C.1 Simple harmonic motion

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Practice exam-style IB Physics questions for Simple harmonic motion, aligned with the syllabus and grouped by topic.

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Question 1

SL • Paper 1A

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SL • Paper 1A

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An object displaced from equilibrium undergoes simple harmonic motion. The necessary relationship between its acceleration $a$ and displacement $x$ is

A.

$a$ is proportional to $x$ and directed away from equilibrium.

B.

$a$ is proportional to $x$ and directed towards equilibrium.

C.

$a$ is proportional to $x^2$ and directed towards equilibrium.

D.

$a$ is constant and directed towards equilibrium.

Question 2

SL • Paper 1A

Easy

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SL • Paper 1A

Easy

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A horizontal mass-spring oscillator is released from rest at maximum displacement. After one quarter of a period, the mass is at equilibrium. The energy of the oscillator at this instant is

A.

all kinetic and zero elastic potential energy.

B.

zero kinetic and zero elastic potential energy.

C.

half kinetic and half elastic potential energy.

D.

zero kinetic and maximum elastic potential energy.

Question 3

SL • Paper 1A

Easy

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SL • Paper 1A

Easy

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A mass on a spring has period $T$ . The mass is replaced by a mass twice as large, using the same spring. The new period is

A.

$0.71T$

B.

$2.00T$

C.

$1.41T$

D.

$0.50T$

Question 4

SL • Paper 1A

Easy

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SL • Paper 1A

Easy

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An oscillator satisfies $a=-\omega^2x$ . The graph of acceleration $a$ against displacement $x$ is

A.

A simple graph with horizontal axis labeled x and vertical axis labeled a. A straight line passes through the origin with a negative gradient. The axes have arrowheads and no grid.

B.

A simple graph with horizontal axis labeled x and vertical axis labeled a. A straight line with negative gradient crosses the vertical axis above the origin rather than passing through the origin. The axes have arrowheads and no grid.

C.

A simple graph with horizontal axis labeled x and vertical axis labeled a. A smooth curve passes through the origin, decreasing for positive x but becoming progressively steeper away from the origin. The axes have arrowheads and no grid.

D.

A simple graph with horizontal axis labeled x and vertical axis labeled a. A straight line passes through the origin with a positive gradient. The axes have arrowheads and no grid.

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Question 5

SL • Paper 1A

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SL • Paper 1A

Easy

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A simple pendulum of length $l$ has period $T$ for small oscillations. The length is changed to $4l$ at the same location. The new period is

A.

$2T$

B.

$\frac{T}{4}$

C.

$4T$

D.

$\frac{T}{2}$

Question 6

SL • Paper 1A

Easy

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SL • Paper 1A

Easy

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An oscillator has angular frequency $12\ \text{rad s}^{-1}$ . Its period is

A.

$0.52\ \text{s}$

B.

$0.083\ \text{s}$

C.

$1.9\ \text{s}$

D.

$75\ \text{s}$

Question 7

HL • Paper 1A

Easy

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HL • Paper 1A

Easy

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Two oscillators have the same frequency. Oscillator $Q$ leads oscillator $P$ by a phase angle of $\pi/2$ . The phasor diagram representing this phase relationship is

A.

A phasor diagram on a plain background with a central origin and equal-length arrows labeled P and Q. The arrows rotate anticlockwise, indicated by a small curved arrow. Phasor P is placed one quarter-turn anticlockwise ahead of phasor Q. No numerical angle labels are shown.

B.

C.

A phasor diagram on a plain background with a central origin and equal-length arrows labeled P and Q overlapping in the same direction. A small curved arrow indicates anticlockwise rotation. No numerical angle labels are shown.

D.

A phasor diagram on a plain background with a central origin and equal-length arrows labeled P and Q pointing in exactly opposite directions. A small curved arrow indicates anticlockwise rotation. No numerical angle labels are shown.

Question 8

HL • Paper 1A

Easy

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HL • Paper 1A

Easy

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For a particle described by $x=x_0\sin(\omega t+\phi)$ , the particle has maximum positive velocity when the phase angle $\omega t+\phi$ is

A.

$\pi$

B.

$\frac{\pi}{2}$

C.

$0$

D.

$\frac{3\pi}{2}$

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Question 9

SL • Paper 2

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SL • Paper 2

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A cart of mass $0.250\ \text{kg}$ is attached to a horizontal spring of spring constant $16.0\ \text{N m}^{-1}$ . Friction is negligible.

Determine the period of oscillation of the cart.

[2]

State the effect on the period if the amplitude is doubled while the spring remains within its elastic limit.

[1]

Question 10

SL • Paper 2

Easy

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SL • Paper 2

Easy

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A pendulum oscillates in air. The maximum displacement reached by the bob decreases after each cycle.

Define damping.

[1]

Explain why the motion is no longer ideal simple harmonic motion.

[2]

Question 11

HL • Paper 1A

Medium

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HL • Paper 1A

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The displacement of a particle in SHM is $x=x_0\sin(\omega t+\phi)$ . At $t=0$ , the particle has displacement $+x_0/2$ and positive velocity. The phase angle $\phi$ is

A.

$\frac{7\pi}{6}$

B.

$\frac{11\pi}{6}$

C.

$\frac{5\pi}{6}$

D.

$\frac{\pi}{6}$

Question 12

HL • Paper 1A

Medium

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HL • Paper 1A

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A particle undergoing SHM has amplitude $0.10\ \text{m}$ and angular frequency $8.0\ \text{rad s}^{-1}$ . At an instant when its displacement is $+0.060\ \text{m}$ , it is moving in the negative direction. Its velocity is

A.

$+0.80\ \text{m s}^{-1}$

B.

$-0.48\ \text{m s}^{-1}$

C.

$-0.64\ \text{m s}^{-1}$

D.

$+0.64\ \text{m s}^{-1}$

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Question 13

HL • Paper 1A

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HL • Paper 1A

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An oscillator of mass $0.50\ \text{kg}$ has angular frequency $6.0\ \text{rad s}^{-1}$ and amplitude $0.20\ \text{m}$ . The kinetic energy when the displacement is $0.10\ \text{m}$ is

A.

$0.36\ \text{J}$

B.

$0.09\ \text{J}$

C.

$0.18\ \text{J}$

D.

$0.27\ \text{J}$

Question 14

HL • Paper 1A

Medium

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HL • Paper 1A

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In an ideal simple harmonic oscillator of amplitude $x_0$ , the kinetic energy and potential energy are equal when the magnitude of the displacement is

A.

$\frac{x_0}{2}$

B.

$\sqrt{2}x_0$

C.

$x_0$

D.

$\frac{x_0}{\sqrt{2}}$

Question 15

SL • Paper 2

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SL • Paper 2

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A motion sensor is used to record the acceleration $a$ and displacement $x$ of a trolley attached to a spring. The graph shows the variation of $a$ with $x$ .

Acceleration varies linearly with displacement for a trolley in SHM.

Outline how the graph shows that the trolley undergoes simple harmonic motion.

[2]

Use the graph to determine the angular frequency of the oscillation.

[2]

Question 16

SL • Paper 2

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SL • Paper 2

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A simple pendulum has length $0.64\ \text{m}$ and is released from a small angular displacement. Use $g=9.81\ \text{m s}^{-2}$ .

A simple pendulum diagram showing a small bob suspended by a light string from a fixed pivot. The string length is labelled l from pivot to the centre of the bob. A dashed vertical equilibrium line is shown, with a small angular displacement of the string from the vertical labelled as theta. The arc of motion is indicated but no numerical answer is shown.

Calculate the period of the pendulum.

[2]

Explain why the period of an ideal simple pendulum is independent of the mass of the bob.

[2]

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Question 17

SL • Paper 2

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SL • Paper 2

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The graph shows the displacement of a small object undergoing simple harmonic motion.

Displacement of an oscillator over two cycles.

Determine the amplitude and period of the oscillation from the graph.

[2]

Describe the energy changes as the object moves from maximum positive displacement to the equilibrium position.

[2]

Question 18

SL • Paper 2

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SL • Paper 2

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A point moves uniformly in a circle of radius $0.050\ \text{m}$ with angular speed $12\ \text{s}^{-1}$ . The horizontal projection of the point is observed on a screen.

A diagram of a point moving uniformly around a circle. The circle radius is labelled, the centre is marked, and a horizontal projection from the point to a straight line or screen is shown. The angular speed label is shown near the circular path. The diagram should illustrate the projection idea without showing displacement-time equations or calculated values.

State the amplitude of the motion of the projection.

[1]

Determine the period of the motion of the projection.

[2]

Outline why the projection undergoes simple harmonic motion.

[1]

Question 19

HL • Paper 2

Medium

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HL • Paper 2

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Two oscillators A and B have the same period, $0.80\ \text{s}$ . Oscillator B reaches each maximum displacement $0.10\ \text{s}$ after oscillator A.

Displacement-time curves for two oscillators.

Determine the phase difference between the two oscillators, stating which oscillator lags.

[2]

State the phase difference one cycle later.

[1]

Question 20

SL • Paper 1B

Medium

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SL • Paper 1B

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A motion sensor records the acceleration $a$ and displacement $x$ of a trolley attached to a horizontal spring. The graph shows $a$ plotted against $x$ .

Measured acceleration against displacement for a trolley on a spring.

State two features of the graph that indicate the motion is simple harmonic.

[2]

Determine the angular frequency and the period of the trolley.

[2]

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Question 21

SL • Paper 1B

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SL • Paper 1B

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The displacement of a particle undergoing simple harmonic motion is recorded as a function of time.

Displacement of a particle undergoing SHM as a function of time.

Determine the amplitude of the oscillation.

[1]

Determine the frequency of the oscillation.

[2]

Identify where in the motion the magnitude of the acceleration is greatest.

[1]

Question 22

SL • Paper 1B

Medium

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SL • Paper 1B

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A horizontal mass-spring oscillator moves between positions $P$ and $T$ . Position $R$ is the equilibrium position. The bar chart shows the kinetic energy, elastic potential energy and total mechanical energy at different positions in one oscillation.

Bar chart of kinetic, elastic potential and total mechanical energy at five positions.

Identify the position at which the speed of the mass is greatest.

[1]

Describe the energy changes as the mass moves from position $P$ to position $R$ .

[2]

The experiment is repeated with significant air resistance. Explain the expected change to the total mechanical energy of the oscillator.

[1]

Question 23

HL • Paper 2

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HL • Paper 2

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A particle undergoing simple harmonic motion has amplitude $0.080\ \text{m}$ and angular frequency $6.0\ \text{s}^{-1}$ . Its displacement is described by $x=x_0\sin(\omega t+\phi)$ . At $t=0$ , $x=+0.040\ \text{m}$ and the particle is moving in the positive direction.

Determine the phase angle $\phi$ at $t=0$ .

[2]

Determine the velocity of the particle at $t=0$ .

[2]

Question 24

HL • Paper 2

Medium

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HL • Paper 2

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An oscillator has amplitude $0.120\ \text{m}$ and angular frequency $4.5\ \text{s}^{-1}$ . At one instant its displacement is $-0.050\ \text{m}$ and it is moving towards the equilibrium position.

Determine the velocity of the oscillator at this instant.

[3]

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Question 25

HL • Paper 2

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HL • Paper 2

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A particle of mass $0.30\ \text{kg}$ undergoes simple harmonic motion with amplitude $0.050\ \text{m}$ and angular frequency $8.0\ \text{s}^{-1}$ .

Determine the total mechanical energy of the particle.

[2]

Determine the kinetic energy when the displacement is $0.030\ \text{m}$ from equilibrium.

[2]

Question 26

HL • Paper 2

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HL • Paper 2

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The displacement of a particle undergoing simple harmonic motion is given by

$x=0.020\sin(15t+1.2)$

where $x$ is in metres and $t$ is in seconds.

Determine the maximum speed of the particle.

[1]

Determine the acceleration of the particle at $t=0$ .

[2]

State the phase difference between displacement and acceleration.

[1]

Question 27

SL • Paper 1B

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SL • Paper 1B

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A student investigates a vertical mass-spring oscillator. The period $T$ is measured for different attached masses $m$ . The graph shows $T^2$ plotted against $m$ .

Measured values of period squared plotted against attached mass for a vertical mass-spring oscillator.

Explain why plotting $T^2$ against $m$ should produce a straight line.

[1]

Use the graph to determine the spring constant.

[2]

Suggest why changing the amplitude would not change the measured period in the ideal model.

[1]

Question 28

SL • Paper 1B

Medium

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SL • Paper 1B

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A simple pendulum is tested using small angular displacements. The graph shows $T^2$ plotted against the length $l$ of the pendulum. Measurements are repeated with two bobs of different mass.

T² against pendulum length for two bob masses.

Use the graph to determine the gravitational field strength $g$ .

[2]

State the conclusion about the effect of bob mass on the period.

[1]

Explain why the same pendulum would not be expected to follow the graph accurately at large angular amplitudes.

[1]

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Question 29

HL • Paper 1B

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HL • Paper 1B

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Two oscillators, $A$ and $B$ , undergo simple harmonic motion with the same amplitude and frequency. Their displacement-time graphs are shown.

Two SHM displacement-time graphs for oscillators A and B.

Determine the period of the oscillations.

[1]

Determine the phase difference between the oscillations and state which oscillator leads.

[2]

State the phase difference between the velocities of the two oscillators.

[1]

Question 30

HL • Paper 2

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HL • Paper 2

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A particle undergoing simple harmonic motion has amplitude $0.180\ \text{m}$ . At one displacement, its kinetic energy is equal to its potential energy.

Determine the magnitude of the displacement at which the two energies are equal.

[2]

Explain why this displacement is not half the amplitude.

[1]

Question 31

SL • Paper 1B

Hard

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SL • Paper 1B

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A simplified model treats a bond in a greenhouse gas molecule as an oscillator. The graph shows the absorption of infrared radiation by the molecule as a function of radiation frequency.

Relative infrared absorption of a molecular oscillator as a function of frequency, with one resonance peak.

Determine the resonant frequency of the molecular oscillator.

[1]

Calculate the period and angular frequency corresponding to this resonance.

[2]

Explain how this oscillator model is related to the enhanced greenhouse effect.

[2]

Question 32

HL • Paper 1B

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HL • Paper 1B

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The displacement of an oscillator is described by $x=x_0\sin(\omega t+\phi)$ . The graph shows its displacement near $t=0$ .

The amplitude is $0.080\ \text{m}$ and $\omega=10\ \text{s}^{-1}$ . At $t=0$ , $x=0.040\ \text{m}$ and the oscillator is moving in the negative direction. Determine the phase angle $\phi$ .

[2]

Calculate the velocity at $t=0$ .

[1]

Calculate the acceleration at $t=0$ .

[1]

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Question 33

HL • Paper 1B

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HL • Paper 1B

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For an oscillator undergoing simple harmonic motion, the square of its speed $v^2$ is plotted against the square of its displacement $x^2$ .

Square speed plotted against square displacement for an SHM oscillator.

Use the graph to determine the angular frequency of the oscillator.

[2]

Determine the maximum speed of the oscillator.

[1]

Determine the amplitude of the oscillator.

[1]

Explain the significance of the two possible signs in $v=\pm\omega\sqrt{x_0^2-x^2}$ .

[1]

Question 34

HL • Paper 1B

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HL • Paper 1B

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A phase-space graph for a particle undergoing simple harmonic motion is shown. The graph plots velocity $v$ against displacement $x$ . The mass of the particle is $0.20\ \text{kg}$ .

Determine the amplitude and maximum speed of the particle.

[2]

Determine the angular frequency and period of the motion.

[2]

Calculate the total energy of the oscillator.

[1]

Question 35

SL • Paper 2

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SL • Paper 2

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A trolley is attached to a horizontal spring and moves on a low-friction track. A motion sensor records the acceleration $a$ and displacement $x$ of the trolley from its equilibrium position. The graph shows the variation of $a$ with $x$ .

Experimental acceleration–displacement data for a trolley on a spring.

Use the graph to evaluate whether the motion is simple harmonic.

I.

State the graphical feature that shows the acceleration is proportional to displacement.

[1]

II.

Explain the significance of the sign of the gradient.

[2]

III.

Conclude whether the trolley executes simple harmonic motion.

[1]

The magnitude of the gradient of the graph is $18\ \text{s}^{-2}$ . Calculate the period of the oscillation.

[2]

Suggest why the points at the largest displacements deviate slightly from the straight line.

[1]

Question 36

SL • Paper 2

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SL • Paper 2

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A simple pendulum of length $0.72\ \text{m}$ is displaced through a small angle and released from rest. Air resistance is negligible.

A simple pendulum diagram with a fixed pivot, a light string of length labelled $l$, and a small bob displaced to one side by a small angular displacement labelled $\theta$. The equilibrium vertical line is shown as a dashed line. The tangential component of weight acts towards the equilibrium position, but no final formula is written on the diagram.

Explain why the pendulum can be modelled as a simple harmonic oscillator for small angular displacements.

I.

Identify the force component that provides the restoring effect.

[1]

II.

Use the small-angle approximation to show that the acceleration has the form required for simple harmonic motion.

[3]

Calculate the time taken for the bob to move from one extreme position to the other extreme position.

[2]

Discuss the effect on the period if the bob is replaced by another bob of twice the mass but the same size.

[1]

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Question 37

SL • Paper 2

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SL • Paper 2

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A horizontal spring-block oscillator is pulled to one side and released from rest. The surface is first very smooth and then the experiment is repeated on a rougher surface.

A horizontal spring attached to a fixed wall and to a block on a track. Three positions are indicated: left extreme, equilibrium, and right extreme. The equilibrium position is marked by a dashed vertical line through the block's central position. The amplitude is indicated from equilibrium to an extreme. No energy values are shown.

Discuss the energy changes during one complete oscillation on the very smooth surface.

I.

State where the kinetic energy is maximum and where it is zero.

[2]

II.

Describe the variation of elastic potential energy and total mechanical energy.

[2]

On the rougher surface the amplitude decreases after each oscillation. Explain why the motion is no longer ideal simple harmonic motion.

[2]

Sketch the variation of kinetic energy with time for one complete oscillation on the very smooth surface, starting from the instant of release at an extreme position.

[2]

Question 38

SL • Paper 2

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SL • Paper 2

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The shadow of a point moving with uniform circular motion is projected onto a straight horizontal screen. The projected shadow oscillates between two end points.

A point moving uniformly around a circle with radius labelled $x_0$. A lamp projects the point onto a horizontal screen so that the shadow moves along a straight line. The centre of the circle projects to the equilibrium point on the screen. A rotating radius is shown, but no equations are printed.

Compare the circular motion of the point with the simple harmonic motion of its shadow.

I.

State the relationship between one revolution of the point and one oscillation of the shadow.

[1]

II.

Explain how the radius and angular speed of the circular motion relate to the amplitude and angular frequency of the shadow.

[2]

The point completes $12$ revolutions in $8.0\ \text{s}$ . Calculate the angular frequency of the shadow.

[2]

Explain why the shadow has zero acceleration at the centre of its motion.

[1]

Question 39

HL • Paper 1B

Hard

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HL • Paper 1B

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A mass of $0.40\ \text{kg}$ undergoes simple harmonic motion. The graph shows the kinetic energy $E_k$ and potential energy $E_p$ as functions of displacement $x$ .

Kinetic and potential energy versus displacement for an SHM oscillator.

Determine the total energy of the oscillator.

[1]

Use the potential-energy graph to determine the angular frequency of the oscillator.

[2]

Calculate the speed of the mass when $x=0.060\ \text{m}$ .

[2]

Question 40

HL • Paper 1B

Hard

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HL • Paper 1B

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A computer model represents an oscillator by the equation $x=x_0\sin\theta$ , where $\theta=\omega t+\phi$ is the phase angle. At one instant the model displays the state of the oscillator.

Quantity / unit	Value
Amplitude, $x_0$ / m	0.090
Angular frequency, $\omega$ / $\text{s}^{-1}$	12
Displacement, $x$ / m	0.054
Velocity direction	positive

The model gives $x_0=0.090\ \text{m}$ , $\omega=12\ \text{s}^{-1}$ and $x=0.054\ \text{m}$ , with the velocity positive. Determine the phase angle $\theta$ at this instant.

[2]

Calculate the velocity at this instant.

[2]

Determine the fraction of the total energy that is potential energy at this instant.

[1]

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Question 41

SL • Paper 2

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SL • Paper 2

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A student investigates vertical oscillations of different masses attached to the same spring. The student measures the time for ten oscillations for each mass and plots $T^2$ against mass $m$ .

Measured and best-fit relation between squared period and mass for a spring-mass system.

Use the mass-spring model to explain the expected relationship between $T^2$ and $m$ .

I.

Derive an expression for $T^2$ in terms of $m$ and the spring constant $k$ .

[2]

II.

State what the gradient of the graph represents.

[1]

The gradient of the best-fit line is $2.8\ \text{s}^2\ \text{kg}^{-1}$ . Calculate the spring constant.

[2]

Evaluate one reason why the graph may have a small positive vertical intercept.

[2]

Question 42

SL • Paper 2

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SL • Paper 2

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The bonds in a carbon dioxide molecule can vibrate about stable separations. A simplified model treats one vibrational mode as a small oscillator that can absorb infrared radiation.

A simplified molecular oscillator diagram showing a central carbon atom and two oxygen atoms connected by bonds represented as springs. The atoms are labelled C and O. Arrows show the atoms vibrating along the molecular axis about equilibrium separations. An incoming infrared wave arrow is shown approaching the molecule, but no energy-level values are included.

Discuss why a vibrating molecular bond can be approximated as a simple harmonic oscillator for small displacements.

I.

State the restoring condition that must be satisfied.

[2]

II.

Explain why the approximation is expected to fail for large molecular displacements.

[1]

Explain how absorption and re-radiation by many such molecular oscillators can contribute to the enhanced greenhouse effect.

[2]

Suggest one limitation of modelling the molecule as an ideal undamped oscillator.

[1]

Question 43

HL • Paper 2

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HL • Paper 2

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An oscillator has displacement described by $x=x_0\sin(\omega t+\phi)$ . A data logger records that at $t=0$ the oscillator is at $x=+0.060\ \text{m}$ and moving in the negative direction. The amplitude is $0.10\ \text{m}$ and the period is $0.80\ \text{s}$ .

Evaluate the phase information for this oscillator.

I.

Calculate the angular frequency.

[1]

II.

Determine the possible values of the phase angle $\phi$ at $t=0$ from the displacement information.

[2]

III.

Use the direction of motion to choose the correct value of $\phi$ .

[2]

Calculate the velocity of the oscillator at $t=0$ .

[2]

Explain why radians rather than degrees must be used in the argument of the sine function for the standard equations of simple harmonic motion.

[1]

Question 44

HL • Paper 2

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HL • Paper 2

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A particle of mass $0.20\ \text{kg}$ undergoes simple harmonic motion with amplitude $0.080\ \text{m}$ and angular frequency $12\ \text{s}^{-1}$ .

Use the kinematic equations to discuss the speed of the particle at different displacements.

I.

Calculate the maximum speed.

[1]

II.

Calculate the speed when the displacement is $0.050\ \text{m}$ .

[2]

III.

Explain why two velocities are possible at this displacement.

[1]

Calculate the total mechanical energy of the oscillator.

[2]

Determine the kinetic energy of the particle when $x=0.050\ \text{m}$ and comment on its relation to the speed found in (a)(ii).

[2]

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Question 45

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HL • Paper 2

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A horizontal oscillator is modelled by $x=x_0\sin\omega t$ . The total energy is $0.48\ \text{J}$ and the amplitude is $0.120\ \text{m}$ .

Evaluate how the energy is shared between kinetic and potential forms.

I.

Determine the displacement at which the kinetic energy equals the potential energy.

[3]

II.

Calculate the potential energy at $x=0.060\ \text{m}$ .

[1]

Sketch, on the same axes, the variation of kinetic energy and potential energy with displacement from $-x_0$ to $+x_0$ .

[2]

Explain why the points where the two curves cross are not at half the amplitude.

[1]

Question 46

HL • Paper 2

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HL • Paper 2

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A particle undergoes simple harmonic motion according to $x=0.035\sin(20t+1.10)$ , where $x$ is in metres and $t$ is in seconds.

Use the equation to explain the motion of the particle.

I.

State the amplitude, angular frequency and phase angle at $t=0$ .

[3]

II.

Determine the acceleration of the particle when $x=0.020\ \text{m}$ .

[1]

Calculate the first time after $t=0$ at which the particle passes through equilibrium moving in the negative direction.

[2]

Explain why the velocity is a quarter cycle out of phase with the displacement.

[1]

Question 47

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two identical oscillators, P and Q, have the same amplitude and frequency. Their displacement-time graphs are sinusoidal. Oscillator Q reaches each maximum displacement $0.12\ \text{s}$ after oscillator P. The period of both oscillators is $0.80\ \text{s}$ .

Displacement-time curves for oscillators P and Q.

Compare the phase of the two oscillators.

I.

Calculate the phase difference between P and Q in radians.

[2]

II.

State which oscillator leads and explain your answer.

[2]

At one instant P is at positive maximum displacement. Determine the displacement and direction of motion of Q at that instant, as fractions of the amplitude.

[2]

Explain why a constant phase difference requires the two oscillators to have the same frequency.

[1]

Question 48

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A small glider of mass $0.50\ \text{kg}$ on an air track is attached to a spring. The glider is released from rest at $x=+0.090\ \text{m}$ . The spring constant is $32\ \text{N m}^{-1}$ .

A glider on a horizontal air track attached to a spring fixed at one end. The equilibrium position is marked by a dashed vertical line, and the release position is shown to the right at displacement $+x_0$. The glider is shown at rest at the release position. No formulae are written on the diagram.

Evaluate the motion using the SHM model.

I.

Calculate the angular frequency of the glider.

[2]

II.

Write a displacement equation of the form $x=x_0\sin(\omega t+\phi)$ for the motion.

[2]

Calculate the speed of the glider when it first reaches $x=+0.030\ \text{m}$ .

[2]

Compare the kinetic energy and elastic potential energy at $x=+0.030\ \text{m}$ .

[2]

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