Deep Dive: Friction Equilibrium and Motion on a Rough Plane

Provenance — 2026-09-29. Read in full before writing: research/veridian/WME01-verified-facts.md (all 317 lines, the primary provenance-checked source, searched specifically for every friction/4.4/5.3 entry it contains); the existing lesson content/veridian/curriculum/wme01/friction-equilibrium-and-motion-on-a-rough-plane.ts (all 739 lines, so this document does not re-run any of its worked scenarios); and all three existing technique-hub documents for this paper — forge-exam-technique-guide.md, codex-formula-and-method-bank.md, signal-frequency-and-boundaries.md — in full. content/veridian/question-bank/wme01.ts was searched for friction/rough-plane content; it holds none specific to this topic, so it is not cited further below. No new primary-source PDF was fetched this pass — every claim below traces to one of the five documents just named, cited inline by document and, where that document itself names one, by real series and question number.

#CODEX Deep-Dive · Friction — Equilibrium and Motion on a Rough Plane

#Pearson Edexcel IAL Mechanics 1 (WME01/01) — spec 4.4 / 5.3

#1. What this document is for, and what it deliberately does not repeat

This topic already has more real, verified material behind it than most WME01 topics: a full nine-block lesson, a dedicated method skeleton in codex-formula-and-method-bank.md §2.11, and several named entries in forge-exam-technique-guide.md. Re-deriving any of that here would be restatement, not addition, so this document does two things instead. Section 3 below names every distinct real variant of this topic this course's research has actually confirmed — including the ones already fully worked elsewhere, each with a one-line pointer rather than a repeated derivation — and builds fresh, VERIDIAN-original worked micro-examples only for the two variants that exist so far only as a quoted examiner-report line, not as a worked illustration anywhere in this repo. Section 4 then does something that does not exist anywhere yet for this specific topic: a single table listing every real series and question this course has confirmed tests friction/4.4/5.3, so a revision session can see the whole confirmed record in one place rather than assembling it from four separate files.

#2. The core model, compressed

The WME01 spec states this topic as two separate guidance lines under two separate top-level sections — 4.4, "An understanding of F = μR when a particle is moving," and 5.3, "An understanding of F ≤ µR in a situation of equilibrium" (both quoted verbatim in the facts bank §1). The facts bank's own inference, stated as an inference and not a printed spec fact (§7): Pearson's question-writing treats these as one scenario family, split only by whether the along-plane resultant is zero. The mechanics of the split: resolve perpendicular to the plane first for the normal reaction R (using every force that has a perpendicular component, not weight alone — the general resolution rule, §5 below, is what most of this topic's real errors actually violate); then resolve along the plane. If the particle is in equilibrium, F is whatever value the along-plane balance requires, subject to the ceiling F ≤ μR. If the required value would exceed that ceiling, equilibrium is impossible and the particle accelerates, at which point F stops being solved-for and becomes the fixed value μR, substituted directly into Newton's second law. Direction is decided once, before any resolution: friction opposes actual sliding (moving case) or the tendency to slide (equilibrium case) — never an applied force's own direction. The single most cross-series-confirmed error on this whole topic is getting that direction backwards; the facts bank names it directly against Oct 2022 Q3, and the lesson's diagram, worked-chain and two MCQs are all built to drill it.

#3. Every distinct real variant confirmed for this topic

3.1 — Non-limiting equilibrium under gravity alone (F < μR, no applied force). The general case spec 5.3's own inequality describes. Fully worked already: lesson prequestion fr-pq-1 and MCQ fr-mcq-1.

3.2 — Limiting equilibrium under gravity alone, μ found by dividing the two resolved equations (μ = tanθ). Verified as a real, credited method, not just a description — Jan 2022 Q5, described in the facts bank as "resolving parallel/perpendicular, eliminating R to solve for μ" (§4, topic 4.2). Fully worked already: lesson worked-chain stage 4 and MCQ fr-mcq-4.

3.3 — The moving case, F = μR substituted directly into F = ma. The direct application of spec 4.4's own guidance. Verified against the real Oct 2022 Q3(a)/(b) two-part shape — a "show F = 0" part followed by a limiting-friction part, deliberately split to test whether F ≤ μR vs F = μR is understood, not just recalled (facts bank §4, topic 4.4). Fully worked already: lesson diagram and marked-solution.

3.4 — An applied force parallel to the plane. Doesn't affect R at all; only ever enters the along-plane equation. Fully worked already: lesson worked-chain (the rope-and-crate scenario).

3.5 — An applied force NOT parallel to the plane, of unknown magnitude, resolved into two simultaneous equations. Verified directly from the real Oct 2022 Q3(b) mark scheme (facts bank §9, the 2026-09-12 audit entry for this lesson): the applied force is horizontal, unknown, and must be resolved into both a perpendicular component (adding to R) and a plane-parallel component, giving two equations both containing the unknown, solved by substitution rather than by the single-step division that works for the gravity-alone case. Fully worked already: lesson MCQ fr-mcq-5.

3.6 — An applied force affecting R that is given symbolically as a multiple of mg, not as a bare number. Genuinely new — no worked illustration of this exists anywhere in this repo yet. forge-exam-technique-guide.md §4 records a real, distinct error from January 2025 Q6(a)/(b), a box on an incline held by a taut string whose tension is given as 0.75mg: "a few candidates used 0.75 instead of 0.75mg for the tension." The same question is also named there as independently reproducing the "R = mg cos α" omission — the string's own tension, not being parallel to the plane, also contributes to R, and some candidates dropped that contribution entirely. Both findings come from FORGE's own summary of a mark scheme this document did not itself re-fetch; the real question's exact geometry (which way the string is inclined, and to what) is not independently confirmed here, so the worked micro-example below is a VERIDIAN-original construction built to carry both verified error mechanisms, not a reconstruction of the real stem.

Micro-example. A block of mass m rests in equilibrium on a rough plane inclined at θ, where sinθ = 8/17 and cosθ = 15/17, held by a light string inclined at 20° to the plane's own surface, pulling up the slope and away from the surface, with tension T = 0.6mg. μ = 0.25. Perpendicular: R + T sin20° = mg cosθ, so R = mg cosθ − T sin20° = mg(15/17) − 0.6mg sin20° = 0.677mg (3 s.f.) — carrying the 0.6mg symbolically through the resolution, not just the coefficient 0.6, and not silently dropping the string's own contribution to reach the wrong R = mg cosθ = 0.882mg. Ceiling: μR = 0.25 × 0.677mg = 0.169mg (3 s.f.). Along the plane: T cos20° − mg sinθ + F = 0 (F assumed up-slope) → F = mg sinθ − T cos20° = mg(8/17) − 0.6mg cos20° = −0.0932mg. The negative sign means the string's up-slope pull (T cos20° ≈ 0.564mg) actually exceeds gravity's down-slope component (mg sinθ ≈ 0.471mg), so the block's tendency is to slide up, and friction of magnitude 0.0932mg acts down the slope to oppose it. Check: required friction 0.0932mg ≤ ceiling 0.169mg, so equilibrium genuinely holds. (All four decimal values independently checked in Python before being written in.)

3.7 — A force condition changing mid-question, so R (and hence F, and hence the acceleration) has to be recomputed from scratch. Verified against the real Jun 2024 Q6 mark scheme (a rough horizontal plane, not an incline): removing a rope's vertical tension component changes the normal reaction and therefore the friction and deceleration; the examiner report's own words: candidates who "did not recognise that the system was subject to a different acceleration" (facts bank §4/§9). Fully worked already: lesson marked-solution and the matching trap-taxonomy item.

3.8 — "Does it move?" decided by an explicit numeric comparison, never by prose alone. Verified against January 2025 Q6(c): the examiner report describes candidates who wrote an "essay" with no calculations and scored nothing (forge-exam-technique-guide.md §4). Fully worked already: codex-formula-and-method-bank.md §2.11's own short check (a 4 kg block, sinα = 7/25, cosα = 24/25, μ = 0.3, comparing 10.98 N against 11.29 N).

3.9 — A "show that" target for this topic, requiring the exact trig fractions carried through, not decimals. Genuinely new as a worked illustration — the underlying convention is named in forge-exam-technique-guide.md §1.1/§2 and codex-formula-and-method-bank.md §1.1, sourced to the real June 2025 mark scheme: "the final given-answer mark can only be earned from a fully correct solution using the exact fractions... a decimal approximation substituted along the way, even one that rounds correctly, breaks the 'show that' chain." Neither document builds a worked example of it for this topic specifically; this is the same limiting-equilibrium relationship as §3.2 above, recast as a "show that" target purely to isolate that discipline, since §3.2's own mechanism (divide the two equations) doesn't by itself require exact fractions.

Micro-example. "A particle is in limiting equilibrium on a rough plane inclined at θ to the horizontal, where sinθ = 12/37 and cosθ = 35/37, held only by friction. Show that μ = 12/35." R = mg cosθ = mg(35/37). At limiting equilibrium, F = μR = mg sinθ = mg(12/37). So μ(35/37) = 12/37, giving μ = 12/35 exactly, by cancelling the common mg/37 factor — no decimal ever enters the chain. A solution that instead substitutes sinθ ≈ 0.3243 and cosθ ≈ 0.9459 partway through reaches μ ≈ 0.3429, the same 3 s.f. value 12/35 rounds to — but per the real June 2025 mark-scheme convention just quoted, that decimal route does not itself satisfy a "show that" target stated as an exact fraction; the exact-fraction chain is what the given-answer mark actually checks, not merely a numerically correct final decimal.

3.10 — Friction as one ingredient inside a larger multi-topic question, not a standalone question. forge-exam-technique-guide.md §1 tags January 2025 Q3 (12 marks) as "suvat + friction (F = μR) + momentum/impulse in one three-part question (3.1/4.4/4.3)" — confirmed as a real paper-structure fact, but no source read this pass details which specific friction sub-mechanism that question's own middle part tests beyond "F = μR" being one ingredient, so this is named here as a structural fact only, not expanded into its own worked case.

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

Every row below is a named series and question this course's own research has independently confirmed tests spec 4.4 and/or 5.3, drawn from the facts bank and the FORGE guide (no row is estimated or inferred beyond what those documents themselves state).

Two different, non-interchangeable counts exist for how often this topic recurs, and neither should be quoted as the other. signal-frequency-and-boundaries.md §1 reports that spec 4.4 and spec 5.1–5.3 were each seen in 2 of the 2 series this course has fully, question-by-question mark-mapped (January and June 2025) — a real, narrow, two-series finding, not a claim about the wider archive. Separately, combining the facts bank's own 17-series archive with the table above, this topic has a specific, named, confirmed instance in at least 5 of those 17 series (Oct 2022, Jan 2022, Jun 2024, Jan 2025, Jun 2025) — the other 12 series were not specifically checked for friction content by any document this pass read, so their absence from this count means "not checked," not "absent." Neither number is a percentage, and no likelihood or grade-boundary figure is attached to this topic specifically anywhere in this course's research — signal-frequency-and-boundaries.md §2's grade-boundary table is whole-paper, not per-topic, and is not repeated here for that reason.

#5. What this document leaves unverified, flagged rather than filled in

• §3.6 and §3.9's citations (January 2025 Q6, June 2025 Q6) both trace to forge-exam-technique-guide.md's own prior deep-read of those mark schemes, not to a fresh primary-source PDF fetch by this pass — the micro-examples built around them are VERIDIAN-original constructions carrying the verified error mechanism, explicitly not claimed reconstructions of either question's real stem or geometry.
• June 2025 Q6's fuller real structure (FORGE's own table names suvat and an added horizontal force alongside the "show that" step) is known here only at that structural level; no source read this pass mines what the suvat or added-force parts of that specific question actually require, so §3.10-style expansion of Q6 itself was not attempted.
• The 12 archive series not named in Section 4's table were not specifically checked for friction/4.4/5.3 content by any document this pass read — this is a gap in what has been checked, not a finding that those series omit the topic.