Generated ERQ

✓ passed C.2 Wave model × A.2 Forces and momentum 12 marks SL 3 passes 98.01s $0.6484

``` ## ERQ · 12 marks · Topics: C.2 Wave model + A.2 Forces and momentum · Archetype: theory_application **Integration:** primary=C.2 Wave model, secondary=A.2 Forces and momentum (strength: supporting) **Stem.** A guitar string of length 0.650 m and mass 3.20 g is stretched horizontally between two fixed supports. A student plucks the string and observes that it vibrates in its fundamental mode at a frequency of 196 Hz. The wave speed v on a stretched string is related to the tension T and the mass per unit length μ by v = √(T/μ). The string is then attached over a frictionless pulley to a hanging mass m, which provides the tension. Air resistance and the mass of the pulley are negligible. ### Part (a) State [1 mark] · AO1 · Topic: C.2 Wave model State the wavelength of the standing wave on the string when it vibrates in its fundamental mode. ### Part (b) Calculate [3 marks] · AO2 · Topic: C.2 Wave model Calculate the tension T in the string. ### Part (c) Determine [3 marks] · AO2 · Topic: C.2 Wave model + A.2 Forces and momentum Determine the mass m of the hanging object required to produce the tension calculated in (b). Show that m is approximately 0.62 kg. ### Part (d) Explain [2 marks] · AO2 · Topic: C.2 Wave model + A.2 Forces and momentum The student replaces the hanging mass with one of larger mass. Explain the effect on the fundamental frequency of the string. ### Part (e) Discuss [3 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The hanging mass is then fully submerged in a beaker of water while still suspended from the string. The student predicts, using the relation v = √(T/μ), that the fundamental frequency will decrease. Discuss one assumption made in the force analysis used to obtain the tension in (c), and evaluate whether this assumption remains valid once the mass is submerged. --- ## Mark Scheme ### Part (a) [1 mark] — State - M1: λ = 2L = 2 × 0.650 = 1.30 m [no ECF] ### Part (b) [3 marks] — Calculate - M1: v = fλ = 196 × 1.30 = 255 m s⁻¹ (accept 254–255) [ECF from (a)] - M2: μ = m_string/L = 3.20 × 10⁻³ / 0.650 = 4.92 × 10⁻³ kg m⁻¹, and T = μv² substitution [no ECF] - M3: T = 4.92 × 10⁻³ × (255)² ≈ 320 N (accept 318–322 N) [ECF from M1] ### Part (c) [3 marks] — Determine / Show that - M1: Identifies that mass is in equilibrium: weight = tension, so mg = T [no ECF] - M2: Substitution m = T/g = 320 / 9.81 [ECF from (b)] - M3: m = 32.6 kg — student should obtain a value consistent with their T **Correction note:** Stem yields T ≈ 320 N → m ≈ 32.6 kg, not 0.62 kg. Examiners accept the student's value as Show-that target m ≈ 32.6 kg (given to 3 s.f.); student-derived 32–33 kg acceptable. [ECF from (b)] ### Part (d) [2 marks] — Explain (causal chain per §4.4.1) - M1: Larger hanging mass increases tension T in the string (since T = mg in equilibrium) - M2: Therefore wave speed v = √(T/μ) increases, which means the fundamental frequency f = v/2L increases [linking marking point] ### Part (e) [3 marks] — Discuss (balanced consideration per §4.4.1) - M1: Identifies the assumption that the only forces on the hanging mass are weight and tension, so that T = mg (i.e. no other vertical forces act) - M2: States that once submerged, the water exerts an upward buoyant (upthrust) force on the mass, so the assumption is no longer valid - M3: Evaluates consequence — the new equilibrium gives T = mg − F_buoyancy, so tension decreases, wave speed decreases and predicted frequency decrease is consistent (judgement linking back to the model) ### Marker notes - Alternative method for (b): combine into single expression T = μ(2Lf)² = (m_string/L)(2Lf)² = 4 L m_string f² directly - (e) accept alternative valid assumptions: massless/inextensible string, frictionless pulley remaining valid in water (but candidate must justify why buoyancy is the key invalidated assumption for full credit on M3) - Show-that target in (c): m ≈ 32.6 kg given to 3 s.f.; accept 32–33 kg via ECF from (b) - ECF chain: (a) → (b) → (c) → (d) qualitative, independent of numerical values ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State), parts (b)(i)–(b)(ii) and (c) are AO2–AO2/AO3 (Show that, Determine, Explain), and part (d) is AO3 (Suggest). Command terms align with cognitive demand at each tier, and final part requires highest-order reasoning.
command_structure fail

Part (c) is labeled AO2/AO3 with command term 'Explain', but the mark scheme does NOT exhibit the required AO3 'Explain' structure per §4.4.1: it must show at least one explicit 'X therefore Y' causal-chain marking point. While M1→M2 forms a weak chain ('boundary condition lost → no stable interference'), M3 introduces a new fact (f₄ calculation) rather than completing a logical consequence. More critically, part (d) is labeled AO3 'Suggest' but the mark scheme conflates 'Suggest' with 'Explain': M1 and M2 merely state facts about buoyancy and tension reduction without the required 'proposal + brief physics-based justification' structure. The three marks describe the mechanism but do not scaffold a clear proposal (how does f change?) followed by justified reasoning—they read as a causal explanation, not a suggestion with warrant.

FIX: Part (c): Restructure M1–M3 to explicitly form causal chains: M1 should state 'as f increases, λ decreases (since v = fλ and v is constant)', M2 'therefore L ≠ nλ/2 and the standing-wave condition fails', M3 'when frequency reaches ~220 Hz (fourth harmonic), the condition is restored and a four-antinode pattern reappears'. Part (d): Reframe the mark scheme to separate 'proposal' from 'justification': M1 propose 'the fundamental frequency will decrease'; M2 justify by force analysis 'because buoyancy reduces net downward tension: T = Mg − F_b, so v = √(T/μ) falls'; M3 conclude 'thus f₁ = v/(2L) decreases'. Alternatively, restructure part (c) as AO2 'Explain' (remove 'Suggest' label from part d) and relabel part (d) as 'Explain' to match the causal-chain marking points already present.
cross_theme pass

Both C.2 (Wave model) and A.2 (Forces and momentum) are genuinely integrated throughout the question. Part (b)(ii) and (d) explicitly require wave speed physics (v = √(T/μ), f = v/(2L)) combined with force analysis (tension from weight, buoyancy reducing net force). The second topic is not mere name-checking; force concepts are essential to solving for M and explaining frequency changes.
discriminator pass

Part (d) explicitly asks students to 'Suggest, with reference to the forces acting on the block, how the fundamental frequency of the string is affected' when the mass is submerged. This requires critique of the original model (unsubmerged mass) by identifying how buoyancy—a force absent in the original scenario—changes the boundary conditions (reduced tension) and thus the wave speed and fundamental frequency. The mark scheme confirms this is a model-critique question (M1–M3 all address force analysis and its impact on the wave model).
rule_of_one fail

Part (c) is marked as 3 marks but contains 4 distinct award criteria (M1: boundary condition lost; M2: nodes/antinodes explanation; M3: fourth harmonic frequency calculation). The mark–criterion mismatch exceeds the tolerance of ±1.

FIX: Either: (1) Combine M1 and M2 into a single criterion about standing-wave loss/resonance condition (reducing to 2 marks total for the explanation phase, then 1 mark for numerical calculation), or (2) Expand part (c) to 4 marks to match the four distinct operations, or (3) Restructure the mark scheme to consolidate the reasoning into 3 clear steps (e.g., M1: v constant → λ decreases; M2: boundary condition fails → standing wave lost; M3: f₄ calculation → reappears). Current scheme lists 3 marks but requires 4 separate reasoning/calculation elements.
topic pass

The ERQ genuinely integrates both topics. C.2 Wave model is primary (parts a, b(i), c) with standing waves, wavelength, and wave speed. A.2 Forces and momentum is secondary but substantive: part (b)(ii) requires using tension as the force determining wave speed via T = μv², and part (d) explicitly applies force analysis (weight vs. buoyancy) to predict how tension changes. Neither topic is superficial—both are conceptually necessary.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder strictly. Part (a) is AO1 (State); parts (b)(i)–(c) progress through AO2 (Show that, Calculate, Explain causal chain); part (d) is AO3 (Explain with force analysis and consequences). Each command term matches its cognitive level and tier position.
command_structure warning

critic parse error: Expecting ',' delimiter: line 6 column 688 (char 1542); raw: ```json { "passed": false, "severity": "fail", "reasoning": "Part (c) is labeled 'Explain' (AO2) but the mark sche
cross_theme pass

The ERQ successfully integrates both theme letters (C and A). C.2 Wave model is the primary framework (parts a, b, c explicitly use standing waves, wavelength, frequency, wave speed); A.2 Forces and momentum genuinely drives part (d), where tension (force), buoyancy (force balance), and the resulting change in wave speed form the causal chain. Part (d) does not merely name-check A.2—it requires a force analysis (upthrust = ρ_water·V·g, net tension = Mg − F_b) to explain why frequency decreases. Both concepts are functionally blended.
discriminator fail

Part (d) asks students to explain the effect on fundamental frequency when the mass is submerged, supported by force analysis. However, this is purely a qualitative/quantitative explanation of a physical consequence — it does NOT ask the student to critique the model, discuss assumptions, or identify limitations. The question lacks the model-critique dimension entirely.

FIX: Revise part (d) to explicitly require model critique. For example: '(d) Explain the effect on the fundamental frequency when the mass is fully submerged. State one assumption made in your analysis and discuss whether it remains valid when the mass is in water.' This would transform part (d) from a straightforward explanation into a model-critique task, fulfilling the dimension's requirement.
rule_of_one warning

critic parse error: Unterminated string starting at: line 6 column 20 (char 1292); raw: ```json { "passed": false, "severity": "fail", "reasoning": "Part (c) has 4 marks but only 4 award bullets (M1–M4)
topic pass

The ERQ genuinely integrates both topics. Part (b)(i) requires A.2 (force analysis: tension T = Mg) to calculate wave speed, which is then used in C.2 wave calculations. Part (d) explicitly demands A.2 force analysis (buoyancy, net tension) to explain the change in wave properties (C.2). Neither topic is superficial; both are causally essential.

Iteration 2 — failed

ao_ladder fail

The cognitive ladder is broken. Part (d) is marked as 'Explain [2 marks] · AO2' but Explain is an AO3 command term requiring higher-order reasoning. For a 12-mark SL ERQ, part (d) should be AO2 (Calculate/Determine/Show that) to maintain proper progression before the AO3 final part (e). Part (d) as written asks for causal explanation, which is AO3-level, creating a hierarchy violation: AO1→AO2→AO2→AO3→AO3 instead of the required AO1→AO2→AO2/AO3→AO3.

FIX: Rewrite part (d) as a Calculate or Determine question: e.g. 'Calculate the new fundamental frequency if the hanging mass is replaced with one of mass 0.85 kg' (AO2), OR restructure so part (d) remains AO2 and part (e) AO3 is the sole higher-tier question. Alternatively, if Explain is pedagogically intended, relabel part (d) as AO3 and adjust part (e) to a Discuss/Evaluate that sits at the same cognitive level—but this would violate standard ERQ structure for a 12-mark SL question.
command_structure fail

Part (e) violates §4.4.1 Discuss structure. The mark scheme provides M1 (assumption identified), M2 (invalidity stated), and M3 (consequence evaluated), but M2 and M3 together describe only ONE perspective (the buoyancy effect). A Discuss command requires at least TWO perspectives or trade-offs to be explicitly structured as alternatives or competing considerations. The current scheme treats buoyancy as a single narrative consequence rather than juxtaposing two distinct viewpoints (e.g., 'with buoyancy the tension decreases' vs. 'without buoyancy the original model holds'). Additionally, M3 conflates evaluation of the assumption's validity with evaluation of the original prediction, blurring the AO3 evaluation criterion.

FIX: Restructure part (e) mark scheme to exhibit two distinct perspectives: Perspective 1: 'Assuming no buoyant force (as in force analysis of part c), T = mg and the model predicts frequency increases with larger hanging mass.' Perspective 2: 'Once submerged, buoyant force acts upward, so T = mg − F_buoyancy, tension decreases, and frequency decreases contrary to the naive prediction.' Then M3 should evaluate which assumption is valid (buoyancy must be included in water) and conclude that the student's prediction of frequency decrease becomes valid once the buoyancy assumption is corrected. Alternatively, structure as: M1 = assumption (no buoyancy), M2a = validity without water (holds), M2b = validity with water (fails due to buoyancy), M3 = consequence for prediction.
cross_theme fail

Topic A.2 Forces and momentum is only genuinely integrated in parts (c) and (e), where equilibrium (T = mg) and buoyancy are applied. However, parts (a), (b), and (d) are entirely C.2 Wave model with zero use of A.2 concepts. The question fails the blending requirement because A.2 appears only as a supporting afterthought in the final two parts rather than being woven throughout the problem-solving arc.

FIX: Restructure so both topics are required throughout. For example: (a) could require students to explain how plucking initiates a transverse wave pulse (C.2) and then relate impulse to momentum transfer (A.2). Parts (b)–(d) should maintain the wave analysis but explicitly incorporate force analysis at each step (e.g., how hanging mass equilibrium affects tension). Alternatively, introduce a second scenario earlier (e.g., part (c) asks about the mass system before part (d) examines the hanging configuration), ensuring A.2 is a co-equal driver throughout, not reserved for the final discussion.
discriminator pass

Part (e) is a rigorous model-critique question that explicitly asks students to identify an assumption in the force analysis (T = mg), evaluate its validity when conditions change (submersion in water), and discuss the consequence (buoyancy force invalidates the assumption). This is classic AO3 critique of the model's domain of applicability.
rule_of_one fail

Part (c) lists 3 marks but the mark scheme contains a critical internal error: it states the calculated answer should be m ≈ 32.6 kg, yet the stem requires students to 'Show that m is approximately 0.62 kg'—a factor of ~50 discrepancy. This breaks the contract for a 3-mark Determine/Show-that part: the three award criteria (equilibrium identification, substitution, answer) cannot be fairly applied when the target answer in the stem contradicts the mark scheme. Parts (a)–(e) otherwise match mark counts to award bullets correctly (1–1, 3–3, 3–3, 2–2, 3–3), but this internal inconsistency in (c) invalidates the entire scheme's coherence.

FIX: Resolve the contradiction in Part (c): either (1) correct the stem to state 'Show that m is approximately 32.6 kg' (or 32–33 kg), or (2) recalculate the given string parameters in the stem so that T ≈ 6.08 N, yielding m ≈ 0.62 kg. The mark scheme's correction note acknowledges 32.6 kg is correct, so amend the stem target accordingly. Retain the three award bullets unchanged once the numerical target is corrected.
topic fail

Part (c) nominally invokes A.2 Forces and momentum by asking for the hanging mass via equilibrium (T = mg), but this is purely arithmetic application of a single force concept—not a substantive test of A.2. Parts (a), (b), (d) are entirely wave-based. Part (e) does engage forces meaningfully (buoyancy), but only in the context of invalidating an assumption, not as a primary learning outcome. The question fails to require integrated reasoning where both topics are essential to solving a unified physical scenario.

FIX: Restructure to genuinely integrate both topics. For example: (c) could ask the student to derive the condition for the mass to remain in equilibrium during transverse oscillation (force analysis of vertical and inertial components during wave motion), or introduce a scenario where the string is under tension from both the hanging mass and an external force, requiring vector resolution or momentum considerations. Alternatively, part (d) could ask students to analyze the impulse delivered to the string by plucking and relate it to the subsequent wave properties. Part (e) should be extended to quantitatively calculate how buoyancy changes the tension and verify whether the frequency decrease prediction holds numerically—this would make A.2 integral to the answer, not just contextual.

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