Generated ERQ

✓ passed C.1 Simple harmonic motion × A.1 Kinematics 10 marks SL 3 passes 101.14s $0.5946

## ERQ · 10 marks · Topics: C.1 Simple harmonic motion + A.1 Kinematics **Stem.** A student investigates vertical oscillations of a 0.50 kg brass cylinder suspended from a light helical spring fixed to a laboratory clamp. The cylinder is pulled vertically downward by 4.0 cm from its equilibrium position and released from rest. A motion sensor mounted directly below records the cylinder's vertical displacement against time. From the trace, the student measures the time for ten complete oscillations to be 20.0 s. The student assumes the oscillation is simple harmonic, that air resistance is negligible, and that the amplitude remains constant throughout the recording. ### Part (a) State [2 marks] · AO1 · Topic: C.1 State the two conditions that must be satisfied by the acceleration of the cylinder for its motion to be classified as simple harmonic. ### Part (b)(i) Calculate [2 marks] · AO2 · Topic: C.1 Calculate the angular frequency of the oscillation. ### Part (b)(ii) Determine [2 marks] · AO2/AO3 · Topic: C.1 Determine the spring constant of the spring. ### Part (c) Show that [2 marks] · AO3 · Topic: C.1 + A.1 Using a kinematic argument based on the variation of acceleration with displacement, show that the maximum speed of the cylinder occurs at the equilibrium position and has magnitude v_max = ωA. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The student records oscillations for 5 minutes and finds that the calculated v_max = ωA overestimates the measured maximum speed during later cycles. Suggest two distinct physical reasons, each linked to a specific assumption stated in the stem, that account for this discrepancy. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Acceleration is directed towards (or proportional to the negative of) a fixed equilibrium position / acceleration is opposite in direction to displacement [ECF: no] - M2: Magnitude of acceleration is directly proportional to the displacement from equilibrium [ECF: no] ### Part (b)(i) [2 marks] - M1: Period T = 20.0/10 = 2.0 s AND use of ω = 2π/T [ECF: no] - M2: ω = 3.14 rad s⁻¹ (accept 3.1 rad s⁻¹, 2 s.f.) [ECF: yes] ### Part (b)(ii) [2 marks] - M1: Use of ω² = k/m (or T = 2π√(m/k)) with substitution k = (3.14)² × 0.50 [ECF: yes from b(i)] - M2: k = 4.9 N m⁻¹ (accept 4.8–5.0 N m⁻¹, 2 s.f.) [ECF: yes] ### Part (c) [2 marks] - M1: States that since a = −ω²x, acceleration is zero at x = 0; therefore between any other point and x = 0 the cylinder is accelerating in the direction of motion, so speed is still increasing — maximum speed must occur at x = 0 [ECF: no] - M2: Derives v_max = ωA by energy conservation (½kA² = ½mv²_max ⇒ v_max = A√(k/m) = ωA) OR by integrating a = −ω²x to give v² = ω²(A² − x²) and setting x = 0 [ECF: no] ### Part (d) [2 marks] - M1: Identifies that air resistance (stated as negligible) provides a velocity-dependent damping force; this dissipates mechanical energy each cycle so the true amplitude A decreases over time, hence the measured v_max < ωA₀ (where A₀ is the initial amplitude used in the calculation) [ECF: no] - M2: Identifies that the assumption of constant amplitude fails because, in addition to damping, internal friction within the spring coils (hysteresis / inelastic behaviour of the spring material) removes energy on each cycle, further reducing the amplitude below the initial value substituted into v_max = ωA [ECF: no] ### Marker notes - (b)(i): Allow ω from T = 2.0 s direct; do not penalise carrying 3.142 through. - (b)(ii): Alternative route via k = mω² accepted; ECF from any reasonable ω in (b)(i). - (c): Accept clear graphical/phase-diagram argument that v and x are π/2 out of phase, provided the kinematic link a = −ω²x → v² = ω²(A² − x²) is shown. Pure assertion without derivation scores M1 only. - (d) discriminator: M1 must explicitly name air resistance/drag as the damping mechanism and link to amplitude decay. M2 must name a *distinct second mechanism* (spring internal friction, hysteresis, anelastic behaviour, or thread/clamp losses). Do NOT award two marks for two restatements of "damping". If candidate instead argues non-Hookean spring response at large extension causing a ≠ −ω²x, award M2 provided the link to v_max = ωA breaking down is explicit.

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ strictly obeys the cognitive ladder: (a) is AO1 State with command term appropriate to recall conditions; (b)(i)–(ii) are AO2 Calculate/Determine requiring standard substitution and rearrangement; (c) is AO3 Show that requiring derivation via calculus and inference of position; (d) is AO3 Suggest requiring evaluation of real experimental discrepancies. The progression from knowledge → procedural application → reasoning is logically ordered and well-differentiated by command term and mark allocation.
cross_theme fail

Part (c) nominally lists both C.1 and A.1, but kinematics (A.1) is only superficially invoked. The question asks students to use kinematics (differentiation: v = dx/dt) to derive v_max, which is the correct physics. However, the alternative mark scheme explicitly permits an energy method (part of C.2 Gravitational potential energy, not A.1), and the energy route bypasses kinematics entirely. More critically, the question structure treats kinematics as a *method* to solve a purely SHM problem, not as a genuinely blended concept where both oscillation AND kinematic relationships are essential to the physics being tested. The two themes are not integrated; A.1 is merely the calculus tool used to manipulate the SHM equation.

FIX: Redesign part (c) to require both SHM and kinematic reasoning as distinct physics contributions. For example: 'A block falls freely from rest through height h and lands on the spring. Using kinematics, find the impact speed. If the collision brings the block to rest and then releases it into SHM, calculate the new amplitude and maximum speed.' This forces genuine interplay: kinematics (free fall) determines the energy input, which then governs the SHM amplitude and velocity. Alternatively, construct a part where students must use constant-acceleration kinematics to analyze the block's motion during a brief external force application, then compare the subsequent SHM oscillation.
discriminator fail

Part (d) asks students to suggest reasons why experimental values differ from calculated values, but this is a factual explanation task (AO3), not a model-critique task. It does not ask students to evaluate, discuss, or critique the assumptions or limitations of the SHM model itself (e.g., 'Discuss the assumption that air resistance is negligible' or 'Explain why treating the spring as ideal is a limitation'). The question stops at explaining discrepancies rather than evaluating the model's validity.

FIX: Reframe part (d) to explicitly require model critique. For example: 'Discuss two assumptions made in the model that contribute to the difference between the calculated and experimental maximum speeds' or 'Explain how violations of the two conditions for SHM (from part a) could account for the observed discrepancies.' This would require students to link experimental observations back to the validity and limitations of the SHM model itself.
rule_of_one fail

Part (a) awards 2 marks but lists only 2 bullet points that are each testing the same concept (restoring force and proportionality to displacement). These should be treated as a single integrated criterion. The note allows both marks for a single equation F = −kx with both properties identified, but the mark scheme itself shows 2 separate bullets for what is logically one condition. More critically, part (d) awards 2 marks but provides 5 distinct acceptable answers with no clear internal discrimination structure—the rubric 'award 1 mark for each distinct reason (max 2)' is ambiguous about which 2 of the 5 are being formally assessed. This violates the dimension's requirement that mark count align with the number of award criteria.

FIX: Restructure part (a) to award 1 mark for 'restoring force directed towards equilibrium' and 1 mark for 'magnitude proportional to displacement (F ∝ −x)', presented as sequential independent criteria. For part (d), either: (i) limit the mark scheme to exactly 2 distinct, prioritized reasons with clear scoring (e.g., M1 = damping/energy loss; M2 = sensor resolution/sampling), or (ii) explicitly list all 5 acceptable mechanisms with a transparent scoring rule (e.g., any two distinct from the list scores full marks, with exemplars). The current hybrid approach is unreviewable.
topic pass

Part (c) explicitly requires kinematics (differentiation of the displacement equation x = x₀cos(ωt) to find v = dx/dt) to derive the maximum speed formula, while the entire context and parts (a), (b) rely on simple harmonic motion concepts (restoring force, angular frequency, spring constant). Both topics are genuinely integrated in the derivation, not superficial.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State the conditions), part (b) is AO2/AO3 (Calculate k, Determine v_max), part (c) is AO3 (Show that with reasoning about consistency), and part (d) is AO3 (Discuss with explicit reference to conditions from (a)). The progression from recall through application to analysis/evaluation is maintained throughout.
cross_theme pass

The ERQ genuinely blends C.1 (Simple Harmonic Motion) and A.1 (Kinematics). Part (c) requires students to use kinematic equations for free fall (A.1) to calculate impact speed, then explicitly compare this with the SHM maximum speed (C.1), creating an authentic physics tension that drives the subsequent analysis. Both concepts are essential to solving the question, not merely name-checked.
discriminator pass

Part (d) explicitly requires the student to discuss modelling assumptions by referencing the two SHM conditions from (a), making a clear model-critique demand. This is the final part and directly asks students to explain why violations of the ideal SHM model account for observed discrepancies between predicted and measured speeds.
rule_of_one fail

Part (d) violates the mark-count/award-criteria contract. The part is worth 2 marks but the mark scheme lists 4 distinct award criteria (M1, M2, and two alternatives), with a conditional "maximum 2" rule. This is ambiguous and breaks the IB standard that each mark should correspond to exactly one clear, independent operation. The conditional phrasing and alternative M-marks make the scheme non-transparent.

FIX: Restructure Part (d) to list exactly 2 independent, mutually exclusive award criteria. For example: M1: correctly identifies and links one specific violation (e.g. non-linear spring response) to failure of the a ∝ −x condition, explaining why v_max = ωA no longer holds. M2: correctly identifies and links a second specific violation (e.g. gravitational work during free fall/compression) to failure of fixed-equilibrium condition, explaining why the actual amplitude differs from measured oscillation amplitude. Remove the 'alternative' M-marks and 'maximum 2' qualifier to ensure clarity and transparency.
topic pass

The ERQ authentically integrates both topics. Part (c) requires A.1 kinematics (free fall v² = u² + 2gs) to find impact speed, then directly compares it against the C.1 SHM maximum speed from (b)(ii), forcing students to reconcile why these differ. Part (d) explicitly demands discussion of SHM conditions (from part a) and their violations, grounding the comparison in physics principles from both domains. Neither topic is superficial or nominal; both are operationally essential.

Iteration 2 — passed

ao_ladder pass

The ERQ obeys the cognitive ladder precisely. Part (a) is AO1 (State) with a straightforward recall of SHM conditions. Parts (b)(i)–(b)(ii) are AO2/AO3 (Calculate/Determine) requiring procedural competence. Part (c) is AO3 (Show that) requiring derivation from first principles. Part (d) is AO3 (Suggest) requiring synthesis and evaluation of assumptions against experimental evidence. Difficulty and demand increase monotonically; no tier is skipped or inverted.
cross_theme pass

Part (c) genuinely integrates both topics: it uses the kinematic relationship a = −ω²x (A.1 kinematics: acceleration-displacement relation) to derive the SHM result v_max = ωA (C.1). The kinematic argument is central to the derivation, not merely name-checked. Topic letters C and A differ, and both physics concepts are authentically used in solution.
discriminator pass

Part (d) excellently satisfies the model-critique dimension. It requires students to identify why the calculated formula (v_max = ωA) overestimates the measured value by critiquing specific assumptions from the stem (negligible air resistance, constant amplitude). This demands evaluation of the model's limitations and physical reasoning about energy dissipation mechanisms.
rule_of_one pass

Each part's mark allocation matches the number of distinct award criteria. Part (a): 2 marks, 2 bullets (two SHM conditions). Part (b)(i): 2 marks, 2 bullets (period calculation + angular frequency result). Part (b)(ii): 2 marks, 2 bullets (formula/substitution + numerical answer). Part (c): 2 marks, 2 bullets (kinematic argument for max speed at equilibrium + derivation of v_max = ωA). Part (d): 2 marks, 2 bullets (air resistance damping + distinct second mechanism). No divergences exceed 1.
topic pass

Part (c) explicitly requires kinematic reasoning (a = −ω²x, integration, or phase analysis) to derive v_max = ωA, firmly anchoring A.1 Kinematics. Parts (a), (b), and (d) thoroughly test C.1 SHM (defining conditions, ω, spring constant, damping effects). Both topics are genuinely integrated, not superficially mentioned.

Planner output (stage 1)

Archetype: experimental setup
Scenario: A student investigates the motion of a mass suspended from a spring by recording its vertical displacement over time. The mass is pulled downward from equilibrium and released, oscillating freely. The student measures the time for ten complete oscillations and the maximum displacement from equilibrium. The investigation aims to verify simple harmonic motion and determine the spring constant, but the student assumes air resistance is negligible and that the amplitude remains constant throughout the motion.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State the defining condition for simple harmonic motion in terms of force and displacement
b(i)	2	AO2	Calculate the angular frequency if the period of oscillation is measured as 2.0 s
b(ii)	2	AO2	Determine the spring constant of the spring if the suspended mass is 0.50 kg
c	2	AO3	Show that the maximum velocity occurs at the equilibrium position and derive an expression for it in terms of amplitude and angular frequency
d	2	AO3	Explain how the presence of air resistance and friction in the spring would affect the motion, and evaluate whether the student's assumption of constant amplitude is valid over an extended observation period

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