Dimensional analysis

Dimensional analysis checks the physical dimensions of quantities and identifies combinations unchanged by the choice of units. It catches impossible equations and reveals scaling before a full dynamical calculation. Adding two quantities requires the same physical dimension; replacing metres by centimetres must change numerical values consistently while preserving physical predictions.

For the ideal simple pendulum, suppose its oscillation period $T$ depends only on mass $m$, length $\ell$, gravitational acceleration $g$, and dimensionless angular amplitude ${\theta }_0$. Write $T=m^a{\ell }^b{g}^{c}F({\theta }_0)$. Since $[m]=M$, $[\ell ]=L$, and $[g]=L{T}^{-2}$, matching physical dimensions gives $a=0$, $b+c=0$, and $-2c=1$. ConsequentlyThe angular amplitude is dimensionless when expressed in radians. Dimensional analysis establishes the length/gravity scaling and the absence of a mass factor under these assumptions, but cannot determine $F$. The harmonic oscillator equation gives $F\to 2\pi$ at small amplitude; the full simple pendulum supplies its finite-amplitude correction. A model with drag or a finite massive support has additional parameters, so the same dimensional argument does not automatically apply.

The result is useful for comparison: at fixed angular amplitude and $g$, lengths in the ratio $1:4$ give periods in the ratio $1:2$. The ratio $T\sqrt{g/\ell }$ should be the same for both in this ideal model. A failure to collapse onto the same value points to measurement error or an omitted parameter. Correct physical dimensions are a necessary consistency check, while the coefficient and the adequacy of the model require more information.

A geometrically similar taut membrane of radius $r$, uniform isotropic tension $T$ per unit length and areal mass density $\sigma$ has normal mode frequencies $f=C{r}^{-1}\sqrt{T/\sigma }$. Matching mass, length and time dimensions fixes these exponents; the dimensionless constant depends on the normal mode and boundary shape. The result does not apply unchanged when bending stiffness adds an independent material parameter.

If mass m, confinement radius R and Planck constant h are the only dimensional inputs to a nonrelativistic kinetic energy, matching mass, length and time dimensions gives $K=C{h}^2/(m{R}^2)$. The dimensionless coefficient C depends on the boundary condition and state. Ratios of energies therefore scale as inverse mass and inverse radius squared only when that dimensionless physics is unchanged. This separates a robust scaling law from a claim about an unspecified confinement spectrum.

If pump power depends only on fluid mass density $\rho$, impeller diameter $D$ and angular velocity $\omega$, dimensional analysis fixes $P=C_P\rho {\omega }^3{D}^5$, where $C_P$ is dimensionless. Comparing pumps with the same $C_P$ gives the displayed scaling. Geometrical or flow-regime changes can change $C_P$; the law is not a universal comparison of arbitrary pumps.

Rescaling dimensional quantities by chosen characteristic scales produces dimensionless variables. The remaining dimensionless ratios identify independent parameters and possible asymptotic regimes.

In the simple pendulum equation, choose the time scale $t_{\ast }=\sqrt{\ell /g}$ and put $\tau =t/t_{\ast }$. The chain rule gives $\ddot{\theta }=(g/\ell )d^2\theta /d{\tau }^2$, so the equation becomes $d^2\theta /d{\tau }^2+\sin \theta =0$. This removes $\ell$ and $g$ from the equation, while the initial conditions still include angular amplitude and dimensionless initial angular speed. A solution can therefore be calculated once and rescaled to many physical lengths and gravity values with the same dimensionless initial data.

The physical dimension of a quantity records its powers of chosen base dimensions, independently of units. For example, velocity has dimension $L{T}^{-1}$ and force has dimension $ML{T}^{-2}$. Dimensional analysis compares these expressions in physical equations.

Natural units choose units so that selected fundamental constants have numerical value one. In relativistic quantum field theory the standard choice is $\hslash =c=1$: energy, mass and inverse length then have the same units. This is a convention, not a physical change of the constants; other natural-unit systems may also normalize other constants.

Planck units use fundamental constants to set natural units of mass, length and time. The displayed convention sets the ordinary Planck mass to one. The related reduced Planck units convention instead normalizes the reduced Planck mass.

Here the reduced Planck mass $M_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}$ is one, so the Friedmann equation for an energy density $\rho$ is $3H^2=\rho$. This differs by a factor of $\sqrt{8\pi }$ from setting the ordinary Planck mass to one; the convention must be stated in inflationary amplitudes.

Dimensional homogeneity requires all terms added in a physical equation, and both sides of that equation, to have the same physical dimension. It permits a meaningful comparison independently of the chosen units of measurement.