PHY.1:4 - Solution

Choose the result to transfer → recover its physical causes and conditions → express their relative contributions → construct compatible comparison settings → resolve consequential mismatches → use the result and revise it when the question changes.

PHY.1:4.1 - Choose the physical result and the proposed change

Name the original arrangement and the comparison arrangement. Say what each does and what the comparison is intended to reveal. Distinguish a proposed experiment from an observed one.

Specify the result at the detail needed for its use: total extension under a load, the location of a maximum, a response after a stated duration, or a distribution of repeated outcomes. Include the location, interval and preparation when they affect that result. A request for average deformation differs from a request for deformation at every point.

Identify what may change between the arrangements and what is fixed by the proposed work. The available fluid, gravitational acceleration, material, instrument range or support can constrain the construction. A control may be a time history rather than one setting. Preserve any known dependence between settings: changing temperature can alter both viscosity and density.

If the existing physical account already yields a sufficient answer, use it. Constructing another experiment is worthwhile when it supplies a contribution the work still needs. C.11.DUA helps choose between acting on a sufficient answer and obtaining more information.

PHY.1:4.2 - Recover the interactions that can change the requested result

Follow the physical route by which the proposed input affects the output. Identify the participants, interactions and constraints on that route. Include an exchange across the chosen boundary when it contributes to the result. In a deformation problem, distinguish the force applied at an end from weight distributed through the body. In a transport problem, distinguish material carried by motion from material spread by diffusion.

Use the applicable theory to express those contributions. B.5.TU supplies the passage from theory to the encountered case; A.3.3.TR helps when several interactions determine the same evolving state. State the physical premises still needed rather than filling an unknown interaction with a convenient equation.

Estimate the relative size or time scale of competing effects. Derive that comparison from their laws: which quantity multiplies each term, and what spatial or temporal variation makes the term large? In a regime where one effect is negligible for the output, retain the reason for that simplification and the conditions under which it can fail.

Bring the preparation into the same account. Geometry, contacts, constraints and initial conditions can alter the solution even when the material equations are unchanged. For a field, a boundary condition applies over a surface or interval; matching its value at one point may leave the intended problem different elsewhere.

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.

PHY.1:4.4 - Construct settings that supply the needed correspondence

Write the original and proposed dimensionless relations together. Set equal the coefficients and preparation features needed by the intended transfer, then solve these conditions jointly with the available physical settings. MMP.10 supplies the constraint formulation when several choices interact.

The result may be a recipe: change length by one factor, forcing by another and reading time by a third. Check that the selected properties describe an available material or realizable arrangement. A material’s stiffness and density, for example, may not be independently adjustable. Changing a support or adding a mass changes the physical account as well as a number.

When the scaled equations, domains and preparation coincide, express how a solution in the comparison variables becomes a solution of the original model. Include the output scale. C.29.1 supplies that transfer argument. If the model permits several solutions, the correspondence relates the permitted solutions; identifying one realized history requires the relevant physical preparation or selection conditions. Statistical predictions require the corresponding statistical account.

For a question about one output, try a weaker construction when reproducing the whole problem is unnecessary. Derive how that output depends on the settings and match the dependence needed by the question. Additional loading can reproduce total extension without reproducing local strain, as :5.2 shows. Keep the resulting comparison tied to the output it determines.

PHY.1:4.5 - Resolve a mismatch by its effect on the answer

If the conditions conflict, identify which physical contribution changes and how that could change the requested result. The conflict can itself rule out the proposed experiment at the chosen settings.

Choose the next move from the remaining physical possibilities. You can change a free setting, construct a different arrangement, retain a simpler physical regime, or derive how the unmatched contribution modifies the output. If several mismatched experiments are to be combined, construct the relation that permits that combination; a fitted relation retains its assumptions and the range over which it can be used. MMP.9 and MMP.11 can supply reduction and constrained model construction for these returns.

A small coefficient can support an approximation when its influence on the output is controlled. Inspect where that argument could fail. Thin boundary layers, a threshold, resonance or a long observation interval can make a nominally small contribution consequential. Derive a useful bound, compare an applicable limiting solution, or obtain a discriminating observation when it can settle the use. C.29.1 carries a bound through the later inference.

Use an existing result when it resolves the mismatch at the required strength. A conditional result or a limit can be enough. Additional measurement is selected by what its possible answers would change, including its cost, through C.11.DUA.

PHY.1:4.6 - Use the comparison and follow a changed question

Translate the obtained result into the original quantity, position and time. State the physical conditions on which that translation depends. A short derivation can supply the whole explanation; another contributor needs only the settings, correspondence and unresolved premises that change their use.

If a material experiment performs the comparison, include the effects of preparing, driving and reading it where they affect the inference. C.29.3 supplies this connection when the physical evolution performs a computation. A numerical solution has its own discretization and calculation errors under C.29.2. Keep those errors separate from a changed physical mechanism.

For a changed question, revisit the result first. Asking about a different location, time, range or intervention can make a previously omitted contribution relevant. Retain the conditions that still apply and reconstruct the affected comparison. B.5.MPC.R helps locate a failure across the physical account, mathematical representation and computation.

The useful continuation can be a proposed experiment, a design choice, an interpreted observation, a narrower claim or a new physical question. When the comparison suggests a different way for a team to obtain its result, use ME.7 to describe the proposed operations and their relations. ME.12 checks the claims on which that composition and its description rely and returns a correction to the affected contribution.