Generated ERQ

✓ passed C.1 Simple harmonic motion × A.3 Work, energy and power 14 marks HL 3 passes 129.31s $0.8140

``` ## ERQ · 14 marks · Topics: C.1 Simple harmonic motion + A.3 Work, energy and power · Archetype: experimental_analysis **Integration:** primary=C.1 Simple harmonic motion, secondary=A.3 Work, energy and power (strength: supporting) **Stem.** A student investigates the vertical oscillations of a mass attached to a light helical spring suspended from a clamp stand. The mass m = 0.250 ± 0.002 kg is displaced downwards by an amplitude x₀ = 4.5 ± 0.2 cm from the equilibrium position and released. A motion sensor placed beneath the mass records position and velocity throughout the motion. From repeated trials the student determines a period T = 0.628 ± 0.010 s. At the equilibrium position the motion sensor records a maximum speed v_max = 0.428 ± 0.015 m s⁻¹. The student wishes to test whether the system satisfies conservation of mechanical energy, by comparing the measured kinetic energy at equilibrium with the elastic potential energy predicted from SHM at maximum displacement. Air resistance and damping over a single oscillation are assumed negligible. ### Part (a) Define [2 marks] · AO1 · Topic: C.1 State the defining condition for simple harmonic motion and write down the relationship between maximum speed, angular frequency and amplitude. ### Part (b)(i) Show that [3 marks] · AO2 · Topic: C.1 Show that the angular frequency of the oscillation is approximately 10.0 rad s⁻¹, and hence that the SHM-predicted maximum speed is about 0.45 m s⁻¹. ### Part (b)(ii) Calculate [3 marks] · AO2 · Topic: C.1 + A.3 Using the measured value v_max = 0.428 m s⁻¹, calculate the kinetic energy of the mass at the equilibrium position and its fractional uncertainty. ### Part (c) Compare and Evaluate [4 marks] · AO3 · Topic: C.1 + A.3 The elastic potential energy stored in the spring at maximum displacement, predicted from the SHM model, is E_p = ½mω²x₀² = 0.0152 J. Compare this value with the kinetic energy obtained in (b)(ii), and evaluate whether the experiment is consistent with conservation of mechanical energy within experimental uncertainty. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The student's analysis assumes that the spring obeys Hooke's law throughout the motion and that the motion is undamped. Suggest one physical reason, other than random measurement error, that could account for the discrepancy identified in (c). --- ## Mark Scheme ### Part (a) [2 marks] - M1: Acceleration is proportional to displacement from equilibrium AND directed towards equilibrium (both required) [no ECF] - M2: v_max = ωx₀ (or equivalent with A for amplitude) [no ECF] ### Part (b)(i) [3 marks] — Show that - M1: ω = 2π/T substituted: ω = 2π / 0.628 [no ECF] - M2: ω = 10.005 rad s⁻¹ ≈ 10.0 rad s⁻¹ [no ECF] - M3: v_max = ωx₀ = 10.0 × 0.045 = 0.450 m s⁻¹ (≈ 0.45 m s⁻¹) [no ECF] ### Part (b)(ii) [3 marks] — Calculate - M1: KE = ½mv² = ½ × 0.250 × (0.428)² [no ECF] - M2: KE = 0.0229 J (accept 0.0228–0.0230 J) [no ECF] - M3: Fractional uncertainty Δ(KE)/KE = Δm/m + 2Δv/v = 0.008 + 2(0.035) ≈ 0.078 (≈ 8%) [ECF from M1] ### Part (c) [4 marks] — Compare (similarity + difference) and Evaluate (position + justification) - M1 (Compare — similarity): By conservation of mechanical energy, the KE at equilibrium and the elastic PE at maximum displacement should represent the same total mechanical energy of the oscillator [ECF from (b)(ii)] - M2 (Compare — difference): The measured KE (0.0229 J) exceeds the SHM-predicted E_p (0.0152 J) by approximately (0.0229 − 0.0152)/0.0152 ≈ 50% [ECF from (b)(ii)] - M3 (Evaluate — position): The result is NOT consistent with conservation of mechanical energy within experimental uncertainty [ECF from (b)(ii)] - M4 (Evaluate — justification): The combined fractional uncertainty in the comparison is at most ~8% (from KE) plus ~9% (from 2Δx₀/x₀ in E_p), giving ≤ ~17%, which is far smaller than the observed ~50% discrepancy — therefore the disagreement is significant and points to a systematic effect rather than random scatter [ECF from (b)(ii)] ### Part (d) [2 marks] — Suggest (proposal + reasoning) - M1 (proposal): Any one plausible systematic cause, e.g. the spring does not obey Hooke's law at this amplitude / the spring has appreciable mass so effective oscillating mass exceeds 0.250 kg / gravitational PE of the spring's own mass contributes / the motion sensor calibration overestimates v_max - M2 (justification): Brief physics link, e.g. a nonlinear (stiffening) spring would produce a smaller restoring force per unit displacement than ½kx₀² predicts, so true stored PE exceeds the SHM estimate / additional oscillating mass increases KE for the same observed v_max without changing predicted E_p ### Marker notes - Show-that target in (b)(i): ω = 10.005 rad s⁻¹ and v_max = 0.450 m s⁻¹ given to 3 sig figs; student values 9.95–10.05 rad s⁻¹ and 0.44–0.46 m s⁻¹ acceptable. Subsequent parts use v_max = 0.428 m s⁻¹ (measured), NOT the SHM-derived value. - Alternative method accepted for (b)(ii) M3: quadrature combination Δ(KE)/KE = √[(Δm/m)² + (2Δv/v)²] ≈ 0.070 also accepted. - (c) accept consistent ECF: if student obtained KE ≈ 0.015 J in (b)(ii), then "consistent within uncertainty" is the correct position and full ECF marks available provided uncertainty comparison is explicit. - (d) accept any of: nonlinear/anharmonic spring response, effective mass of spring (Rayleigh correction ~ m_spring/3), additional gravitational PE not accounted for, amplitude measurement bias, sensor calibration error, internal friction in spring, energy lost as heat in spring material. Do NOT award "air resistance" alone (stem excludes it over one oscillation) unless explicitly justified. ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ follows a strict cognitive ladder: (a) AO1 Define; (b)(i)–(ii) AO2 Show/Calculate; (c) AO3 Determine with energy judgment; (d) AO3 Explain with causal reasoning. Command terms align correctly with complexity tiers, and experimental uncertainty permeates the design appropriately.
command_structure fail

Part (c) mark scheme violates §4.4.1 AO3 'Determine' structure. The command 'determine...and compare' requires both a Determine action (quantitative result + mechanism) AND a Compare action (at least one similarity AND one difference). M1–M3 correctly establish the two energy values, but M4 conflates comparison with evaluation (uncertainty commentary). The scheme lacks an explicit marking point that states both a similarity (e.g., 'both measured and predicted values represent total mechanical energy') and a difference (e.g., '0.428 cf 0.442 m s⁻¹'; 6% discrepancy in kinetic vs predicted elastic energies), before then evaluating whether the difference is significant. Additionally, Part (d) 'Explain' correctly includes a causal chain (M1→M2), but the scheme's M1 presentation is disjunctive ('OR') rather than unified; while acceptable, the marker notes offer four alternatives without hierarchical clarity.

FIX: Part (c): Split M4 into two distinct marking points: (M4a) Compare: similarity—'both represent total mechanical energy of the system at the stated instant/configuration' AND difference—'measured kinetic energy at equilibrium (0.0229 J) vs SHM-predicted elastic potential energy at amplitude (0.0244 J); ~6% difference in magnitude'; (M4b) Evaluate: 'The ~6% discrepancy lies within the combined experimental uncertainties (~5% in v_max, ~4% in A), therefore energy conservation is supported within the precision of the measurement.' Part (d): Retain the causal structure but in M1 list the single most probable violation first (nonlinear spring response at large extension) with 'or alternatively' as secondary, to clarify the primary teaching point.
cross_theme pass

Both C.1 (SHM) and A.3 (work, energy, power) are genuinely integrated. Part (c) uses energy conservation (kinetic and elastic potential energy from A.3) as the primary method to analyze the oscillating mass, directly comparing measured v_max data with SHM predictions via total mechanical energy. Energy concepts are essential to solving part (c), not merely mentioned in passing.
discriminator pass

Part (d) explicitly asks the student to 'Explain one physical assumption in the SHM model that is likely violated' when the measured v_max is smaller than predicted. This is a clear model-critique question requiring identification and causal explanation of an assumption breakdown (e.g., nonlinear spring behavior, air resistance, or mass of spring). The dimension is satisfied.
rule_of_one pass

Each part's mark count matches its award-bullet count: (a) 2 marks / 2 bullets, (b)(i) 3 marks / 3 bullets, (b)(ii) 3 marks / 3 bullets, (c) 4 marks / 4 bullets, (d) 2 marks / 2 bullets. All Calculate and Show-that sections contain exactly the required formula/substitution/answer chain.
topic pass

The ERQ genuinely integrates both C.1 (SHM) and A.3 (work, energy, power). Parts (a)–(b) establish SHM kinematics; Part (c) explicitly applies energy conservation (A.3) to compare kinetic energy at equilibrium with elastic potential energy at maximum displacement, requiring conceptual understanding of energy in oscillatory systems. Part (d) tests whether assumptions underlying both topics hold at larger amplitudes. Neither topic is superficial.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (Define), part (b) is AO2 (Show that/Calculate), part (c) is AO3 (Determine and Compare with evaluation), and part (d) is AO3 (Explain with causal reasoning). Command terms and cognitive demand progress appropriately through AO1→AO2→AO3 tiers.
command_structure fail

Part (c) mark scheme violates §4.4.1 for the 'Compare' command term. M3 is labeled as a comparison requiring 'similarity AND difference,' but the mark scheme does NOT structure this as a discrete, assessable marking point with clear criteria. The 'Compare' command requires explicit parallel evaluation of at least one similarity AND one difference as separate evaluable elements. Additionally, M4 ('Evaluate') lacks the required structure: it must contain a position + supporting consideration + limiting consideration per §4.4.1, but instead it conflates method (calculating combined uncertainty) with verdict without clearly separating the position statement from supporting and limiting reasoning.

FIX: Restructure Part (c) M3 and M4 as follows: M3 (Compare — similarity): Both the measured kinetic energy at equilibrium and the SHM-predicted elastic potential energy at maximum displacement represent the same total mechanical energy of the system (by conservation of energy, KE at equilibrium = elastic PE at amplitude). M4 (Compare — difference): The measured kinetic energy (0.0229 J) is approximately 50% larger than the SHM-predicted elastic PE (0.0152 J); the values disagree substantially in magnitude. M5 (Evaluate — position): The discrepancy is NOT consistent with conservation of energy within experimental uncertainty. M6 (Evaluate — supporting consideration): Combined experimental uncertainty is ~14% (from part b and amplitude), whereas the observed discrepancy is ~50%, far exceeding measurement precision. M7 (Evaluate — limiting consideration): This suggests a systematic effect such as nonlinear spring response, air resistance, or the measured v_max being anomalously high, rather than random experimental scatter. Alternatively, consolidate to 4 marks by: M3 = explicit similarity; M4 = explicit difference + quantification; M5 = position (consistent/not); M6 = justification combining both uncertainty analysis and physics reasoning.
cross_theme pass

Both C.1 (SHM) and A.3 (work, energy, power) are genuinely integrated throughout part (c), which forms the intellectual core of the question. Part (c) requires students to calculate kinetic energy and elastic potential energy, apply energy conservation explicitly, and evaluate the discrepancy—energy concepts are essential to the analysis, not merely name-checked. Part (d) reinforces the energy theme by explaining why energy conservation fails at large amplitude. The two topics support each other meaningfully.
discriminator pass

Part (d) explicitly asks students to explain why the SHM model fails at large amplitude—a direct critique of model assumptions and limitations. The mark scheme requires students to identify the physical reason (nonlinear spring response, air resistance, damping, etc.) and link it causally to the observed discrepancy, which is precisely the model-critique flavor required by this dimension.
rule_of_one pass

Each part's mark count matches or closely aligns with the number of distinct award criteria. Part (a) 2M/2 bullets; (b)(i) 3M/3 bullets (formula, substitution, answer); (b)(ii) 3M/3 bullets (formula, calculation, uncertainty); (c) 4M/4 bullets (two determinations, comparison with quantification, evaluation); (d) 2M/2 bullets (primary mechanism, causal link to outcome). No divergence exceeds 1 mark.
topic pass

Both topics are genuinely integrated and essential to the ERQ. C.1 (SHM) is the primary framework: defining SHM, deriving ω from T, predicting v_max and total mechanical energy via ω²A². A.3 (Work, energy and power) is the secondary but substantive topic: part (c) requires explicit calculation of kinetic energy (½mv²), elastic potential energy (½kA² or ½mω²A²), energy conservation analysis, and uncertainty quantification; part (d) evaluates energy dissipation and nonlinear breakdown. Neither topic is superficial—both are required for a complete solution.

Iteration 2 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder rigorously. Part (a) is AO1 (Define) with command term 'State'; parts (b)(i)–(b)(ii) are AO2 (Show that / Calculate) focused on procedural reasoning; parts (c)–(d) are AO3 (Compare & Evaluate / Suggest) requiring analysis of experimental data against theory and critical reasoning about systematic effects. The progression from recall → calculation → evaluation is well-ordered and appropriate for a 14-mark HL experimental analysis.
command_structure warning

critic parse error: Unterminated string starting at: line 6 column 20 (char 799); raw: ```json { "passed": false, "severity": "fail", "reasoning": "Part (c) mark scheme violates the Compare–Evaluate AO
cross_theme pass

Both C.1 (SHM) and A.3 (Work, energy and power) are genuinely integrated throughout the question. C.1 provides the theoretical framework (ω = 2π/T, v_max = ωx₀, E_p = ½mω²x₀²), while A.3 is actively used in calculating kinetic energy (½mv²), elastic potential energy, and energy conservation comparisons in parts (b)(ii), (c), and (d). The two topics are not merely named but structurally woven into the experimental analysis.
discriminator pass

Part (d) explicitly asks students to 'Suggest one physical reason, other than random measurement error, that could account for the discrepancy identified in (c).' This is a genuine model-critique task requiring students to identify and justify a systematic limitation or assumption breakdown (e.g., Hooke's law failure, spring mass, nonlinearity) rather than perform another calculation. The mark scheme confirms this evaluative intent.
rule_of_one fail

Part (c) lists 4 mark criteria but requires only 2 distinct operations for a 'Compare and Evaluate' task (comparison + evaluation with justification). The mark scheme conflates structural elements (similarity/difference/position/justification) with independent award criteria, inflating the count. A 4-mark Compare-and-Evaluate should have ~4 substantive criteria only if the evaluate step genuinely requires 2 independent sub-operations; here, it does not—the justification (M4) is subsidiary to the evaluation position (M3).

FIX: Reorganise Part (c) mark scheme to 4 marks with clearer delineation: M1 (state expected equivalence by energy conservation), M2 (calculate percentage discrepancy ~50%), M3 (state not consistent within uncertainty), M4 (justify by comparing uncertainty budget ~17% to observed gap ~50%, concluding systematic effect). Alternatively, reduce to 3 marks by merging M3 and M4 into a single 'Evaluate with justification' criterion. Current structure conflates structural presentation with independent marking operations.
topic pass

The ERQ genuinely integrates both topics. C.1 (SHM) is primary and forms the core conceptual framework: defining SHM, deriving ω from period, using v_max = ωx₀, and predicting elastic PE from SHM energy storage. A.3 (Work, energy and power) is secondary but substantive: parts (b)(ii), (c), and (d) require direct calculation and comparison of kinetic energy, elastic potential energy, and conservation of mechanical energy. The question cannot be solved without invoking energy principles alongside SHM kinematics.

Planner output (stage 1)

Archetype: experimental setup
Scenario: A student investigates the oscillation of a mass suspended from a vertical spring. The mass is displaced from equilibrium and released, executing simple harmonic motion. The student measures the period of oscillation, the amplitude, and records the velocity at various positions during the motion. The experiment aims to verify energy conservation and the characteristics of SHM, including the relationship between maximum velocity, amplitude, and angular frequency.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	Define
b(i)	3	AO2	Show that
b(ii)	3	AO2	Calculate
c	4	AO3	Determine
d	2	AO3	Explain

Generate another →