does its <oscillation period> change when its <length> changes? For small <amplitude>, the ideal model gives $T_0=2\pi\sqrt{\ell/g}$, where $g$ is <gravitational acceleration>. Taking $g=9.81\,\mathrm{m/s^2}$ as the model parameter, a one-metre pendulum has predicted period $2.0061\,\mathrm s$. Increasing its <length> to four metres gives $4.0121\,\mathrm s$: quadrupling the length doubles the period. The point mass cancels from the equation, so changing it alone does not change this model's answer.
The prediction identifies a mechanism and a way to challenge it. The <small-angle approximation> replaces $\sin\theta$ by $\theta$; the full ideal equation predicts a longer period at large <amplitude>. At $60^\circ$, its period is about $1.07318T_0$, already a $7.3\%$ change. Measurements at such amplitudes can reveal the limit of the approximation, while appreciable drag requires a <damped pendulum>. These distinctions are developed in <simple pendulum> and <dimensional analysis>. The small-amplitude model is also derived in https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums[Ling, Sanny and Moebs, University Physics, pendulums].
\Image[/pendulum-period-scaling.png]
{title=Pendulum predictions connect a measurable scaling law with the limit of a small-angle approximation}
Several different experiments can become one calculation. <Nondimensionalization> rewrites the pendulum equation using $\tau=t\sqrt{g/\ell}$, leaving $d^2\theta/d\tau^2+\sin\theta=0$. All lengths share that dimensionless equation; the <amplitude> remains a parameter. Similarly, <conservation laws> and <symmetry> reveal features that many models share. This is one source of the subject's beauty: a compact relation can organize a large range of phenomena.
The connection to evidence remains essential when a model is elegant. In his introductory lectures, <Richard Feynman> emphasizes testing proposed relations by experiment and using idealizations whose accuracy depends on the conditions. https://www.feynmanlectures.caltech.edu/I_01.html[Feynman, Leighton and Sands, Atoms in motion, section 1–1]{ref} A discrepancy can motivate a better measurement, a better approximation, or a different physical model; the mathematical calculation and the empirical assessment are both part of the explanation.
Back to article page
