SCIENCE · INVESTIGATE AND EXPLAIN

Integrate a force along two genuinely different paths

Work is the line integral of F·dr, not the integral of a force magnitude with every sign discarded. In two dimensions its component form is integral(Fx dx+Fy dy). Parameterize each path segment, substitute the coordinates along that segment and include only its actual displacement components.

Learning outcome: Compute each segment integral, reverse one path to form a loop and compare unequal versus equal coefficients. Verify the explicit potential only in the conservative case.

Starting recap

Use vector dot products for work, gradients for conservative force, definite integrals for accumulation and explicit energy/frame boundaries. Check accessible states before assigning trajectories.

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Try

With a=−1,b=1, east-then-north work is2 J, north-then-east−2 J and the chosen loop4 J. Reverse the loop and its work changes sign.

Open the experiment

Predict

For a = 1,b = 2, work on the east-then-north path is:

Your prediction and reason

Experiment

Compute each segment integral, reverse one path to form a loop and compare unequal versus equal coefficients. Verify the explicit potential only in the conservative case.

Compare two paths through the same force field

Worked teaching model · synthetic values · Supplied smooth field F=(a y,b x) N on the entire plane. Compare paths from(0,0) to(1,2) m: east then north versus north then east. Paths are specified; no force-only trajectory solver or physical agent is inferred.

East-force coefficient a: 1 N/m North-force coefficient b: 2 N/m

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Two prescribed paths with the same endpoints

Spatial axes use the same pixels per physical unit, preserving geometry. 1: east then north (solid).2: north then east (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Two prescribed paths with the same endpoints

east then north

east position (m)	north position (m)
0	0
1	0
1	2

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

north then east

east position (m)	north position (m)
0	0
0	2
1	2

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
East-then-north path work	4	J
North-then-east path work	2	J
Forward first path and reverse second loop work	2	J
Curl z component	1	N/m
Conservative field in this full-plane family	0	flag

The closed-loop work is nonzero, so no single endpoint-only potential represents this whole field.

Integrate Fx dx+Fy dy separately on each straight segment. First-path work is2b, second-path work2a. A single pair of equal path integrals would not prove conservativeness of an arbitrary field, but here a=b gives the explicit potential and zero curl everywhere on the supplied full plane. The field can do work along a specified path without by itself making that path a free trajectory.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

The closed-loop work is nonzero, so no single endpoint-only potential represents this whole field.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Integrate a force along two genuinely different paths

For the supplied full-plane field F=(a y,b x), compare paths from(0,0) to(1,2) m. East first along y = 0 gives zero work on the first segment; north at x = 1 contributes2b J. North first along x = 0 gives zero first work; east at y = 2 contributes2a J. Thus a closed loop following the first path forward and the second backward has work2(b−a).

At a = 1,b = 2, the two works are4 and2 J, so the loop work is2 J. The same endpoints do not determine work for this field. This is not missing energy: the specified field is nonconservative, and its source is outside the particle-only account. A given force field also does not make every drawn path a free trajectory; guiding it may require other interactions.

For the special a=b case, an explicit potential is U=−a xy plus any constant. Taking its negative gradient recovers both force components everywhere on the full plane. Equality of just two path integrals would not prove that an arbitrary field is conservative; here the explicit global potential establishes it for this particular family.

Changed-case check

Predict before reading the comparison. With a=−1,b=1, east-then-north work is2 J, north-then-east−2 J and the chosen loop4 J. Reverse the loop and its work changes sign.

Retained investigation

Compute each segment integral, reverse one path to form a loop and compare unequal versus equal coefficients. Verify the explicit potential only in the conservative case. Why is comparing two paths a disproof when unequal but not a universal proof when equal?

Source and boundary

Primary conceptual reference checked30 October2026: OpenStax University Physics Volume1. All functions, numerical packets and teaching prose are original. Bounded supplied models are not measured experiments, apparatus instructions, independent practical skill or institutional credit. Numerical algorithm properties are derived for the disclosed model; do not treat them as universal solver guarantees.

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Why is comparing two paths a disproof when unequal but not a universal proof when equal?

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Previous: Converge a numerical method without confusing it with validation Next: Use the force derivative without discarding a flat minimum

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