PHY.1:4.3 - Express the comparison through meaningful scales

Choose reference quantities from the physical question. A reference length might be a gap rather than the total apparatus length; a response time might be compared with the duration of forcing. Explain that choice in the working terms.

For a quantity q, choose a positive reference magnitude Q with the same units and write q=Q*q_hat. The number q_hat is its value in that scale. When the law concerns a difference from a reference q0, use q=q0+Q*q_hat and carry q0 through the substitution. This matters, for example, when an absolute pressure and a pressure difference enter different relations.

Substitute the scaled quantities into the physical relations. Transform derivatives and integrals too: if x=L*x_hat and t=T*t_hat, then a time derivative contributes a factor 1/T and a spatial derivative a factor 1/L. Divide each equation by an appropriate nonzero reference contribution. The remaining coefficients show which relative effects must be compared.

For example, in a flow whose speed varies by order U over distance L, characteristic inertial acceleration is U^2/L, viscous acceleration is nu*U/L^2, and gravitational acceleration is g. Their ratios to the inertial contribution are nu/(U*L) and g*L/U^2. These ratios compare physical terms. A different physical account can introduce another contribution and another condition.

Scale the geometry and preparation as well as the equations. A prescribed forcing history is compared at corresponding scaled times; a boundary profile at corresponding scaled positions. Keep a separately imposed forcing duration as an independent condition unless the work makes it proportional to the chosen response time.

For many quantities, dimensional analysis can construct dimensionless combinations systematically. Choose a combination whose physical interpretation makes the comparison useful. MATH.11 supports the construction of quantities unchanged by stated transformations. Dimensionless dependence still needs the physical premises that selected the quantities and relations; it can leave an unknown function to be obtained through MMP.11 or other subject work.