Deep Dive: Equilibrium of a Particle Coplanar Forces

Provenance: written 2026-09-29. Read in full for this pass: research/veridian/WME01-verified-facts.md (the 17-series verified-facts bank — §1 spec text, §3 mark-scheme conventions, §4 topic 5, §6–§8, and the lesson-audit entry for this topic at §9, lines 283–293); content/veridian/curriculum/wme01/equilibrium-of-a-particle-coplanar-forces.ts in full (its own worked chain, diagram, method-comparison and MCQs are not reproduced below); all three existing WME01 technique-hub documents in full (codex-formula-and-method-bank.md §1.6/§2.10, forge-exam-technique-guide.md §1–§4, signal-frequency-and-boundaries.md §1–§2); and content/veridian/question-bank/wme01.ts (grepped for "equilibrium/coplanar/Lami/resolv" — no hits, so nothing there to cross-check against for this topic). No new primary-source PDF was fetched for this pass; every claim below traces to one of the five documents just named, each of which already traces its own claims to a real spec, mark-scheme, examiner-report or question-paper PDF.

#CODEX Deep-Dive — Equilibrium of a Particle Under Coplanar Forces

#Pearson Edexcel IAL Mechanics 1 (WME01/01), spec 5.1–5.2

This is not a fourth re-teach of the topic. The lesson file owns the full worked chain, the diagram, the method-comparison block and the MCQs, and none of that is repeated here. codex-formula-and-method-bank.md already gives this topic a formula masterlist entry (§1.6) and a short method skeleton with its own numeric check (§2.10). What none of the four existing documents does is assemble, in one place, the complete real theory this topic actually rests on — including what the spec itself deliberately restricts it to — set out every distinct force-pairing spec 5.2 names, and report several facts about the topic's one fully-mined real anchor question, and about how it fits into the wider exam record, that exist only scattered across the facts bank's own audit trail and are not currently stated in the lesson, CODEX or SIGNAL. That is this document's actual job.

#1. What spec 5.1–5.2 asks for, and why it is a narrow topic by design

Spec 5.1 is six words: "Forces treated as vectors. Resolution of forces." Spec 5.2 names four kinds of force — weight, normal reaction, tension and thrust, friction — and its own guidance narrows the scope explicitly, restricting the topic to "uncomplicated systems" only. That restriction is not incidental: it is the single biggest reason this topic sits outside the facts bank's own list of topics with rich, multi-series verified material (kinematics, moments, momentum, connected particles and vectors all made that list; statics of a particle did not). A spec that tells its own examiners to keep the systems simple produces a smaller space of real question variety to mine in the first place — a fact worth stating up front, because it explains why this document is shorter than the ones written for those richer topics, not because less effort went into checking it.

The four named forces, briefly, without repeating the lesson's own fuller definitions: weight acts vertically down, magnitude $mg$; normal reaction acts perpendicular to a contact surface, away from it, and can only push; tension pulls a taut string toward its anchor point and can only pull; thrust is the same straight-line mechanism as tension but for a rigid rod, which — unlike a string — can push as well as pull. Friction is the fifth force 5.2 names but is deliberately out of scope here: it has its own spec points (4.4/5.3) and its own sibling lesson (friction-equilibrium-and-motion-on-a-rough-plane.ts), and every trap and worked example in that file is built around a rough-plane scenario this document has no reason to duplicate.

#2. The one condition, and why a missing term disqualifies an equation rather than just weakening it

Equilibrium of a particle is the statement that the vector sum of every force acting on it is zero. Resolving into two independent directions and setting each sum to zero is that one condition, applied twice. The general marking guidance's own Mechanics-specific rule is what makes this more than notation: an M mark for a resolution requires the equation to be dimensionally correct, and every force with a component in the resolved direction has to contribute its own term. The one fully verified real trap for this exact spec point is a direct instance of that rule breaking: a real WME01 mark scheme records "incomplete resolutions e.g. 5 = Fcos30 rather than 5 = Fcos30 + Tcos60" (Jan 2022, Q1) — an equation missing a term is not a smaller version of the right equation, it is a different, wrong equation, and it is marked as a method failure (M0), not an accuracy slip. This is already the lesson's own central trap and is not re-derived here beyond this one paragraph of connective tissue; it is repeated only because the rest of this document builds directly on it.

#3. The three credited methods — and a real finding about how evenly they're actually marked

Three methods are confirmed, across two series, as fully creditable for a three-force coplanar equilibrium problem: resolving horizontally and vertically; resolving along (or perpendicular to) one of the unknown forces; and Lami's theorem or an equivalent triangle-of-forces/sine-and-cosine-rule approach (Jan 2022 Q1 names all three; Jan 2023 Q6 independently reconfirms the triangle-of-forces/Lami's alternative). The lesson already teaches all three and compares their trade-offs. What is genuinely additive here is what the real mark scheme does with that choice: for the one anchor question fully mined for this topic, the mark scheme gives a single combined M1/A1 line covering all three routes — "resolving perp to string or from triangle of forces or Lami's Theorem" earns the identical credit, whichever the candidate picks. This course's own adversarial audit of the lesson file originally found, and then removed, two invented B-marks that an earlier drafting pass had assigned for "spotting the 90° relationship" and "recognising three concurrent forces" — marks the real scheme simply does not contain. The practical consequence for a candidate is worth stating plainly, because it is not obvious from "three methods are all creditable" alone: choosing the faster method here buys time, never extra marks. There is no bonus for elegance and no penalty for reliability — the only real cost of the slower resolve-and-eliminate route is the clock, which is exactly the trade-off forge-exam-technique-guide.md's timing sections describe in general but never tie to this specific method choice.

#4. The real anchor in full — Jan 2022 Q1

Every other WME01 document that cites this question quotes fragments of its examiner report. Here is what the facts bank's own adversarial audit trail (research/veridian/WME01-verified-facts.md §9) additionally recovered directly from the mark scheme and the real question paper (Publications Code WME01_01_2201_MS/_ER; question paper P68783A):

• It is two separately-scored 3-mark parts, not one combined question. Part (a) asks for $F$ alone; part (b) asks for $T$ alone. Total: 6 marks.
• Part (a)'s own mark line: $F = 5\cos30°$ oe earns M1 A1; the evaluated value, "4.3 or better," earns a second, separate A1.
• Part (b)'s own mark line: $T = 5\sin30°$ oe earns M1 A1; $T = 2.5$ N earns a second A1.
• A specific condition on the M1 if a candidate resolves horizontally and vertically instead of using one of the direct methods: they must eliminate $T$ to obtain that mark — the two-unknown horizontal equation alone, with no elimination step, does not itself earn it.
• No B mark exists anywhere in this question. Every mark is M or A.
• The particle's own weight is confirmed as 5 N. The real question paper describes the setup as a particle "attached to one end of a light inextensible string" — a genuine physical detail not stated in the lesson itself, which deliberately declined to assert what its reconstructed $F$ and $T$ represent, since the string's other end, any second force, and the full stem remain unrecovered. That caution still holds: this document adds one confirmed fact (the 5 N is the particle's weight, on a string) without pretending the rest of the scene is now known.
• The weight-mass confusion trap sits in the same report paragraph as the missing-term trap: some candidates wrote "weight to be confused with mass resulting in the use of 5g." With the weight identity above now confirmed, this trap can be stated slightly more precisely than before: it is not a hypothetical "if this were a weight" caution, it is a real substitution of $mg$ for a force already correctly given in newtons, on a question where that force genuinely is a weight.

#5. Second-series corroboration — honestly scoped, and no further than that

Jan 2023 Q6 independently confirms the triangle-of-forces/Lami's-theorem alternative as credited on a different real question: the examiner report names "a triangle of forces" using "the sine and/or cosine rule (or Lami's Theorem)" as a genuine alternative method. That is the entire extent of what the archive reviewed for this course has recovered from that series for this topic — no mark breakdown, no scenario numbers, no examiner-named trap. It corroborates the method, not the topic's trap taxonomy, and should not be read as a second data point of equal weight to Jan 2022 Q1's fully-mined breakdown above.

#6. Does a clean, friction-free three-force question keep recurring? A frequency finding, narrowly scoped

signal-frequency-and-boundaries.md tags every question of the two series this course has fully mark-mapped question-by-question (Jan 2025, Jun 2025) against the spec. In both, every appearance of 5.1–5.3 is bundled with either friction on an incline (4.4/5.3) or Newton's second law (4.1) — never a standalone, friction-free three-force problem of the kind Jan 2022 Q1 (and, by its own report language, Jan 2023 Q6) appears to be. That gives four real data points in tension with each other: two older series reading as bare three-force equilibrium questions, and the two most recently fully-mapped series folding this spec point into a different scenario every time. Four data points, from a 17-series archive that was never tagged topic-by-topic outside those two fully-mapped series, is not enough to call this a trend — but it is enough to say plainly that a learner should not assume the bare, smooth three-force question is the only shape this spec point takes, and should practise the friction-blended version (the sibling lesson's own territory) at least as much as the clean one this topic's own lesson teaches.

#7. The force-pairings spec 5.2 actually names, and where the evidence runs out

Spec 5.2 names weight, normal reaction, tension and thrust as the non-friction forces a particle-equilibrium question can draw on. The one real anchor recovered (§4) pairs weight with at least one string. Beyond that:

• Normal reaction paired with tension, no weight-bearing configuration at all — a particle held against a smooth surface by a string — is not illustrated anywhere in the lesson, CODEX or the facts bank, though it is a direct reading of spec 5.2's own named force list. No real WME01 question of this exact shape was found in the archive reviewed, so the following is a short, clearly VERIDIAN-original illustration of the mechanism only, not a claim that this exact configuration has been examined: a particle of weight 6 N rests against a smooth vertical wall, held by a string inclined at 35° to the wall. Resolving parallel to the wall (vertical): $T\cos35° = 6 \Rightarrow T \approx 7.33$ N. Resolving perpendicular to the wall (horizontal): $R = T\sin35° \approx 4.21$ N. Nothing about this check is claimed as exam-representative; it exists only to confirm the same resolving mechanism applies when the second force is a reactive push rather than an applied pull.
• Thrust as its own tested scenario — a rod under compression rather than tension, specifically within this spec point — was not found tested anywhere in the archive reviewed. The lesson defines thrust correctly (its own MCQ item), but no real mark scheme or examiner report checked for this course names a thrust-specific equilibrium question. This is recorded as a gap, not filled with an invented one.
• Exactly two forces in equilibrium is a trivial corollary of the same vector-sum-zero condition — two forces in equilibrium must be equal in magnitude and exactly opposite in direction — but this is stated here as elementary reasoning from the definition, not as an examined case; nothing in the archive reviewed suggests a WME01 question is ever built around only two forces, which would leave nothing to solve for.

#8. What remains unconfirmed

Stated plainly rather than left implicit: the second force's identity in Jan 2022 Q1, the exact wording of its stem, and any diagram it carried are still not recovered — only the weight identity of the 5 N force is newly confirmed here, from the question paper's own wording, nothing more. Jan 2023 Q6's own numbers and mark allocation were never recovered, only its one-sentence method confirmation. Whether the sine rule itself sits in the P1/P2 formula-booklet pages available to M1 candidates remains unverified (CODEX's own flag, not resolved by this pass). Whether the bare, friction-free three-force question type still appears in any of the series between Jan 2023 and Jan 2025 is genuinely unknown, since those series were never tagged topic-by-topic. No likelihood percentage and no grade boundary is stated anywhere above for this specific spec point — the "seen in 4 of 4 examined instances, drawn from 2 fully-mapped and 2 partially-mapped series" framing in §6 is a count of series actually read, not a probability, and WME01's overall grade-boundary table (SIGNAL §2) is a whole-paper figure that says nothing about this topic specifically, so it is not repeated here as if it did.

#Sources

• research/veridian/WME01-verified-facts.md — spec text (§1), mark-scheme conventions (§3), topic 5 (§4), topics flagged thin (§7), and the equilibrium-lesson audit entry (§9, lines 283–293), itself sourced to WME01_01_2201_MS/_ER and question paper P68783A, and to WME01_01_2301 (Jan 2023 Q6) examiner-report prose.
• content/veridian/technique-hub/wme01/codex-formula-and-method-bank.md §1.6, §2.10 — formula status and method skeleton for this spec point.
• content/veridian/technique-hub/wme01/forge-exam-technique-guide.md §1–§2 — general M/A/B conventions and the dimensional-completeness rule applied here.
• content/veridian/technique-hub/wme01/signal-frequency-and-boundaries.md §1 — the two fully mark-mapped series' own topic tags, used for §6 above.
• content/veridian/curriculum/wme01/equilibrium-of-a-particle-coplanar-forces.ts — the lesson this document deliberately does not duplicate.