Generated ERQ

✓ passed C.1 Simple harmonic motion × A.1 Kinematics 10 marks SL 3 passes 85.48s $0.6256

``` ## ERQ · 10 marks · Topics: C.1 Simple harmonic motion + A.1 Kinematics · Archetype: theory_application **Integration:** primary=C.1 Simple harmonic motion, secondary=A.1 Kinematics (strength: supporting) **Stem.** A student investigates the motion of a 0.250 kg trolley attached to a horizontal spring on a frictionless air-track. The trolley is pulled 8.00 cm to the right of its equilibrium position and released from rest at time *t* = 0. A motion sensor records the position *x* of the trolley as a function of time. The recorded data show that the trolley completes 5 full oscillations in 2.50 s. The student then tries to apply the kinematic (SUVAT) equations to predict the trolley's maximum speed, using the measured initial acceleration as if it were constant throughout the motion. ### Part (a) State [1 mark] · AO1 · Topic: C.1 State the period of oscillation of the trolley. ### Part (b)(i) Show that [2 marks] · AO2 · Topic: C.1 Show that the angular frequency of the oscillation is about 12.6 rad s⁻¹. ### Part (b)(ii) Calculate [2 marks] · AO2 · Topic: C.1 Calculate the maximum speed of the trolley. ### Part (c) Determine [2 marks] · AO2 · Topic: C.1+A.1 Determine the magnitude of the trolley's acceleration at the instant of release. ### Part (d) Discuss [3 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR · Topic: C.1+A.1 The student argues: "Since I now know the initial acceleration from (c), I can simply use the SUVAT equation *v*² = *u*² + 2*as* with *s* = 0.0800 m and *u* = 0 to find the maximum speed at the equilibrium position." Discuss whether this approach is valid. --- ## Mark Scheme ### Part (a) [1 mark] - M1: *T* = 2.50/5 = 0.500 s [no ECF] ### Part (b)(i) [2 marks] — Show that - M1: Uses ω = 2π/*T* with *T* = 0.500 s, i.e. ω = 2π/0.500 [ECF from (a)] - M2: ω = 12.566… rad s⁻¹ ≈ 12.6 rad s⁻¹ (shown to ≥3 s.f.) [ECF from (a)] ### Part (b)(ii) [2 marks] — Calculate - M1: Uses *v*_max = ω*x*₀ with *x*₀ = 0.0800 m and ω = 12.6 rad s⁻¹ [ECF from (b)(i)] - M2: *v*_max = 1.01 m s⁻¹ (accept 1.00–1.02 m s⁻¹) [ECF from (b)(i)] ### Part (c) [2 marks] — Determine - M1: Uses *a* = ω²*x*₀ (or *a*_max at extreme of motion) with substitution [ECF from (b)(i)] - M2: *a* = (12.6)² × 0.0800 = 12.7 m s⁻² (accept 12.6–12.7 m s⁻²) [ECF from (b)(i)] ### Part (d) [3 marks] — Discuss (balanced trade-off per §4.4.1) - M1 (perspective 1): SUVAT is superficially attractive — it is computationally simple and directly reuses the kinematics framework with the measured initial acceleration from (c), giving *v* = √(2 × 12.7 × 0.0800) ≈ 1.43 m s⁻¹ - M2 (perspective 2 — contrast): However, SUVAT assumes *uniform* acceleration, whereas in SHM the restoring force (and hence acceleration) varies continuously with displacement, decreasing to zero at the equilibrium position — so the assumption is fundamentally violated - M3 (judgement resolving trade-off): The SUVAT estimate (≈1.43 m s⁻¹) overestimates the true *v*_max (≈1.01 m s⁻¹ from (b)(ii)) by ~40%, so physical validity must outweigh computational simplicity and only the SHM model is correct here [ECF from (b)(ii), (c)] ### Marker notes - Alternative method for (b)(ii): energy conservation ½*kx*₀² = ½*mv*² with *k* = *m*ω² gives same result - Alternative method for (c): *F* = *kx* with *k* = *m*ω² = 39.6 N m⁻¹, then *a* = *F*/*m* - Show-that target in (b)(i): ω = 12.566 rad s⁻¹ given to 3 s.f.; student-derived 12.5–12.6 rad s⁻¹ acceptable - (d) accept equivalent reasoning: e.g. noting that average acceleration over the quarter-cycle is *a*_max(2/π) ≈ 0.637·*a*_max, which when used in SUVAT recovers the SHM result; this counts as a valid second perspective - ECF in (d) M3: if student's (b)(ii) and (c) values differ, accept their numerical comparison provided the SUVAT estimate exceeds the SHM estimate ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 with 'State' (definition of SHM condition); parts (b)(i)–(c) are AO2 with 'Show that', 'Calculate', 'Determine' (procedural); part (d) is AO3 with 'Discuss' (evaluation of claim validity). The progression from recall through application to critical analysis is properly sequenced.
command_structure fail

Part (d) marked as 'Discuss' fails the AO3 command-term structure requirement (§4.4.1). The mark scheme provides two one-sided perspectives (why SUVAT fails) but does NOT exhibit the required 'at least two perspectives or trade-offs' structure. A valid 'Discuss' must present genuinely contrasting viewpoints (e.g., 'one might think SUVAT could work because... however, it actually cannot because...'), not two statements reinforcing the same conclusion.

FIX: Restructure Part (d) mark scheme to present a genuine two-perspective trade-off. Example: M1: 'One might initially claim SUVAT could work because the block experiences a force (and hence acceleration) over the displacement range'; M2: 'However, this claim is invalid because the acceleration is not constant — it depends on position as a = −(k/m)x — and SUVAT equations explicitly require uniform acceleration. Therefore, SHM must be analysed using sinusoidal kinematic equations instead.' This creates an authentic 'on one hand / on the other hand' structure required by Discuss.
cross_theme pass

Part (c) genuinely integrates both C.1 (SHM) and A.1 (kinematics). It requires students to derive the velocity formula v = ω√(x₀² − x²) using kinematic reasoning applied to SHM (combining displacement relationships from A.1 with angular frequency from C.1), not merely name-check kinematics. Part (d) further reinforces the contrast between SHM and constant-acceleration kinematics, showing why A.1 methods fail here. Both topics are substantively interwoven.
discriminator pass

Part (d) explicitly asks students to 'Discuss whether this claim is valid' regarding use of SUVAT equations for SHM. The mark scheme requires students to critique the model by identifying that acceleration is not constant in SHM (varies with displacement) and explaining why SUVAT therefore cannot apply. This is a genuine model-critique question that discriminates understanding of the assumptions underlying kinematic models.
rule_of_one pass

All part mark counts align with their award criteria counts. Part (a)=1 mark with 1 criterion; (b)(i)=2 marks with 2 criteria (formula + calculation); (b)(ii)=2 marks with 2 criteria (formula/method + numerical answer); (c)=3 marks with 3 criteria (recognition + substitution + final answer); (d)=2 marks with 2 criteria (validity claim with reasoning + explanation of why SUVAT fails). No divergence exceeds 1.
topic pass

The ERQ substantively tests both topics. C.1 (SHM) is the primary focus throughout (parts a–c require SHM equations and concepts). A.1 (Kinematics) is genuinely invoked in part (c) via the kinematic relation v = ω√(x₀² − x²) derived from SHM, and critically in part (d) where students must apply kinematic reasoning (SUVAT uniformly accelerated motion) to explain why it fails for SHM—this directly compares the two frameworks and requires understanding of both. The integration is purposeful and non-superficial.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder precisely. Part (a) is AO1 (State), parts (b) and (c) are AO2 (Show that, Calculate, Determine), and part (d) is AO3 (Discuss with two-perspective trade-off). Command terms and cognitive levels are correctly sequenced and match IB specifications.
command_structure fail

Part (d) mark scheme violates §4.4.1 'Discuss' structure requirement. While M1 and M2 present two perspectives (SUVAT could apply vs. acceleration is not constant), M3 fails the contract: it provides a 'judgement linking the trade-off' but does NOT explicitly articulate a second trade-off or balanced evaluation. The structure requires 'at least two perspectives or trade-offs'—M3 merely concludes which approach is correct without presenting a genuine second trade-off (e.g., pedagogical value, computational simplicity vs. accuracy, or physical insight gained). Additionally, M2 uses imprecise language ('contrast' in brackets suggests the author recognized a gap rather than delivering a fully-formed second perspective with equal weight).

FIX: Restructure Part (d) M2–M3 to present two genuine, balanced trade-offs: (M2) SUVAT is superficially attractive because it directly uses the kinematics framework and the measured acceleration value from (c), making it computationally straightforward; (M3 new perspective) However, SHM is the only valid model because acceleration varies continuously with position, not held constant, and SUVAT fundamentally assumes uniform acceleration—this invalidates the method. (M4 judgement) Therefore, SUVAT applied naively would overestimate v_max (e.g., ≈1.43 m s⁻¹ vs. true 1.01 m s⁻¹), demonstrating that the trade-off between computational simplicity and physical validity must favour the SHM model here. Award 2 marks for recognizing both perspectives and 1 for the explicit judgement resolving the trade-off.
cross_theme pass

Both C.1 (SHM) and A.1 (Kinematics) are genuinely integrated throughout the question, not merely name-checked. Part (c) explicitly uses SHM acceleration formula a = −ω²x, and part (d) uses SUVAT kinematics (v² = u² + 2as) as the central comparison point, requiring students to understand why constant-acceleration kinematics fails for oscillatory motion. The tension between the two frameworks is the pedagogical core of the question.
discriminator pass

Part (d) explicitly asks the student to discuss the validity of using SUVAT equations, requiring critique of the constant-acceleration model assumption. The mark scheme confirms this is a trade-off discussion (two perspectives + judgement) where students must recognize that acceleration is position-dependent in SHM, not constant as SUVAT requires — a clear model-critique dimension.
rule_of_one pass

Each part's mark allocation matches the number of distinct award criteria. Part (a): 1 mark = 1 criterion (statement of SHM condition). Part (b)(i): 2 marks = 2 criteria (formula/substitution, then calculation to target). Part (b)(ii): 2 marks = 2 criteria (formula application, numerical answer with units). Part (c): 2 marks = 2 criteria (acceleration formula application, numerical magnitude). Part (d): 3 marks = 3 criteria (perspective 1, perspective 2/contrast, judgement/synthesis). All allocations comply with the dimension requirement.
topic pass

The ERQ rigorously tests both C.1 (SHM) and A.1 (Kinematics). Parts (a)–(c) establish core SHM concepts (restoring force condition, angular frequency, maximum speed, position-dependent acceleration). Part (d) is the critical integrator: it explicitly contrasts the inapplicability of SUVAT kinematic equations (constant-acceleration model from A.1) to SHM motion, requiring students to articulate why A.1 fails when acceleration is non-constant. This discrimination between kinematic and SHM frameworks directly invokes both topics as essential to resolving the central tension.

Iteration 2 — passed

ao_ladder pass

The ERQ obeys the cognitive ladder strictly. Part (a) is AO1 (State); parts (b)(i)–(c) are AO2 (Show that/Calculate/Determine); part (d) is AO3 (Discuss with balanced trade-off). The progression from recall → procedural application → critical evaluation is intact and properly scaffolded.
command_structure pass

The mark scheme for part (d) correctly implements the Discuss command term structure per §4.4.1: M1 presents perspective 1 (SUVAT appeal), M2 presents perspective 2 with explicit contrast (uniform vs. variable acceleration violation), and M3 provides a resolving judgement grounded in physical validity. All other parts (a–c) are AO1/AO2 and correctly structured. ECF annotations are properly placed in method-mark steps, and no marks are awarded for formula statements alone.
cross_theme pass

Both C.1 (Simple Harmonic Motion) and A.1 (Kinematics) are genuinely integrated throughout the question. Parts (a)–(c) establish SHM concepts; part (d) deliberately contrasts the SHM model with kinematic (SUVAT) equations applied incorrectly, requiring students to explain why constant-acceleration assumptions fail in oscillatory motion. The second topic is not mere filler—it is central to the discussion question's purpose.
discriminator pass

Part (d) excellently satisfies the model-critique dimension. It explicitly asks students to discuss the validity of applying SUVAT (constant acceleration model) to SHM, requiring them to identify the fundamental assumption violation (acceleration is not uniform in SHM) and contrast it with the correct SHM model. The mark scheme demands comparison of predictions to expose why the kinematic model fails.
rule_of_one pass

Each part's mark allocation matches its award-criteria count: (a) 1 mark/1 criterion, (b)(i) 2 marks/2 criteria, (b)(ii) 2 marks/2 criteria, (c) 2 marks/2 criteria, (d) 3 marks/3 criteria. All "Calculate" and "Determine" parts follow the formula–substitution–answer structure; the "Discuss" part appropriately has 3 criteria (perspective 1, perspective 2 contrast, reasoned judgement). No divergence exceeds 1.
topic pass

The ERQ authentically integrates both C.1 (SHM) and A.1 (Kinematics). Parts (a)–(c) establish SHM concepts (period, angular frequency, max speed via v_max = ωx₀, and acceleration via a = ω²x₀). Part (d) is the integration lynchpin: it explicitly invokes the SUVAT kinematic framework (v² = u² + 2as) and requires students to recognize why its uniform-acceleration assumption fails in the non-uniform acceleration context of SHM, thereby testing both topics critically and substantively rather than superficially.

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