Deep Dive: Connected Particles Pulleys and Lifts

Provenance: written 2026-09-29. Primary source read and grepped for this topic: research/veridian/WME01-verified-facts.md (§1 spec text, §3 mark-scheme conventions, §4 topic 4.2 and its neighbours, §5 source archive, §6–§8, §9 per-lesson audits, all re-read in full this session — no new primary-source PDF was fetched). Cross-checked against the real existing lesson content/veridian/curriculum/wme01/connected-particles-pulleys-and-lifts.ts (read in full) so nothing below duplicates its full worked exemplars, and against this paper's other technique-hub documents (forge-exam-technique-guide.md, codex-formula-and-method-bank.md §2.8, signal-frequency-and-boundaries.md) and content/veridian/question-bank/wme01.ts, all read in full, to confirm this document is additive rather than a restatement. No Pearson textbook was used; every quotation below traces to the facts bank's own already-verified mark-scheme/examiner-report/spec extracts.

#CODEX Deep-Dive · Connected Particles, Pulleys, Pegs, Lifts, and Cars with Trailers

#WME01 (Mechanics 1), spec item 4.2 — with its spec 1.1 vocabulary and spec 5.2 vocabulary

#Why this document exists alongside the lesson

connected-particles-pulleys-and-lifts.ts already teaches this topic end to end, with full-length marked solutions for four scenario families and a two-stage pulley-then-free-motion combination. This document does a different job: it is a single-topic reference that lays out the complete real theory as a catalogue of distinct sub-cases — including two the lesson does not build a dedicated example for (a bare two-particle pulley with no incline, friction, or lift attached, and a smooth-incline-plus-pulley combination) — with short micro-examples (a few lines each, not exam-length workups) and a synthesis of exactly which real exam angles are evidenced for each sub-case, and which are not. Every numeric micro-example below uses fresh, independently-chosen numbers, distinct from both the lesson's own worked numbers and the real past-paper figures quoted in it.

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#1. What spec 4.2 actually asks for, and the vocabulary it leans on

Spec item 4.2 ("Simple applications including the motion of two connected particles") gives its own guidance in three lettered parts, paraphrased here rather than reproduced at length: (i) two connected particles moving in a straight line or under gravity with constant forces on each, where a smooth fixed pulley or peg may be involved; (ii) motion under a force that changes from one fixed value to another — the spec's own example is "a particle hitting the ground"; (iii) motion directly up or down a smooth or rough inclined plane. That is the entire scope of this topic as Pearson defines it — three problem shapes, not one blurred idea.

Two spec items outside 4.2 supply the vocabulary this topic depends on. Spec 1.1 (mathematical models) lists the relevant modelling objects — particle, rigid body, light rod, inextensible string, and, among the supports, a "light smooth pulley" and a peg. Spec 5.2 (equilibrium) names the force types a connector can carry: "Weight, normal reaction, tension and thrust, friction." Put together, these give the two connector types this topic ever uses — a light inextensible string, which can only pull (tension, never thrust), and a light rigid rod (a car's tow-bar is one), which can push or pull (thrust or tension) — and the two kinds of direction-changing support that don't add resistance of their own: a light smooth pulley and a peg.

A lift is not named anywhere in this vocabulary list, and it is worth being explicit about why it still belongs in this topic: a lift's cage and its occupant(s) are not joined by a string or rod at all, but by ordinary surface contact, so the connecting force is a normal reaction, not a tension or thrust. The mechanism below (§2) still applies to it, once "the connector" is understood as the contact itself rather than a string.

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#2. The two mechanisms that generate every result in this topic

2.1 — Shared acceleration magnitude, from inextensibility. A string or rod that cannot stretch or compress forces every particle attached to it to travel the same distance, in the same time, as every other particle on the same connector. This is a pure length/geometry argument — it never mentions force — and it is why every connected-particles equation of motion uses one symbol, $a$, for every particle at once, once each particle's own direction of motion has been taken as that particle's own positive direction. Over a pulley or peg, one particle's positive direction can be "upward" while the other's is "downward"; the shared quantity is the magnitude, not a shared sign convention.

2.2 — One connector, one internal force, from masslessness. Apply $F=ma$ to the connector itself, treated as a particle of mass zero: whatever its own acceleration, $F=0$ follows immediately, since $ma=0\times a=0$ regardless of $a$. For a string this forces the pull at both ends to be numerically equal — one tension, the same number, throughout that string's own length. A light smooth pulley or peg does not break this argument; it only bends the string's path without adding any force along it, which is exactly why the spec's own vocabulary calls it "smooth." A rigid rod runs the identical zero-net-force argument, except the internal force can carry either sign: positive (tension, ends pulled together) or negative (thrust, ends pushed apart). The argument is scoped to one connector's own length. A second, physically separate string run the same zero-net-force argument along its own length independently, with no reason to produce the same numeric tension as the first string — the two tensions are related only through whichever single particle both strings happen to touch, via that one particle's own equation of motion.

2.3 — What changes when the connection itself changes. Both mechanisms above assume the connector stays taut and intact throughout the motion being analysed. Spec 4.2(ii)'s own guidance — motion under a force that changes value, with the spec's own example being "a particle hitting the ground" — names the case where that assumption fails mid-question. Two distinct ways this happens in the reviewed evidence: (a) an external force on the system changes value while the connector itself stays intact (a driving force is removed, but the tow-bar or string is still there, still enforcing one shared acceleration on whatever remains connected); (b) the connector itself stops enforcing anything, because one particle it was attached to has landed or otherwise stopped, so the string running to the other particle goes slack. In case (b) the surviving particle is, from that instant, a free particle under gravity alone — acceleration $g$, not whatever shared value the connected system had — and needs a fresh suvat treatment starting from the velocity it had at the exact instant the connector went slack.

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#3. The sub-case catalogue

Each entry gives the physical setup, the equations §2 produces for it, a short numeric illustration, and what the reviewed archive does and does not confirm about it being examined. $g=9.8$ m/s² throughout, matching this course's own standing convention.

#3.1 — Two particles connected in series, no pulley (spec 4.2(i)'s own base case)

Setup. Particle P hangs below particle Q on one string (tension $T_1$); Q is itself attached to a second, separate string (tension $T_2$, e.g. a winch cable lowering both). No pulley or peg anywhere. §2.1 gives one shared $a$; §2.2 gives $T_1$ and $T_2$ as two genuinely independent unknowns, linked only through Q's own equation.

Micro-example. P (5 kg) hangs below Q (3 kg); Q is lowered by a winch cable holding tension $T_2=70$ N. Whole system (downward positive): $8g-70=8a \Rightarrow a=(78.4-70)/8=1.05$ m/s² downward. P's own equation: $5g-T_1=5a \Rightarrow T_1=5(9.8-1.05)=43.75$ N. Check via Q: $3g+T_1-T_2=3a \Rightarrow 29.4+43.75-70=3.15=3(1.05)$ ✓.

Evidence. Confirmed directly: Jun 2024 Q3 — two particles connected by "a light inextensible string," with a second, separate string attached to the upper one, both described in the real stem as taut and vertical. Verified error: "many seemed unfamiliar with this type of situation involving two tensions," and specifically "missing the 3mg term when giving the equation for Q" or substituting the given tension into the wrong equation. Independently corroborated in Jan 2022 Q7 for a second, separate finding: both series' mark schemes require the solved tension to be given in terms of $mg$, "must be of form $kmg$" — not as a decimal, on the identical dependent mark, in two different series. That is 2 of the 17 series this course has checked for this exact configuration, both used as sources.

#3.2 — Two particles over a single smooth fixed pulley, nothing else attached (the plain Atwood pair)

Setup. Two particles hang from opposite ends of one string over one smooth fixed pulley — the textbook base case §2.2's pulley/peg clause describes, with no incline, friction, or lift anywhere in it.

Micro-example. A (5 kg) and B (8 kg), released from rest. $a=(m_B-m_A)g/(m_A+m_B)=3(9.8)/13\approx2.26$ m/s². $T=m_A(g+a)\approx5(12.06)\approx60.3$ N; check via B: $m_Bg-T\approx78.4-60.3=18.1\approx8(2.26)=18.1$ ✓.

Evidence — a genuine, named gap. This exact bare configuration is spec-guidance vocabulary ("problems involving smooth fixed pulleys...may be set") and it is the mechanism every other pulley sub-case below builds on, but no named series in the 17 reviewed for this course cites a real WME01 question of precisely this shape — every real pulley-adjacent question the facts bank actually names (3.1, 3.6, 3.7 below) attaches the pulley to something else (a lift cable, or a stage that changes). Treat this sub-case as foundational theory, not as an independently exam-confirmed configuration in its own right.

#3.3 — Peg as a direction-changing connector

A peg does the identical job as a smooth pulley in §2.2's argument — it bends a string's path without adding force along it — so no separate set of equations exists for it; every result in 3.2, 3.4, and 3.7 below applies unchanged if "pulley" is read as "peg." Evidence. Spec-guidance vocabulary only (spec 1.1 and 4.2(i) both name it). No named series in the reviewed archive was found to test a peg specifically as distinct from a pulley — an honest gap, not a claim that pegs are untested; only that this course's own reviewed evidence does not confirm one.

#3.4 — A hanging particle connected via a pulley to a particle on an inclined plane (4.2(i) + 4.2(iii) combined)

Setup. Particle C hangs vertically; the string passes over a smooth pulley at the top of a plane inclined at $\theta$ to particle D resting on the plane. If the plane is rough, D's equation gains a friction term (§3.4's own micro-example below uses a smooth plane specifically because the lesson's own worked example already covers the rough case in depth — this is the shorter, friction-free half of the same combination).

Micro-example. C (2 kg) hangs; D (5 kg) on a smooth plane, $\sin\theta=7/25$. Released from rest, C falls, D slides up the plane. Whole system: $m_Cg-m_Dg\sin\theta=(m_C+m_D)a \Rightarrow 2(9.8)-5(9.8)(0.28)=19.6-13.72=5.88=7a \Rightarrow a=0.84$ m/s². D's own equation (up-slope positive): $T-m_Dg\sin\theta=m_Da \Rightarrow T=5(0.84+9.8\times0.28)=5(3.584)=17.92$ N. Check via C: $m_Cg-T=2(9.8)-17.92=1.68=2(0.84)$ ✓.

Evidence — an honest boundary to name. Spec 4.2(i) and 4.2(iii) are adjacent guidance points under the same numbered spec item, and this combination is the natural reading of pairing them, but the reviewed archive's own confirmed 4.2(iii) citations — Oct 2022 Q3 and Jan 2022 Q5 (below) — are both single-particle incline problems (a block held or driven by an applied force at an angle, not a second particle over a pulley). No named series in this course's reviewed archive was found confirming this exact two-body pulley-and-incline shape as a real past-paper question; it is a theoretically consistent combination of two separately-confirmed spec points, not itself an independently exam-confirmed configuration. This document states that distinction plainly rather than implying a citation that does not exist.

#3.5 — Car and trailer, rigid tow-bar

Setup. A rigid rod (§2.2) replaces the string, so the connector's own internal force can be a tension or a thrust; everything else about §2.1–2.2 is unchanged.

Micro-example (also carries into 3.7(a) below). Tractor (1200 kg, resistance 300 N) tows a trailer (800 kg, resistance 150 N) on the flat, driving force 2000 N. Whole system: $2000-450=2000a \Rightarrow a=0.775$ m/s². Isolate the trailer for the tow-bar force $F$: $F-150=800(0.775)=620 \Rightarrow F=770$ N (tension — the bar is being stretched).

Evidence. Confirmed: Oct 2019 Q3. Verified method note: the most successful candidates wrote "the equation of motion for the trailer" first and used its acceleration in the car's own equation — isolate the simpler particle first, exactly as above. Verified trap, independent of the method chosen: "omission of the weight components" on the inclined version of this question, an error the report notes can cancel between the two equations and still produce a numerically plausible but uncredited answer, since a mark scheme checks each equation's own terms.

#3.6 — A lift with one or more occupants

Setup. The connector is a cable (string mechanics, §2.1–2.2, always in tension — a cable cannot push) between the lift and whatever it hangs from; each occupant is connected to the lift floor by a normal reaction $N$, a genuine Newton's-third-law pair appearing once (upward) in the occupant's own equation and once (downward, by reaction) in the lift's own equation whenever the lift is isolated on its own.

Micro-example — a variant the lesson's own worked example does not show. A lift (250 kg) with one occupant (60 kg) is moving downward but decelerating at 0.4 m/s² — i.e. the net acceleration is upward even though the motion is downward. Cable tension (whole system, upward positive since $a$ is upward): $T-(310)(9.8)=310(0.4) \Rightarrow T=310(9.8+0.4)=3162$ N. Occupant's own equation: $N-60(9.8)=60(0.4) \Rightarrow N=60(10.2)=612$ N — the floor pushes up on the occupant with more than their own weight (612 N > 588 N), which is the general rule this micro-example is chosen to illustrate: an occupant feels heavier than their own weight whenever the lift's net acceleration is upward (rising and speeding up, or falling and slowing down), and lighter whenever it is downward (rising and slowing down, or falling and speeding up) — a genuine four-way combination (direction of travel × direction of acceleration) worth naming explicitly, since only one of the four (rising, accelerating) is what the lesson's own worked example shows.

Evidence. Confirmed, two separate series: Jan 2023 Q7, verified error using "the mass of the lift alone...instead of the whole-system" mass in the system equation; Oct 2022 Q4, verified error where the lift's own equation "omitted the reactions from the two occupants." Two named series independently confirming this is a repeat Pearson question type, not a one-off.

#3.7(a) — Force changes value, connector stays intact (spec 4.2(ii), external-force variant)

Continuing 3.5's numbers. The driving force is removed. New whole-system deceleration: $450/2000=0.225$ m/s². Trailer alone: $F-150=800(-0.225)=-180 \Rightarrow F=-30$ N — negative, meaning the tow-bar has flipped from tension (+770 N, pulling the trailer forward) to a 30 N thrust (pushing it), because the trailer's own resistance alone would decelerate it faster than the shared value now requires. The sign is read off the numbers on each side of the connector, not fixed by which phase of the question is being solved.

Evidence. Same anchor as 3.5 (Oct 2019 Q3): the question's own second part removes the driving force and asks for the new tow-bar force, which is exactly this mechanism.

#3.7(b) — Force changes value, connector itself changes (spec 4.2(ii), "a particle hitting the ground")

Setup. A pulley system runs for a first stage; one particle lands (or otherwise stops), the string running to the other particle goes slack, and that surviving particle becomes a free particle under gravity alone for a second stage, needing its own fresh suvat treatment from the velocity it had at the instant of handover (§2.3(b)).

Micro-example. A (3 kg) rises, B (4 kg) falls, over a smooth pulley; B starts 0.7 m above the ground. Stage 1: $a_1=(4-3)g/7=1.4$ m/s². B lands after falling 0.7 m: $v^2=2(1.4)(0.7)=1.96 \Rightarrow v=1.4$ m/s; time $t_1=v/a_1=1$ s. Stage 2: string slack, A continues rising from 1.4 m/s, decelerating at $g$. Additional height gained: $h_2=v^2/(2g)=1.96/19.6=0.1$ m; time to that highest point $t_2=v/g=1.4/9.8=1/7$ s.

Evidence. This is the reviewed archive's own confirmed instance of the spec's named example ("a particle hitting the ground"): Jan 2025 Q7, added to this course's material in a 2026-09-25 gap-audit pass. Verified failure mode: candidates "failed to realise that there are two parts to the motion," using one equation throughout instead. Jun 2025 Q8 independently confirms the same structural shape per this paper's own FORGE guide, though that series was checked for shape only, not deep-mined line-by-line the way Jan 2025 was — 2 of 17 series, both named, one only shape-confirmed. FORGE itself flags this as a two-series pattern, short of the 3-series bar this course otherwise sets before calling something a standing convention, and this document keeps that same hedge rather than upgrading it.

#3.8 — A chain of three or more particles, multiple separate strings

Generalisation, not a new mechanism. §2.1's shared-$a$ argument and §2.2's per-connector-own-tension argument both extend to any number of particles joined in sequence: with $n$ particles and their own external forces $f_1,\dots,f_n$ (each taken positive in that particle's own direction of motion), the whole-system shortcut generalises cleanly to

$$a=\frac{f_1+f_2+\cdots+f_n}{m_1+m_2+\cdots+m_n},$$

still valid regardless of how many separate strings, pulleys, or friction terms sit between the particles, because every internal tension still cancels in pairs when every equation is summed. Recovering any one internal tension, though, now needs the chain "cut" at exactly that string — isolate the sub-chain on one side of it and apply $F=ma$ to that sub-chain alone, exactly as an isolated single particle would be treated, repeating once per tension wanted. This course's own lesson already builds a full three-particle, two-pulley, one-friction-term worked chain-drill on exactly this structure (table + two smooth pulleys + one particle on each side), so no new numeric example is duplicated here; the point this entry adds is the general $n$-particle formula and the explicit "cut the chain at the tension you want" method, stated once rather than left implicit in that one worked case.

#3.9 — Motion directly up or down an inclined plane, single particle (spec 4.2(iii) on its own)

A genuine boundary with this course's friction lesson, stated plainly. Spec 4.2(iii) is grouped under the same numbered spec item as the two-body scenarios above, but the reviewed archive's own confirmed citations for it are single-particle problems, not connected-particles ones: Oct 2022 Q3, verified error: friction drawn "acting down rather than up the plane"; Jan 2022 Q5, an equilibrium-versus-motion incline question resolving parallel/perpendicular and eliminating $R$ to solve for a coefficient of friction. A third citation once attributed to this family, Jun 2024 Q6, was independently re-verified against its real mark scheme and found to be set on a rough horizontal plane, not an incline — that correction already lives in friction-equilibrium-and-motion-on-a-rough-plane.ts's own audit trail and is not repeated as incline evidence here. This document names 3.9 for completeness of spec 4.2's own numbering, but its worked depth (friction direction, $F=\mu R$ at the moving limit) belongs to this course's spec-4.4/5.3 friction lesson, not to a second full treatment here.

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#4. The whole-system shortcut, and its one hard limit, stated once and generally

Every sub-case above uses the same shortcut: sum every particle's own equation of motion and every internal connector force cancels, because it pulls each of the two particles it touches toward each other with equal and opposite force (§2.2), leaving the general $n$-particle form given in §3.8. This is the fastest route to a shared acceleration whenever no internal force is wanted, and it is provably incapable of yielding one whenever an internal tension, thrust, or reaction force is wanted — those are exactly the quantities that were summed to zero to get the shortcut in the first place. Every micro-example above that needed an internal force (3.1's $T_1$, 3.4's $T$, 3.5/3.7(a)'s tow-bar force, 3.6's cable tension and normal reaction) therefore falls back to at least one single-particle equation after using the shortcut for $a$ — never the shortcut alone.

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#5. Exam-angle synthesis, grounded in named series

Where this topic sits in the whole paper, per the two fully mark-mapped recent series (Jan 2025, Jun 2025 — already tagged by this paper's own SIGNAL document, not re-derived here): spec 4.2 anchored the paper's own final and heaviest question in both — 15 of 75 marks in January, 17 of 75 in June — 2 of 2 series checked this way. SIGNAL's own hedge is repeated here rather than softened: that is two data points confirming a pattern, not seventeen.

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#Sources

research/veridian/WME01-verified-facts.md (§1, §3, §4 topic 4.2 and its 4.2/4.4 neighbours, §5, §6–§8, §9's connected-particles and friction-lesson audit entries) — itself checked against qualifications.pearson.com-sourced mark schemes and examiner reports for Oct 2019 Q3, Jun 2024 Q3, Jan 2022 Q5/Q7, Oct 2022 Q3/Q4, Jan 2023 Q7, Jan 2025 Q7, per that document's own provenance trail. content/veridian/curriculum/wme01/connected-particles-pulleys-and-lifts.ts (read in full, to avoid duplicating its worked exemplars). content/veridian/technique-hub/wme01/forge-exam-technique-guide.md and signal-frequency-and-boundaries.md (read in full, for the Jun 2025 Q8 shape-confirmation and the final-question mark-weight finding). content/veridian/technique-hub/wme01/codex-formula-and-method-bank.md §2.8 (read, to keep this document's own micro-examples numerically distinct from its pulley worked check). content/veridian/question-bank/wme01.ts (read, to confirm only one existing item touches this topic and to keep this document's own numbers distinct from it).