PHY.6:4 - Solution

Choose the consequence → identify participants and exchanges → supply response laws → connect the relations → make the preparation consistent → derive and use the needed consequence.

PHY.6:4.1 - Choose the consequence and physical participants

State what the answer will let someone interpret, choose or do. Identify the quantity, interval and preparation that matter. A peak response, a total transferred amount and a settled value can require different accounts of the same arrangement.

Choose the physical participants and their boundaries at a useful resolution. Locate the interactions that can affect the answer, including supports, surrounding media and measurement when relevant. Distinguish a quantity stored in a participant from a quantity passing through its boundary. Include storage in a contact or field when omitting it would change the consequence.

Use PHY.5 to resolve a consequential choice of scale or omitted interaction. A spatial field, a few aggregate variables and individual particles are possible descriptions; their physical adequacy depends on the question and regime.

PHY.6:4.2 - Construct the balances with physical meanings

Use C.29.BB to choose the additive quantity, common interval, signs, transfers and internal production. Write each term with its physical meaning and compatible units. For vector quantities, use a common frame or an explicit transformation between frames.

For a fixed spatial region, a useful form is

rate of stored quantity = inward transfer - outward transfer + internal production.

A local continuity equation expresses the same relation for a density and its flux. The choice of quantity determines the production term: a chemical species can be consumed while the atoms it contains remain in reaction products. A closed boundary alone does not make every selected quantity constant.

Identify which terms remain undetermined. A momentum balance may still need forces; a species balance may need transport and reaction rates. If the requested total already follows without those details, retain that consequence and avoid completing an unnecessary model.

PHY.6:4.3 - Supply the response and configuration relations

For each unresolved interaction that affects the answer, state how its response depends on the physical conditions. Use an applicable theory, an established material response, an available measurement or a stated working hypothesis. PHY.4 helps constrain a law whose form is unknown; MMP.11 helps represent its remaining freedom.

Recover the conditions under which the relation is used. Does it describe the current state, dependence on earlier states, a spatial gradient, an average response or fluctuations? Does it assume a settled contact, a constant material parameter or a particular preparation? Supply the configuration relations needed to connect that response to the retained variables.

Keep physical restrictions that the chosen description relies on. A passive damping law transfers mechanical energy into other forms; it does not destroy total energy. If that heating changes the response during the intended use, include the resulting dependence. A fitted or learned response can also supply a relation, but its physical restrictions and usable range must come from its construction or grounds, not from the fact that it produces numbers.

When several physically plausible laws remain, derive what each changes in the requested consequence. A bound or a conditional answer may be sufficient. Seek another observation or a more detailed account when its possible result can change the work; C.11.DUA helps decide that inquiry.

PHY.6:4.4 - Connect the physical relations before choosing a solving order

State what a connection makes common and what it transfers. Match quantities, units, frames and orientations at that connection. Equal values require a physical reason: two locations in contact can still have a finite resistance or an intervening store. If the connection has its own consequential dynamics, describe those dynamics.

Combine the part balances through the common transfer. Transfers internal to the combined boundary cancel when they describe the same exchange over the same interval. Retain conversions between quantities and energy forms. For an energetic connection, derive its power expression from the physical variables; variable names such as potential and flow alone do not establish their product as power.

Keep the equations as simultaneous relations while constructing the physical account. They can include derivatives, algebraic constraints, spatial dependence or statistical response. In a differential-algebraic description, some relations constrain values while others describe change. Which variable is solved for is a later mathematical or computational choice; changing that order need not change the physical interaction.

Eliminate a variable only while retaining the relation or reconstruction needed by later use. MMP.9 supplies mathematical reduction and MMP.10 supplies a formulation for the chosen analysis. If two parts disagree at their connection, return to their quantity meanings and physical assumptions before changing a solver.

PHY.6:4.5 - Make the preparation compatible with the connected account

Supply the independent initial, boundary and driving conditions needed by the requested evolution. Substitute the proposed preparation into the connected relations. Solve for dependent initial values and reactions. Preserve the difference between a physical initial condition and a numerical starting guess.

A motion constraint also constrains admissible initial velocity. An electrical connection can constrain initial potentials. A spatial description needs boundary conditions appropriate to its transport and response. Determine which conditions can be chosen freely and which follow from the others.

An equation count can expose an omitted relation, but equal counts do not establish a consistent or uniquely determined problem. Examine the dependence of the actual relations and the existence conditions needed for the intended result. C.29.2 supplies the subsequent computational formulation; a solver diagnostic can help locate a problem without deciding whether the physical preparation should change.

If the preparation is incompatible, identify the conflict and its physical alternatives. Correct a mistaken initial value, relax an unjustified ideal constraint, or describe the interaction that establishes the new state. Choose among these from the actual preparation. Do not silently substitute an easier initial state.

PHY.6:4.6 - Derive the consequence and return through the implicated premise

Obtain the first result at the resolution the question needs. This may be an initial derivative, an integrated balance, a limiting state or a computed evolution. B.5.MPC connects the mathematical result and its conditions to the physical question.

Use checks that can distinguish a wrong construction for this use. Combining part balances can expose a duplicated exchange; an energy calculation can expose a sign error or an omitted conversion; substituting the preparation can expose an impossible constraint. A limiting case or available observation can test a disputed physical premise. Passing one such check establishes only what it examines.

Distinguish a consequence of the stated equations from evidence that those equations describe the intended situation. Numerical accuracy concerns how the chosen consequence was obtained. Physical adequacy concerns the premises, preparation and interactions represented. Keep those qualifications with the result, without requiring new evidence when the conditional result already serves the work.

When the question, preparation or connection changes, revisit the affected relations. Retain contributions whose conditions still hold. An unresolved exchange law directs work to physical theory or response characterization; inconsistent coupled quantities direct work to the connection; an adequate account with an inaccurate computed result directs work to the computation. B.5.MPC.R supplies the broader repair when physical, mathematical and computational contributions need to change together.