Deep Dive: Moments Rods on Two Pivots

Provenance: written 2026-09-29. Read in full for this pass: research/veridian/WME01-verified-facts.md (§1 spec text, §3 mark-scheme conventions, §4 topic 6 "Moments," §6–§8, and the lesson-audit entry for this exact topic at §9, lines 224–232); content/veridian/curriculum/wme01/moments-rods-on-two-pivots.ts in full (its worked chain, diagram, method-comparison, marked-solution and four MCQs are not reproduced below); all three existing WME01 technique-hub documents in full (codex-formula-and-method-bank.md §1.7/§2.12, forge-exam-technique-guide.md §1 and its trap-digest section, signal-frequency-and-boundaries.md §1); and content/veridian/question-bank/wme01.ts (grepped for "moment/pivot/tilt/mrp-" — no hits, so nothing there to cross-check for this topic). No new primary-source PDF was fetched for this pass. The Jan 2025 Q4 and Jun 2025 Q5 material below is sourced via FORGE's own fresh pdftotext read of WME01_01_2501_MS/_ER and WME01_01_2506_MS (FORGE's provenance line, dated 2026-09-28) — this pass relied on FORGE's citation rather than re-fetching those PDFs itself, and says so at each point that matters.

#CODEX Deep-Dive — Moments: Rods on Two Pivots

#Pearson Edexcel IAL Mechanics 1 (WME01/01), spec 6.1

This is not a fourth re-teach of the topic. The lesson file owns the full worked chain (a 4 m rod, weight 100 N, supports 1 m and 3 m from A), the diagram, the method-comparison block, the marked-solution with its common-wrong-path, and four MCQs — none of that is repeated here. codex-formula-and-method-bank.md already gives this topic a formula-masterlist entry and a short method skeleton with its own numeric check. What none of the four existing documents does is: state the complete model this topic rests on in one place; work through, with real numbers, the two sub-cases the lesson currently only gestures at (an MCQ, a trap-taxonomy footnote) rather than fully solving; and fold in a fifth and sixth real data point — Jan 2025 Q4 and Jun 2025 Q5 — that postdate the facts bank's own moments section and exist only in FORGE's later, fresher read. That is this document's actual job.

#1. The complete model

Modelling vocabulary. Spec item 1.1 supplies the vocabulary this whole topic runs on: a rigid body does not deform, so "the rod is about to tilt about D" describes one well-defined event, not a rod flexing somewhere along its length. A rod is a rigid body with a length worth measuring distances along and no width to worry about. A light rod contributes no weight of its own. A uniform rod's weight is spread evenly, so it can be treated as one force acting at the geometric midpoint. A non-uniform rod's weight is not evenly spread; its centre of mass sits somewhere else, and a question either states that point or asks the candidate to find it.

Two independent equilibrium conditions. A rigid body in equilibrium needs, first, resultant force zero — which spec 6.1's own restriction to "coplanar parallel forces acting on a body and conditions for equilibrium" (spec 6.1 guidance) collapses to a single vertical resolve, since every force in this topic's scope is vertical — and second, resultant moment about any point zero. The word "any" is the whole reason pivot choice is a real, examinable technique rather than a convenience: equilibrium holds about every point in the plane, so the candidate is free to pick the one point that does the most work.

The mechanism, stated once and derived, not asserted. A moment is force × perpendicular distance from the chosen point to the force's line of action. If a force acts exactly at that point, its distance is zero, so its moment is zero regardless of the force's own magnitude — true even for a completely unknown reaction, because the multiplication by zero happens on the distance, not on the force. Taking moments about a support therefore removes that support's own reaction from the equation structurally. This is the entire content of "choose the pivot that eliminates an unknown," and it is why three independent series (Jan 2023 Q4, Jun 2024 Q4, Oct 2023 Q1 — verified in the facts bank) converge on exactly the same rule and the same two error modes.

Sign convention. Every moment is signed by physical intuition — a downward force to one side of the pivot turns that side down if nothing resists it — the same way real mark schemes present it, not via a vector cross-product (outside M1's own scope). What the mark scheme actually requires is internal consistency: one convention, held throughout one equation.

#2. The real distinct sub-cases

Spec 6.1 itself has no lettered sub-parts (facts bank §1) — it is one undivided guidance sentence. The sub-cases below are not spec numbering; they are the distinct scenario shapes the exam record itself actually produces, each attached to a named series.

#Case A — known rod weight, two unknown reactions (baseline)

A rod of known weight rests on two supports; both reactions are unknown. One moments equation about either support, plus one vertical resolve, is the efficient, three-series-confirmed method (§1 above). The lesson's own worked chain and marked-solution fully own this case's arithmetic; nothing is added here beyond naming it as the base case every other case below is a variation on.

#Case B — "about to tilt about X": which reaction goes to zero

Adding a load increases until one support is on the point of losing contact. "About to tilt about X" names the support that stays in contact — the reaction that reaches zero belongs to the other support. This is the single strongest documented finding on this spec item, confirmed verbatim in Jan 2023 Q4 ("one of the reactions had to be zero") and independently in Jun 2024 Q4 ("an equation with M as the only unknown"). The lesson's own marked-solution, diagram and two MCQs (mrp-mcq-2, the pre-question pair) already give this a full worked treatment; it is named here only as the case the two genuinely new sub-cases below (C and E) both build on.

#Case C — the rod's own weight and its centre-of-mass position both unknown

This is Jan 2023 Q4's real structure (facts bank §9, lines 228–229), and the lesson currently tests recognition of it with one MCQ (mrp-mcq-4) but never solves it end to end with numbers. Here, neither reaction is the unknown of interest — the rod's own weight $W$ and the distance $x$ from A to its centre of mass are both unknown, found from two separate "about to tilt" conditions, each giving one equation in $W$ and $x$, solved simultaneously. This is the one confirmed exception to Case A's "one moments equation is enough" rule: with two rod-intrinsic unknowns instead of two reactions, a single equation cannot separate them, and the real mark scheme's own dependent mark (DM1, "dependent on both M's, solve simultaneous equations") confirms both scenarios are required together.

Worked micro-example (VERIDIAN-original — no real spec-6.1 question's own $W$/$x$ figures survive in the archive reviewed; see facts bank §8). A non-uniform plank $AB$, 5 m long, rests on supports at $C$ (1 m from A) and $D$ (4 m from A). Its own weight $W$ acts at an unknown distance $x$ from A.

• With a 12 N load hung at $B$, the plank is on the point of tilting about $D$ (so $R_C = 0$). Taking moments about $D$: $W(4-x) = 12 \times 1 = 12$.
• With a 24 N load hung at $A$, the plank is on the point of tilting about $C$ (so $R_D = 0$). Taking moments about $C$: $W(x-1) = 24 \times 1 = 24$.

Adding the two equations: $4W - Wx + Wx - W = 36$, so $3W = 36$, $W = 12$ N; substituting back, $12(x-1) = 24$, so $x = 3$ m from A. Independent check: with both loads removed, $R_C = 4$ N and $R_D = 8$ N (from the plain baseline equilibrium) — both positive, a physically sensible rod resting on two supports — and re-solving the full three-force system for each loaded condition separately confirms $R_C = 0$ and $R_D = 0$ exactly at the stated loads, not approximately.

#Case D — a rod/beam held by tensions rather than resting on supports

The same technique (two unknown parallel forces, found via one moment equation plus one resolve) applies whether the two forces are upward reactions from supports or upward tensions from two strings. The one confirmed trap specific to this variant is not a mechanics error at all but a reading error: Jan 2020 Q2's examiner report records "the misinterpretation of the given information about the tensions," with candidates writing pairs such as "T and 20T, T and 20, or T and T" (Jan 2020, Q2) where the stem actually meant one tension was 20 N more than the other, not a multiple of it or an unrelated second unknown.

Worked micro-example (VERIDIAN-original numbers; the real Q2's own figures were not recovered in the archive reviewed). A light rod $AB$, 6 m long, is held horizontal by two vertical strings at $C$ (1 m from A) and $D$ (5 m from A). A 50 N weight hangs from a point $P$ on the rod. The tension at $D$ is 20 N more than the tension at $C$: $T_D = T_C + 20$.

Resolve vertically: $T_C + T_D = 50$. Substituting: $T_C + (T_C + 20) = 50$, so $2T_C = 30$, $T_C = 15$ N, $T_D = 35$ N. Taking moments about $C$ then locates $P$: $T_D \times 4 = 50 \times (d - 1)$, so $140 = 50(d-1)$, $d = 3.8$ m from A — between the two strings, as it must be. The trap sits entirely in the first line: reading "20 N more than" as $T_D = 20T_C$ or as $T_C = T_D = 20$ (both seen, per the quote above) makes the resolve equation unsolvable or wrong before any mechanics has even been applied.

#Case E — a multi-part question where an earlier force must be dropped

This case does not appear in the facts bank at all; it is FORGE's own addition from a direct read of the January 2025 mark scheme and examiner report, not independently re-verified against the primary PDF in this pass. FORGE records that on Jan 2025 Q4's final part, unsuccessful candidates took moments about the correct new pivot but kept "the tension at C still 600N or the particle P still attached" (Jan 2025, Q4) — a force that an earlier part's own physical scenario had established, carried forward into a later part where the scenario had changed (a support losing contact, a particle removed) and that force no longer belonged. This is a different failure from Case B's "wrong reaction zeroed": the pivot choice is correct, and the mistake is failing to update which forces are still present once the physical picture has moved on.

Worked micro-example (VERIDIAN-original throughout — the real Q4's own numbers are not known beyond the "600N"/"particle P" fragment quoted above, so no attempt is made to reconstruct that question; this only illustrates the general failure mode). A uniform rod $AB$, 4 m long, weight 80 N, rests on supports at $C$ (1 m from A) and $D$ (3 m from A). Part (a): a particle of weight 20 N is attached at $E$, 3.5 m from A; find $R_C$ and $R_D$. Moments about $C$: $R_D \times 2 = 80 \times 1 + 20 \times 2.5 = 130$, so $R_D = 65$ N; resolving, $R_C = 100 - 65 = 35$ N. Part (b): the particle at E is removed, and a new load $M$ is attached at $A$ instead; given the rod is now on the point of tilting about $C$, find $M$. Correct working drops the removed particle: with $R_D = 0$, moments about $C$ give $M \times 1 = 80 \times 1$, so $M = 80$ N (checked independently: resolving gives $R_C = 80 + M = 160$ N, and moments about A confirms $R_C \times 1 = 80 \times 2$, i.e. $160$, matching). The stale-force error carries the removed 20 N particle forward at its old distance from $C$ (2.5 m): $80 \times 1 + 20 \times 2.5 = M \times 1$, giving the wrong $M = 130$ N — a plausible-looking number, not an obviously broken one, which is exactly why this trap survives to the final line rather than being caught earlier.

#A thinner sixth data point, reported honestly short

signal-frequency-and-boundaries.md §1 tags Jun 2025 Q5 (6 marks, spec 6.1) as "a rod carrying several unknown forces at once" — FORGE's own topic-tag from its fresh mark-scheme read, not a quoted examiner-report sentence. No mark code, quote, or figure from that question was mined beyond this tag. It is recorded here as a genuine sixth confirmed appearance of this spec item, and as a hint that a rod can carry more than the one added load every case above uses — but it is deliberately not built into a seventh worked case, because there is nothing beyond the tag to build one from. Per this document's own governing rule, the material ends where the real evidence does.

#3. Scope boundary: what spec 6.1 does not cover here

Spec 6.1's own guidance restricts this topic to coplanar parallel forces — in practice, every force in every case above is vertical. A rod resting against a wall with friction, or pinned at a hinge with both a horizontal and a vertical reaction component, is a non-parallel coplanar force system: still rigid-body statics, but outside spec 6.1's own stated scope, and not evidenced anywhere in the archive reviewed as a spec-6.1 question. Thrust (a rod under compression) is named in spec 5.2's own force list but was not found tested as a moments-specific scenario in the archive reviewed either. Both are recorded as genuine gaps, not filled with invented examples, in keeping with this course's standing rule against manufacturing material the real record does not supply.

#4. Synthesis: every real exam angle this topic has actually been tested from

Six named series, out of the 17 with at least one primary WME01 document reviewed for this course (facts bank §5), confirm a spec-6.1 moments question:

That is six of seventeen — a real, named count, not a percentage and not a claim about the eleven series where no spec-6.1 content was recovered one way or the other (most of those simply were not deep-mined for topic-by-topic content at all; the facts bank explicitly never built a full per-question spec-tag across the whole archive, §8). Within the two series FORGE mark-mapped question-by-question in full (Jan 2025, Jun 2025), spec 6.1 anchored a real question in both.

#5. What remains unconfirmed

The Jan 2025 Q4 quotation in Case E is reported at one remove — sourced from FORGE's own fresh PDF read, not independently re-verified against WME01_01_2501_MS/_ER in this pass — and nothing beyond the "600N"/"particle P" fragment is known about that question's actual scenario, so no fuller reconstruction is attempted anywhere above. Jun 2025 Q5 is known only as a topic-tag; its actual mark scheme, mark codes and any trap it carries were never mined by FORGE or this pass. No mark-weighting table (how many of a series' 75 marks spec 6.1 typically carries) exists for this topic, consistent with the facts bank's own scoping decision not to build one (§8). No likelihood percentage and no grade boundary is stated anywhere above — §4's "six of seventeen" is a count of series actually read, not a probability.

#Sources

• research/veridian/WME01-verified-facts.md — spec text (§1), mark-scheme conventions (§3), topic 6 "Moments" (§4), the topics-with-rich-material list (§6), the numeric-figures caveat (§8), and the moments-lesson audit entry (§9, lines 224–232), itself sourced to wme01-01-rms-20230302.pdf/wme01-01-que-20230118.pdf (Jan 2023 Q4) and to the Jun 2024 Q4, Oct 2023 Q1, Jan 2020 Q2 examiner-report prose quoted in §4 topic 6.
• content/veridian/technique-hub/wme01/forge-exam-technique-guide.md — Jan 2025 Q4 and Jun 2025 Q5 topic identification (§1's mark tables) and the Case E quotation (its trap-digest section), sourced there to a fresh pdftotext read of WME01_01_2501_MS/_ER and WME01_01_2506_MS, not re-verified against those PDFs in this pass.
• content/veridian/technique-hub/wme01/codex-formula-and-method-bank.md §1.7, §2.12 — formula status and method skeleton for this spec point.
• content/veridian/technique-hub/wme01/signal-frequency-and-boundaries.md §1 — the two fully mark-mapped series' own topic tags, used for the Jun 2025 Q5 entry in §2 and §4 above.
• content/veridian/curriculum/wme01/moments-rods-on-two-pivots.ts — the lesson this document deliberately does not duplicate.