Generated ERQ

✓ passed C.1 Simple harmonic motion × A.1 Kinematics 10 marks SL 3 passes 102.91s $0.6115

## ERQ · 10 marks · Topics: C.1 Simple harmonic motion + A.1 Kinematics **Stem.** A student investigates the vertical oscillations of a 0.250 kg glider attached to a horizontal spring of spring constant k = 16.0 N m⁻¹, resting on a low-friction air track. The glider is pulled 0.0600 m to the right of the equilibrium position and released from rest at t = 0. A high-speed camera records the displacement x of the glider at 0.0500 s intervals. Three consecutive frames near t = 0.250 s give x(0.200 s) = +0.0600 m, x(0.250 s) = +0.0208 m, and x(0.300 s) = −0.0185 m. The student assumes the motion is simple harmonic with x(t) = A cos(ωt). ### Part (a) Define [2 marks] · AO1 · Topic: C.1 Define simple harmonic motion and state the condition on the restoring force required for it to occur. ### Part (b)(i) Calculate [2 marks] · AO2 · Topic: C.1 Calculate the period T of oscillation predicted by the mass–spring model. ### Part (b)(ii) Determine [3 marks] · AO2/AO3 · Topic: C.1 Using the value of ω from (b)(i), determine the instantaneous velocity of the glider at t = 0.250 s predicted by the SHM model x(t) = A cos(ωt). ### Part (c) Show that [3 marks] · AO3 · Topic: C.1 + A.1 Using the three tabulated displacement values and the kinematic central-difference approximation v(t) ≈ [x(t + Δt) − x(t − Δt)] / (2Δt), show that the experimentally estimated velocity at t = 0.250 s is approximately −0.785 m s⁻¹. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The student also tries to apply the constant-acceleration kinematic equation v² = u² + 2as to the motion between t = 0.200 s and t = 0.300 s. Suggest why this approach is invalid for this system, even over such a short interval. --- ## Mark Scheme ### Part (a) [2 marks] - Mark 1: Acceleration (or restoring force) is proportional to displacement from a fixed equilibrium position [ECF: no] - Mark 2: Acceleration (or restoring force) is directed opposite to the displacement / towards equilibrium [ECF: no] ### Part (b)(i) [2 marks] - Mark 1: Uses T = 2π√(m/k) [ECF: no] - Mark 2: T = 2π√(0.250/16.0) = 0.785 s (accept 0.79 s) [ECF: yes] ### Part (b)(ii) [3 marks] - Mark 1: Identifies v(t) = −Aω sin(ωt) with ω = 2π/T = 8.00 rad s⁻¹ [ECF: no] - Mark 2: Correct substitution: v = −(0.0600)(8.00) sin(8.00 × 0.250) = −0.480 × sin(2.00 rad) [ECF: yes] - Mark 3: v = −0.436 m s⁻¹ (accept −0.44 m s⁻¹; negative sign or "directed towards equilibrium" required) [ECF: yes] ### Part (c) [3 marks] - Mark 1: Applies central-difference formula with correct indexing: v(0.250) ≈ [x(0.300) − x(0.200)] / (2 × 0.0500) [ECF: no] - Mark 2: Numerator = (−0.0185 − 0.0600) = −0.0785 m; denominator = 0.100 s [ECF: yes] - Mark 3: v ≈ −0.785 m s⁻¹ (negative sign indicating motion towards/through equilibrium in the −x direction) [ECF: yes] ### Part (d) [2 marks] - Mark 1: The acceleration during SHM is not constant — it varies continuously as a = −ω²x (changes with position) [ECF: no] - Mark 2: The kinematic equation v² = u² + 2as is derived assuming uniform acceleration, so it cannot be applied to any interval of SHM regardless of duration / would only become approximately valid in the limit Δt → 0 [ECF: no] ### Marker notes - (b)(ii): Accept calculator in radian mode only; degree mode gives ~−0.0167 m s⁻¹ and earns Mark 1 only. - (c): Sign must be explicit for Mark 3. Magnitude 0.785 m s⁻¹ alone scores 2/3. - (c) vs (b)(ii) comparison is not required, but the discrepancy (−0.785 vs −0.436 m s⁻¹) signals that either ω, the phase, or the SHM idealisation needs scrutiny — useful framing for (d). - (d) discriminator: accept any of — (i) a is not constant in SHM; (ii) suvat assumes uniform a but here a ∝ −x; (iii) over [0.200, 0.300] s the glider passes through equilibrium where a changes sign, so no single average a satisfies suvat. Do NOT accept "friction" or "air resistance" alone (track is low-friction by stem).

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 with Define (command word stated explicitly); parts (b)(i)–(b)(ii) and (c) are AO2/AO3 with Calculate/Determine/Show that; part (d) is AO3 with Suggest. Complexity and demand escalate appropriately from recall through calculation to analysis and evaluation of assumptions.
cross_theme fail

Part (c) uses only C.1 (SHM kinematics: x(t) = A cos(ωt) and average velocity calculation). A.1 (Kinematics) is not genuinely integrated into the solution—it appears only as a foil in Part (d), where the student is asked why constant-acceleration kinematics FAILS. A.1 is name-checked but not actually used to solve the question; it serves only as a discriminator about what doesn't work, not as a complementary physics concept needed to answer (a)–(c).

FIX: Restructure Part (c) to require students to use both SHM equations AND kinematic analysis. For example: 'Using the displacement and time data, (i) verify the SHM model x(t) = A cos(ωt) is consistent; (ii) calculate the instantaneous velocity at t = 0.0500 s using v = dx/dt from the SHM equation; (iii) independently confirm this velocity using kinematic methods (e.g. numerical differentiation or energy conservation), then explain why constant-acceleration SUVAT would give a different answer and why.' This forces genuine integration of both topics into the core solution, not relegation of A.1 to a 'what fails' discussion.
discriminator pass

Part (d) explicitly asks students to critique the kinematic model (constant-acceleration equation v² = u² + 2as) by identifying why it fails in SHM and defining the narrow conditions under which it could approximate. This is authentic model critique with clear AO3 discriminator intent.
rule_of_one fail

Part (c) has 3 marks but the mark scheme lists 3 award criteria that do not cleanly separate into distinct operations. Mark 2 is malformed: it offers two contradictory routes ("Computes... OR student must reconcile...") without clear demarcation of which single operation earns the mark. This violates the strict 1-mark-per-operation contract. The "Show that" instruction compounds this by mixing verification with calculation, making it ambiguous which operation is being assessed.

FIX: Restructure Part (c) as two separate parts, or clarify Mark 2 as exactly one operation: either (i) 'Substitutes ωt into x = A cos(ωt) to predict x = 4.81 cm' OR (ii) 'Recognises discrepancy between prediction (4.81 cm) and measurement (3.71 cm) and justifies as within experimental uncertainty (~20%).' Then explicitly state which route earns the mark. Alternatively, if both routes must be accepted, allocate 4 marks total (one per operation: compute ωt, predict x, reconcile/justify, calculate average velocity).
topic pass

The ERQ authentically integrates both C.1 (SHM) and A.1 (Kinematics). Parts (a)–(c) directly test SHM concepts (definition, period, maximum speed, kinematic equation of motion). Part (d) is the critical discriminator: it explicitly requires understanding why constant-acceleration kinematics (A.1) fails in SHM because acceleration is not constant—this forces the student to contrast the two topics' domains of validity. The question does not merely mention both topics; it demands reasoning about their incompatibility in this context.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 with Define (state definition + relationship); parts (b)(i)–(c) are AO2/AO3 with Calculate/Determine (requiring substitution and numerical work); part (d) is AO3 with Suggest (requiring discrimination between two distinct physical mechanisms). The progression from foundational knowledge → calculation → explanation is sound.
cross_theme pass

The ERQ successfully blends C.1 (Simple Harmonic Motion) and A.1 (Kinematics) as genuine dual themes. Part (c) explicitly requires kinematic finite-difference analysis of raw data, and part (d) asks students to explain the physical discrepancy between the SHM model's instantaneous velocity and the kinematic average-velocity estimate—a comparison that demands understanding both frameworks. Neither topic is merely name-checked; both are operationally essential to solving the problem and understanding its resolution.
discriminator pass

Part (d) explicitly asks students to 'Suggest two distinct reasons for this discrepancy' between the kinematic estimate and the SHM model prediction. This is a clear model-critique task: students must identify and explain limitations of either the kinematic approximation method or the SHM model itself (measurement error, non-ideal spring behavior, damping, variable vs. constant acceleration). The mark scheme confirms acceptance of multiple valid critiques (coarse finite-difference approximation, constant-acceleration assumption vs. variable SHM acceleration, sensor uncertainty, non-ideal spring behavior, damping). This fully satisfies the dimension's requirement for a final part that critiques the model rather than merely performing another calculation.
rule_of_one fail

Part (c) is marked as [2 marks] but the mark scheme lists only 1 distinct operation (central-difference formula application); the second mark conflates the numerical answer with result presentation. Part (b)(ii) similarly bundles formula substitution and final evaluation into a single bullet, creating ambiguity about the intermediate step award. These divergences exceed the ±1 tolerance.

FIX: Part (b)(ii): Split into three distinct bullets: (1) correct formula v(t) = −Aω sin(ωt); (2) numerical substitution of Aω and ωt values; (3) final answer with sign and units. Part (c): Explicitly separate (1) identification and application of central-difference formula with correct indices; (2) arithmetic calculation of numerator (−0.0185 − 0.0600) and denominator (0.1000); (3) final velocity answer −0.785 m s⁻¹ with sign/direction. This ensures mark count and award-bullet count align at exactly 2 for each part.
topic pass

The ERQ genuinely requires both C.1 (SHM) and A.1 (Kinematics) concepts. Parts (a)–(b) establish the SHM model using ω, period, and velocity from the sinusoidal equation. Part (c) mandates kinematic estimation via finite differences from raw data, forcing students to apply the definition v = Δx/Δt. Part (d) explicitly requires comparison of the two methods, demanding understanding of why the constant-acceleration kinematic approximation diverges from variable-acceleration SHM. Neither topic is superficial; both are operationally essential to the solution.

Iteration 2 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: Part (a) is AO1 (Define), Part (b) is AO2 (Calculate/Determine), Part (c) is AO3 (Show that/Derive), and Part (d) is AO3 (Suggest/Explain). Command terms and cognitive demand increase appropriately through the progression.
cross_theme pass

The question genuinely integrates both C.1 (Simple Harmonic Motion) and A.1 (Kinematics). Part (c) uses the kinematic central-difference approximation formula [a core A.1 technique] to estimate velocity from displacement data, then part (d) explicitly contrasts kinematic equations (constant-acceleration suvat, A.1) with the variable-acceleration nature of SHM (C.1). Both concepts are operationally necessary to answer the question; A.1 is not mere name-checking or filler.
discriminator fail

Part (d) asks students to suggest why a kinematic equation is invalid, which is a critique of the constant-acceleration model's applicability. However, this is not a critique of the SHM model itself or its assumptions—it's a critique of applying a *different* model (constant acceleration kinematics) to SHM. The ERQ lacks a final part that genuinely critiques the SHM model used throughout (e.g., 'Discuss why the calculated velocity [−0.436 m s⁻¹] differs significantly from the experimental value [−0.785 m s⁻¹]' or 'Suggest a limitation of the model x(t) = A cos(ωt)').

FIX: Replace or augment part (d) with a question that explicitly asks students to critique the SHM model—for example: '(d) The velocity calculated from the SHM model in (b)(ii) differs significantly from the experimental estimate in (c). Suggest one reason why the assumption that the motion follows x(t) = A cos(ωt) may not fully explain this discrepancy.' This would directly evaluate whether students can recognize that the idealized model has limitations.
rule_of_one pass

All parts match their mark counts to the number of distinct award criteria. Part (a): 2 marks, 2 bullets. Part (b)(i): 2 marks, 2 bullets. Part (b)(ii): 3 marks, 3 bullets (formula/ω, substitution, answer-with-units-and-sign). Part (c): 3 marks, 3 bullets (formula application, numerator/denominator calculation, final answer-with-sign). Part (d): 2 marks, 2 bullets (two distinct physical reasons why constant-acceleration assumption breaks down). The scheme is internally consistent and adheres to the 3-operation Calculate structure throughout.
topic pass

Both C.1 (Simple harmonic motion) and A.1 (Kinematics) are genuinely integrated throughout the ERQ. C.1 is tested in parts (a), (b)(i), (b)(ii), and (c)–(d) via the SHM model and its predictions. A.1 is tested in part (c) through the central-difference kinematic approximation for velocity, and in part (d) through critical analysis of why constant-acceleration kinematic equations (v² = u² + 2as) fail in SHM. Neither topic is superficial; both are essential to solving and understanding the question.

Planner output (stage 1)

Archetype: experimental setup
Scenario: A student performs an experiment using a mass–spring system oscillating vertically on a frictionless surface. The mass is displaced from equilibrium and released, undergoing simple harmonic motion. High-speed video analysis allows measurement of displacement and velocity at various times during the motion. The student must analyze the kinematics of the motion using both energy and motion equations to verify SHM predictions and evaluate the validity of the constant-acceleration kinematic model.
Cross-pollination part: d
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1\|AO2	Define simple harmonic motion and state the condition for its occurrence
b(i)	2	AO2\|AO3	Calculate the period of oscillation given mass, spring constant, and amplitude
b(ii)	2	AO2\|AO3	Determine the maximum velocity and maximum acceleration using SHM relationships
c	2	AO2\|AO3	Show that the displacement–time relationship satisfies x = A cos(ωt + φ) using provided experimental data
d	2	AO3	Evaluate whether constant-acceleration kinematic equations (such as v² = u² + 2as) can be validly applied to segments of SHM motion, and explain the limitations

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