Conservative vs non-conservative forces

A conservative force changes potential energy by an amount that depends only on start and end positions. Friction depends on the path and transfers mechanical energy into thermal and other forms.

Think like a programmer

A conservative model can derive force from a reusable potential function. A non-conservative model needs explicit path or velocity-dependent accounting, so total mechanical energy is no longer the only invariant.

Model checklist

Inputs: State, path, and force model.
State: Potential energy, kinetic energy, and any tracked loss.
Rule: Add conservative energy changes; account for non-conservative work separately.
Output: Energy ledger.
Check: Mechanical energy is constant only when the model has no non-conservative work.

Energy as an accounting check

Set mass and height. In this no-loss model, gravitational potential energy becomes kinetic energy at ground level.

Height: 5 mMass: 2 kg

Potential energy 98.07 J; predicted landing speed 9.90 m/s; kinetic energy 98.07 J.

Potential is a reusable function

For a conservative force, define a potential (U(\mathbf r)) and obtain force from its negative gradient:

\[\mathbf F_c=-\nabla U, \qquad W_c(A\to B)=U(A)-U(B).\]

Two routes from A to B should return the same potential difference. In code, keep the potential reference, parameters, coordinate frame, and units with the function. A visual height map may show (U), but an object’s update must call the same force rule or an independently tested derivative of it.

Non-conservative work belongs in an energy-transfer ledger:

\[E_{\mathrm{mech},f}-E_{\mathrm{mech},i}=W_{\mathrm{nonconservative}}, \qquad E_{\mathrm{total},f}-E_{\mathrm{total},i}=0\]

when the receiving thermal/internal reservoirs are included. A kinetic-friction model needs contact normal force, coefficient, sliding direction, distance or velocity history, and a convention for heat transferred into the chosen system.

Ask a path question and a ledger question

Test a potential with two discretized paths between the same endpoints; refine their segments and compare work. Test a frictionless drop against the analytic mechanical-energy value. Then add a fixed friction path: mechanical energy should fall by the recorded dissipative work while the extended ledger closes. A closed path in a conservative fixture has zero net work within tolerance.

State what can change energy: an external drive, a moving support, a thermostat, drag, or numerical constraint impulses. Never label a residual “friction” simply because energy drifted. Keep solver time-step/refinement tests separate from the physical work model, and show the reader which energy terms are mechanical versus untracked internal energy.

Try this experiment

Prediction: The no-loss lab keeps potential and kinetic energy equal; adding friction would lower mechanical energy.

Change height and mass. Then describe a new energy field you would add to record heat generated by friction.

Where this model breaks

Potential energy is defined only for suitable conservative force models and reference choices. A zero reference is conventional; energy differences drive the prediction.

Summary

Use a documented potential function for conservative forces. For friction and drives, record path-dependent work and receiving reservoirs explicitly, then test path independence and the complete energy ledger separately from numerical drift.

Glossary

Conservative force: force whose work is path-independent.
Potential energy: stored energy associated with configuration.
Mechanical energy: kinetic plus potential energy.

Self-check

Which force is path-independent?
What happens to mechanical energy with friction?
Why is a potential reference arbitrary?

Sources

OpenStax, University Physics Volume 1, potential energy chapter.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Conservative vs Non-Conservative Forces, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Energy, Work, and Power, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Conservative vs Non-Conservative Forces as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Conservative vs Non-Conservative Forces into a test

Compare path-independent energy changes with forces such as friction that transfer mechanical energy out of the model.

Name the inputs and units that the energy, work, and power model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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