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Date	May 2022	Marks available	4	Reference code	22M.2.HL.TZ1.6
Level	Higher level	Paper	Paper 2	Time zone	1
Command term	Determine	Question number	6	Adapted from	N/A

Question

A mass–spring system oscillates horizontally on a frictionless surface. The mass has an acceleration $a$ when its displacement from its equilibrium position is $x$ .

The variation of $a$ with $x$ is modelled in two different ways, A and B, by the graphs shown.

Outline two reasons why both models predict that the motion is simple harmonic when $a$ is small.

[2]

Determine the time period of the system when $a$ is small.

[4]

Outline, without calculation, the change to the time period of the system for the model represented by graph B when $a$ is large.

[2]

The graph shows for model A the variation with $x$ of elastic potential energy E_p stored in the spring.

Describe the graph for model B.

[2]

Markscheme

For both models:
displacement is ∝ to acceleration/force «because graph is straight and through origin» ✓

displacement and acceleration / force in opposite directions «because gradient is negative»
OR
acceleration/«restoring» force is always directed to equilibrium ✓

attempted use of $ω^{2} = (-) \frac{a}{x}$ ✓

suitable read-offs leading to gradient of line = 28 « s^-2» ✓

$T = \frac{2 π}{ω}$ « $\frac{2 π}{\sqrt{28}}$ » ✓

$T = 1.2$ s ✓

time period increases ✓

because average ω «for whole cycle» is smaller

slope / acceleration / force at large x is smaller

area under graph B is smaller so average speed is smaller. ✓

same curve OR shape for small amplitudes «to about 0.05 m» ✓

for large amplitudes «outside of 0.05 m» E_p smaller for model B / values are lower than original / spread will be wider ✓ OWTTE

Accept answers drawn on graph – e.g.

Examiners report

This item was essentially encouraging candidates to connect concepts about simple harmonic motion to a physical situation described by a graph. The marks were awarded for discussing the physical motion (such as "the acceleration is in the opposite direction of the displacement") and not just for describing the graph itself (such as "the slope of the graph is negative"). Most candidates were successful in recognizing that the acceleration was proportional to displacement for the first marking point, but many simply described the graph for the second marking point.

This question was well done by many candidates. A common mistake was to select an incorrect gradient, but candidates who showed their work clearly still earned the majority of the marks.

Many candidates recognized that the time period would increase for B, and some were able to give a valid reason based on the difference between the motion of B and the motion of A. It should be noted that the prompt specified "without calculation", so candidates who simply attempted to calculate the time period of B did not receive marks.

Candidates were generally successful in describing one of the two aspects of the graph of B compared to A, but few were able to describe both. It should be noted that this is a two mark question, so candidates should have considered the fact that there are two distinct statements to be made about the graphs. Examiners did accept clearly drawn graphs as well for full marks.

Syllabus sections

Additional higher level (AHL) » Topic 9: Wave phenomena » 9.1 – Simple harmonic motion

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Additional higher level (AHL) » Topic 9: Wave phenomena

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Markscheme

Examiners report

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