Generated ERQ

✓ passed C.1 Simple harmonic motion × A.3 Work, energy and power 12 marks HL 3 passes 113.12s $0.7230

``` ## ERQ · 12 marks · Topics: C.1 Simple harmonic motion + A.3 Work, energy and power · Archetype: data_response **Integration:** primary=C.1 Simple harmonic motion, secondary=A.3 Work, energy and power (strength: supporting) **Stem.** A student investigates a horizontal mass–spring oscillator on a frictionless air track. A glider of mass m = 0.250 kg is attached to a spring of stiffness k and displaced a distance A = 0.080 m from equilibrium, then released. A motion sensor records the glider's speed v at several measured displacements x during one oscillation cycle. The student then plots v² against x² to test the simple-harmonic model and to estimate k from energy conservation. | x / m | v / m s⁻¹ | x² / 10⁻³ m² | v² / m² s⁻² | |--------|-----------|--------------|-------------| | 0.000 | 1.118 | 0.00 | 1.250 | | 0.020 | 1.083 | 0.40 | 1.173 | | 0.040 | 1.044 | 1.60 | 1.090 | | 0.050 | 0.985 | 2.50 | 0.970 | | 0.060 | 0.886 | 3.60 | 0.785 | | 0.070 | 0.686 | 4.90 | 0.470 | A best-fit straight line through the points has y-intercept 1.245 m² s⁻² and gradient −195 s⁻². ### Part (a) Define [2 marks] · AO1 · Topic: C.1 Define simple harmonic motion. ### Part (b)(i) Show that [3 marks] · AO2 · Topic: C.1 Using the SHM relation for v as a function of x, show that a graph of v² against x² should be a straight line with gradient −ω² and y-intercept ω²A². ### Part (b)(ii) Calculate [2 marks] · AO2 · Topic: C.1 Using the gradient of the best-fit line, calculate the period T of the oscillation. ### Part (c) Determine [3 marks] · AO3 · Topic: C.1 + A.3 By equating the total mechanical energy of the oscillator to ½mv²ₘₐₓ (obtained from the y-intercept of the graph), determine a value for the spring constant k and comment on its consistency with the value implied by the gradient of the graph. ### Part (d) Evaluate [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The student notices that the measured y-intercept (1.245 m² s⁻²) is slightly smaller than the value of ω²A² calculated from the gradient. Evaluate whether this discrepancy is consistent with a small amount of energy dissipation in the apparatus. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Acceleration (or restoring force) is proportional to displacement from a fixed equilibrium position [no ECF] - M2: Acceleration (or restoring force) is directed opposite to the displacement (toward equilibrium) [no ECF] ### Part (b)(i) [3 marks] — Show that - M1: States/uses v² = ω²(A² − x²) [no ECF] - M2: Expands to v² = ω²A² − ω²x², explicitly identifying the form y = c + (slope)x² [no ECF] - M3: Identifies y-intercept = ω²A² AND gradient = −ω² [no ECF] ### Part (b)(ii) [2 marks] — Calculate - M1: ω² = 195 s⁻² ⇒ ω = 13.96 rad s⁻¹ (accept 14.0); uses T = 2π/ω [no ECF] - M2: T = 0.450 s (accept 0.45 s) [ECF from gradient interpretation] ### Part (c) [3 marks] — Determine - M1: vₘₐₓ² = 1.245 m² s⁻² (from intercept) and applies energy conservation ½kA² = ½mvₘₐₓ², giving k = mvₘₐₓ²/A² [no ECF] - M2: k = (0.250 × 1.245)/(0.080)² = 48.6 N m⁻¹ (accept 48–49 N m⁻¹) [ECF from (b)(ii)] - M3: Comparison: gradient implies k = mω² = 0.250 × 195 = 48.75 N m⁻¹; the two values agree to within ~0.3%, so the data are consistent with SHM [ECF from M2] ### Part (d) [2 marks] — Evaluate (position + supporting + limiting per §4.4.1) - M1: Proposal/position: yes, consistent — a small frictional/drag loss removes mechanical energy, so the measured vₘₐₓ (and hence the intercept ω²A²) is slightly reduced relative to the ideal value predicted from the gradient - M2: Limiting consideration: the gradient depends only on ω (frequency of oscillation), which is largely unaffected by small damping, whereas the intercept depends on amplitude/energy and is reduced — so the observed pattern (low intercept, gradient-consistent ω) matches weak damping ### Marker notes - Show-that target in (b)(i): identification of both gradient = −ω² and intercept = ω²A² must be explicit. - Accept (b)(ii) answers using ω from rounded value: T in range 0.44–0.46 s. - (c) M3: accept any quantitative statement of agreement within ≤5 %; bare "they are similar" without numbers earns 0. - (d) accept alternative valid framings: energy lost to air drag, sensor friction, or spring internal damping; reject answers attributing the discrepancy to random measurement scatter without reference to energy. ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (Define); parts (b)(i)–(b)(ii) and (c) are AO2/AO3 (Show that, Calculate, Deduce); part (d) is AO3 (Evaluate). Command terms are appropriately sequenced and scaffold from knowledge recall through calculation to critical reasoning.
command_structure fail

Part (c) fails the AO3 'Deduce' command-term structure. Per §4.4.1, Deduce requires 'claim + explicit warrant linking to physical principle', but M1 only presents a derivation linking graph features to energy expressions—it does NOT state an explicit claim about the deduced mass value, nor does it warrant WHY that deduction follows from the gradient/intercept relationship. M3 provides the comparison comment but arrives after the claim is buried in method. Additionally, Part (d) 'Evaluate' structure is incomplete: M1 calculates the energy loss but M2's judgement lacks an explicit 'limiting consideration' (only supporting consideration + position). The evaluation should include both a reason the claim might seem plausible AND a reason it fails.

FIX: Part (c): Restructure M1 to state explicitly: 'Since E_K = ½mω²(A² − x²), the graph gradient = −½mω² and intercept = ½mω²A². Therefore, m can be deduced from m = 2 × gradient / ω².' Then M2 computes m = 0.250 kg as the warrant. M3 compares to stated value. Part (d): Restructure M2 to include both perspectives: 'The claim appears plausible because the air track does eliminate sliding friction; however, this is a limiting consideration—air drag on the glider is NOT eliminated by a frictionless track. The ~21% energy loss over 60 cycles shows dissipation is measurable and non-negligible, so the claim is not justified.' This explicitly separates supporting (track removes friction) from limiting (air drag remains) considerations.
cross_theme pass

The ERQ genuinely integrates both C.1 (Simple harmonic motion) and A.3 (Work, energy and power). Part (b)(i) requires students to extract ω from the E_K vs x² graph by recognizing that E_K = ½mω²(A² − x²), which fundamentally depends on both energy conservation (A.3) and SHM kinematics (C.1). Parts (c) and (d) further blend the two themes: (c) uses the total energy expression ½kA² alongside the gradient-mass relationship to recover experimental parameters, and (d) uses energy loss calculations to evaluate a physical claim. The second topic is not merely name-checked; it is operationally essential to every calculation.
discriminator pass

Part (d) clearly asks the student to evaluate a model claim by critiquing an assumption: whether the 'frictionless air track' justifies treating energy dissipation as negligible. The mark scheme requires students to use experimental amplitude decay data to show the claim is unjustified, explicitly distinguishing between track friction and air drag. This is genuine model critique, not calculation.
rule_of_one fail

Part (c) is marked out of 3 but contains 4 distinct award bullets (M1 linking energy to graph, M2 calculation of m, M3 comment on agreement, plus an implicit route choice). This exceeds the allowed tolerance of 1 divergence. Additionally, M1 is overly complex (presents two alternative routes) which clouds the mark structure; a 3-mark Deduce should have exactly 3 clear, sequential operations.

FIX: Restructure Part (c) to have exactly 3 marks with 3 award criteria: (M1) state that E_total = ½kA² = intercept, calculate k from intercept [linking]; (M2) calculate m using ω² = k/m [substitution]; (M3) comment on agreement with stated value within error [reasoned conclusion]. Remove the alternative gradient→m route from the mark scheme or separate it explicitly into a parallel scheme.
topic pass

The ERQ robustly integrates both C.1 (Simple harmonic motion) and A.3 (Work, energy and power). C.1 is primary: parts (a), (b)(ii), and (d) directly require SHM concepts (definition, ω, amplitude decay). A.3 is secondary but essential: parts (b)(i) and (c) fundamentally depend on energy expressions (E_K = ½mω²(A² − x²), total energy E = ½kA², energy loss calculation). The graph itself (E_K vs x²) embeds the energy–displacement relationship from A.3 into an SHM context, ensuring both topics are genuinely tested, not merely mentioned.

Iteration 1 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) uses AO1 (Define) at the base; parts (b)(i)–(b)(ii) and (c) use AO2–AO3 (Show that, Calculate, Deduce) in the middle; part (d) uses AO3 (Evaluate) at the top. The progression from recall→application→analysis is sound and the command verbs match the assessment objectives at each tier.
command_structure warning

critic parse error: Unterminated string starting at: line 6 column 20 (char 912); raw: ```json { "passed": false, "severity": "fail", "reasoning": "Part (c) mark scheme fails the Deduce (AO3) command-t
cross_theme pass

The ERQ genuinely integrates both C.1 (SHM) and A.3 (Work, energy and power). Part (b)(ii) requires both ω from SHM kinematics AND energy concepts (v_max = ωA₀ derived from E_K = ½mv²). Part (c) is the strongest blend: it uses the SHM energy relation E_total = ½kA₀² (linking displacement amplitude to total energy) combined with the work–energy principle (y-intercept of E_K vs x² graph represents total mechanical energy). Part (d) evaluates energy conservation by comparing initial and final amplitudes, directly invoking both topics. Neither topic is mere name-checking; both are essential to solving the question.
discriminator pass

Part (d) explicitly asks the student to **evaluate** the claim that energy is conserved by using the amplitude data after 60 oscillations. The final task requires the student to quantify the energy loss, identify it as contradicting the claim, and critique the underlying assumption by explaining what dissipative mechanisms (air drag, internal damping) are NOT eliminated by a frictionless track. This is a model-critique task, not a pure calculation.
rule_of_one fail

Part (c) is marked as 3 marks but contains 4 distinct award criteria (M1: linking to y-intercept and formula, M2: numerical k value, M3: comparison to expected k). This exceeds the acceptable divergence of ±1. Part (d) is marked 2 marks with 2 criteria, which is correct.

FIX: Part (c) should be reorganised to align marks and criteria. Option 1: Reduce to 2 marks by merging M1 and M2 into a single 'calculate k from y-intercept' step (formula + substitution + answer), then award M2 for the comparison warrant. Option 2: Keep 3 marks but restructure as: M1 (linking y-intercept to total energy and deriving formula), M2 (calculating k numerically), M3 (comment on consistency). The current scheme's separate 'linking' and 'calculation' steps within M1–M2 blur the criterion boundaries. Alternatively, upgrade part (c) to 4 marks if both the intercept-based and gradient-based routes warrant distinct award steps.
topic pass

The ERQ legitimately integrates both C.1 (Simple harmonic motion) and A.3 (Work, energy and power). C.1 is primary: parts (a), (b)(i) and most of (b)(ii) test SHM definitions, angular frequency, and amplitude. A.3 is secondary but essential: parts (b)(ii), (c) and (d) require energy conservation concepts (kinetic energy vs total mechanical energy, E_K = ½k(A² − x²), fractional energy loss). The graph of E_K vs x² directly invokes the energy relationship E_K = E_total − ½kx² inherent to SHM, making energy analysis inseparable from the oscillator's dynamics. Part (d) forces evaluation of energy dissipation mechanisms, fully integrating the energy topic.

Iteration 2 — failed

ao_ladder pass

The ERQ strictly obeys the cognitive ladder. Part (a) is AO1 (Define); parts (b)(i)–(c) are AO2/AO3 with Show that/Calculate/Determine; part (d) is AO3 Evaluate. Each tier builds progressively and command terms match IB expectations for HL 12-mark extended response.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 Evaluate structure. M1 states a position ('yes, consistent') but lacks an explicit supporting consideration with physics warrant. M2 provides a limiting consideration but is framed as mechanism explanation, not a genuine trade-off or contrasting limitation. The scheme does not exhibit the required [position + supporting consideration + limiting consideration] structure with clear logical separation.

FIX: Restructure Part (d) as follows: M1 (Position + supporting): Yes, the discrepancy is consistent with energy dissipation. Small damping removes mechanical energy from the system, reducing the maximum speed and hence the y-intercept value, because the intercept = ω²A² depends directly on oscillation amplitude/energy. M2 (Limiting consideration): However, the gradient (which determines ω) is largely independent of small damping and depends only on the restoring-force frequency. Therefore, the observed pattern—where the intercept is lower than predicted from the gradient but ω remains nearly constant—is the expected signature of weak damping. Alternatively, expand to three marks if needed: M1: Position—yes, energy dissipation explains the lower intercept. M2: Supporting—damping removes energy, reducing vₘₐₓ and the y-intercept proportionally. M3: Limiting—ω (from gradient) is insensitive to weak damping, so the gradient should remain nearly constant while the intercept falls, matching the observation.
cross_theme pass

The question genuinely integrates both C.1 (Simple Harmonic Motion) and A.3 (Work, Energy and Power). Part (c) is the critical junction: it uses the energy conservation principle (A.3) — specifically ½kA² = ½mv²ₘₐₓ — to determine k from the graph's y-intercept, then cross-checks against the SHM prediction from the gradient. Part (d) further leverages energy concepts (mechanical energy dissipation) to interpret the experimental discrepancy. Both physics concepts are genuinely used in solving, not merely named.
discriminator pass

Part (d) explicitly asks the student to evaluate whether a discrepancy between measured and theoretical values is consistent with energy dissipation, requiring critical analysis of the model's assumptions (frictionlessness, ideal oscillation). This is a genuine model-critique question that goes beyond calculation.
rule_of_one pass

All parts have mark counts matching their award-criterion bullets: (a) 2 marks / 2 bullets; (b)(i) 3 marks / 3 bullets; (b)(ii) 2 marks / 2 bullets; (c) 3 marks / 3 bullets; (d) 2 marks / 2 bullets. Each criterion is distinct and operationally clear (definition, derivation steps, formula application, substitution, numerical answer, comparison, reasoning).
topic pass

The ERQ authentically integrates both topics. C.1 (SHM) is primary and drives the core physics: the student derives v² = ω²(A² − x²) and uses it to extract ω and T from the linear plot. A.3 (Work, energy and power) is genuinely required in part (c), where energy conservation (½kA² = ½mvₘₐₓ²) is the sole method to determine k from the y-intercept. Part (d) further invokes energy dissipation as a physical explanation. Neither topic is superficial or merely nominal.

Planner output (stage 1)

Archetype: real-world device
Scenario: A student investigates a mass–spring system oscillating vertically on a frictionless air table. The spring has stiffness constant k, and a mass m is attached and set into simple harmonic motion with amplitude A. The student measures displacement, velocity, and time data, then uses energy methods to verify the model and assess energy dissipation in the real apparatus.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	Define
b(i)	3	AO2	Show that
b(ii)	2	AO2	Calculate
c	3	AO3	Derive
d	2	AO3	Evaluate

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