C.4 Standing waves and resonance
Practice exam-style IB Physics questions for Standing waves and resonance, aligned with the syllabus and grouped by topic.
Question 1
A standing wave is produced on a string by reflection at one end. What condition is required for the incident and reflected waves to form a stationary pattern?
A.
They travel in opposite directions with different frequencies.
B.
They travel in the same direction with different amplitudes.
C.
They travel in the same direction with the same frequency and wavelength.
D.
They travel in opposite directions with the same frequency and wavelength.
Question 2
Two adjacent nodes in a standing wave on a string are separated by 0.18 m. What is the wavelength of the travelling waves forming the standing wave?
A.
0.09 m
B.
0.72 m
C.
0.18 m
D.
0.36 m
Question 3
Two points lie between the same adjacent pair of nodes in a standing wave. Their distances from the nearest node are different. What is correct about their oscillations?
A.
They have different amplitudes and are in phase.
B.
They have the same amplitude and are in phase.
C.
They have different amplitudes and are in antiphase.
D.
They have the same amplitude and are in antiphase.
Question 4
A string of length (L) is fixed at both ends. What is the wavelength of the third harmonic?
A.
\(3L/2\)
B.
\(3L\)
C.
\(L/3\)
D.
\(2L/3\)
Question 5
A driven oscillator has a natural frequency of (12, ext{Hz}). The driving frequency is slowly increased from (2, ext{Hz}) to (20, ext{Hz}). At which frequency is the steady-state amplitude expected to be largest if damping is light?
A.
10 Hz
B.
2 Hz
C.
20 Hz
D.
12 Hz
Question 6
A wave on a string is reflected from a support.
State what is meant by a standing wave.
[1]
Outline why the positions of nodes remain fixed.
[1]
Question 7
A child is pushed periodically on a swing.
Define resonance.
[1]
Outline why the amplitude becomes large when the pushing frequency is close to the natural frequency.
[1]
Question 8
A pipe is closed at one end and open at the other. The speed of sound is (340, ext{m s}^{-1}) and the pipe length is (0.85, ext{m}). What is the first-harmonic frequency?
A.
400 Hz
B.
100 Hz
C.
200 Hz
D.
300 Hz
Question 9
A resonance curve is plotted for a forced oscillator. The damping is increased. What happens to the maximum amplitude and the frequency at which it occurs?
A.
The maximum amplitude increases and the resonant frequency shifts lower.
B.
The maximum amplitude increases and the resonant frequency shifts higher.
C.
The maximum amplitude decreases and the resonant frequency shifts lower.
D.
The maximum amplitude decreases and the resonant frequency shifts higher.
Question 10
A displaced oscillator returns to equilibrium in the shortest possible time without crossing the equilibrium position. What type of damping is this?
A.
Light damping
B.
Resonant damping
C.
Critical damping
D.
Heavy damping
Question 11
Two points in neighbouring loops of a standing wave have non-zero amplitudes. What is their phase difference?
A.
\(\pi/2\)
B.
0
C.
\(2\pi\)
D.
\(\pi\)
Question 12
A transverse pulse on a string reflects from a fixed end. What happens to the reflected pulse?
A.
It is not inverted and has a phase change of \(\pi\) rad.
B.
It is inverted and has a phase change of \(\pi\) rad.
C.
It is inverted and has no phase change.
D.
It is not inverted and has no reflected energy.
Question 13
A standing wave on a string has adjacent antinodes separated by (0.24, ext{m}). The frequency is (150, ext{Hz}).
Determine the wavelength of the travelling waves forming the standing wave.
[1]
Calculate the wave speed.
[1]
Question 14
Two labelled points, P and Q, are on a string showing a standing wave. P and Q are in adjacent loops.
State the phase difference between P and Q.
[1]
Explain why the amplitudes of P and Q may be different.
[1]
Question 15
A string of length (0.75, ext{m}) is fixed at both ends. The wave speed on the string is (120, ext{m s}^{-1}).
Calculate the first-harmonic frequency.
[1]
State the frequency of the fourth harmonic.
[1]
Question 16
A pipe is closed at one end and open at the other.
State the displacement condition at the closed end.
[1]
Explain why only odd harmonics occur in this pipe.
[1]
Question 17
A door-closing mechanism is adjusted to change its damping.
Distinguish between light damping and critical damping using the motion after release.
[1]
State one practical reason for using critical damping in a door closer.
[1]
Question 18
A student drives a string fixed at both ends and records the frequencies at which clear standing-wave patterns occur. The length and tension of the string are constant.
Identify the harmonic number corresponding to each recorded pattern.
[1]
Use the graph to estimate the first-harmonic frequency.
[1]
Determine the wave speed on the string using the first-harmonic frequency and the string length shown in the stimulus.
[1]
Question 19
The diagram shows the maximum displacement envelope of a standing wave on a string.
Measure or read the distance between adjacent nodes from the diagram.
[1]
Determine the wavelength.
[1]
State the phase difference between particles in adjacent loops.
[1]
Suggest why a node in the experiment may have a small non-zero observed amplitude.
[1]
Question 20
A motion sensor records the displacement of three identical oscillators after each is released from the same displacement. The damping is different for each oscillator.
Identify the curve showing light damping.
[1]
Identify the curve showing critical damping.
[1]
State one feature of the heavy damping curve.
[1]
Question 21
A string has one fixed end and one free end. Its first-harmonic frequency is (45, ext{Hz}). What is the frequency of the next allowed harmonic?
A.
90 Hz
B.
135 Hz
C.
180 Hz
D.
225 Hz
Question 22
A pipe open at both ends and a pipe closed at one end have the same length and contain the same gas. What is the ratio
[
\frac{ ext{first-harmonic frequency of open-open pipe}}{ ext{first-harmonic frequency of closed-open pipe}}?
]
A.
\(2\)
B.
\(1\)
C.
\(4\)
D.
\(1/2\)
Question 23
A pipe of length (0.60, ext{m}) is closed at both ends. The speed of sound in the gas is (300, ext{m s}^{-1}). What is the frequency of the second harmonic?
A.
250 Hz
B.
500 Hz
C.
1000 Hz
D.
125 Hz
Question 24
The third harmonic in a closed-open pipe has frequency (510, ext{Hz}). The speed of sound is unchanged. What is the first-harmonic frequency of the same pipe?
A.
170 Hz
B.
255 Hz
C.
85 Hz
D.
1530 Hz
Question 25
An infrared photon is strongly absorbed by a greenhouse gas molecule. What is the best resonance explanation for this absorption?
A.
The photon amplitude is larger than the molecular amplitude.
B.
The photon speed matches the speed of the molecule.
C.
The photon wavelength is longer than all molecular dimensions.
D.
The photon frequency matches an allowed molecular vibration or rotation frequency.
Question 26
A mass-spring oscillator has natural frequency (f_0). The mass is replaced by a mass four times as large, with the same spring. What is the new natural frequency?
A.
\(2f_0\)
B.
\(f_0/4\)
C.
\(4f_0\)
D.
\(f_0/2\)
Question 27
A forced oscillator is tested first with light damping and then with greater damping.
State the effect of greater damping on the maximum steady-state amplitude.
[1]
Explain, in terms of energy, why this effect occurs.
[1]
Question 28
A string of length (0.90, ext{m}) has one fixed end and one free end. The wave speed is (72, ext{m s}^{-1}).
Calculate the first-harmonic frequency.
[1]
Calculate the frequency of the next allowed harmonic.
[1]
Question 29
A student observes a standing wave on a string. The support at the far end is not perfectly rigid.
State the boundary condition at a perfectly fixed end.
[1]
Explain why the observed node at the support may not have exactly zero amplitude in the real experiment.
[1]
State one systematic error this may introduce when determining the wavelength from the pattern.
[1]
Question 30
A mass-spring oscillator has spring constant (18, ext{N m}^{-1}) and mass (0.50, ext{kg}).
Calculate its natural frequency.
[1]
State how the natural frequency changes if the spring constant is increased with the same mass.
[1]
Question 31
Two pipes have equal length and contain air at the same temperature. Pipe X is open at both ends. Pipe Y is closed at one end and open at the other.
Compare the first-harmonic wavelengths in X and Y.
[1]
Compare the first-harmonic frequencies in X and Y.
[1]
Question 32
A resonance curve of amplitude against driving frequency is obtained for a lightly damped oscillator.
State the phase relation between driver and oscillator near resonance.
[1]
Explain why the amplitude is finite at resonance.
[1]
Question 33
The displacement-time graphs for two damped oscillators are shown. Graph A crosses equilibrium several times with decreasing amplitude. Graph B approaches equilibrium without crossing, more slowly than the fastest possible return.
Identify the damping in graph A.
[1]
Identify the damping in graph B.
[1]
State how graph B differs from critical damping.
[1]
Question 34
A forced oscillator is driven at different frequencies. The steady-state amplitude is measured for two damping conditions.
State which curve represents greater damping.
[1]
Estimate the resonant frequency for the lightly damped oscillator.
[1]
Describe two effects of increasing damping shown by the data.
[1]
Explain why the amplitude does not increase without limit at resonance.
[1]
Question 35
A set of pendulums of different lengths is attached to a horizontal support. The support is driven by a driver pendulum. The maximum amplitude of each pendulum is recorded.
| Pendulum | Length / m | Maximum amplitude / cm |
|---|---|---|
| A | 0.30 | 1.8 |
| B | 0.45 | 3.5 |
| C | 0.56 | 8.2 |
| D | 0.64 | 18.6 |
| E | 0.72 | 9.0 |
| F | 0.90 | 3.1 |
Identify the pendulum most likely to have the same natural frequency as the driver.
[1]
Use the data to describe how response amplitude depends on pendulum length.
[1]
Explain why one pendulum has a much larger amplitude than the others.
[1]
Question 36
A standing wave is formed in a string with two free ends. A high-speed image shows the displacement envelope.
Identify the positions of displacement antinodes.
[1]
Determine the harmonic number of the mode shown.
[1]
Calculate the frequency of this mode using the wave speed given in the stimulus.
[1]
Question 37
Greenhouse gas molecules absorb some infrared radiation strongly but absorb other infrared frequencies weakly.
State what must be true of the radiation frequency for strong absorption.
[1]
Explain the absorption using the idea of resonance.
[1]
State one reason why this resonance model alone is not a complete climate model.
[1]
Question 38
A pipe closed at one end resonates at (170, ext{Hz}) and (510, ext{Hz}), with no resonance between these frequencies. The speed of sound is (340, ext{m s}^{-1}).
State the harmonic numbers for these two resonances.
[1]
Determine the length of the pipe.
[1]
Question 39
A student measures the resonant frequencies of a pipe that is open at one end and closed at the other. The pipe length is constant.
| Resonance | Pipe length / m | Frequency / Hz |
|---|---|---|
| R1 | 1.00 | 85.9 |
| R2 | 1.00 | 258.1 |
| R3 | 1.00 | 429.8 |
| R4 | 1.00 | 602.3 |
| R5 | 1.00 | 773.2 |
Use the data to decide whether the pipe supports even harmonics.
[1]
Determine the first-harmonic frequency from the data.
[1]
Calculate the speed of sound in the pipe.
[1]
Suggest one reason why the calculated speed may differ from the accepted value.
[1]
Question 40
The phase difference between a driver and a driven oscillator is measured as the driving frequency is varied. The amplitude is measured at the same frequencies.
Identify the frequency region where the oscillator is near resonance.
[1]
Describe how the phase difference changes as driving frequency increases.
[1]
Explain why the phase data support the amplitude data.
[1]
Question 41
An oscillator is driven at different frequencies. The table gives the steady-state amplitude for two different added masses attached to the same spring.
| Driving frequency / Hz | Amplitude, 0.20 kg / cm | Amplitude, 0.45 kg / cm |
|---|---|---|
| 2.0 | 1.2 | 2.8 |
| 2.5 | 1.5 | 5.6 |
| 3.0 | 2.0 | 8.9 |
| 3.5 | 3.0 | 5.4 |
| 4.0 | 5.7 | 3.0 |
| 4.5 | 9.1 | 2.1 |
| 5.0 | 5.8 | 1.6 |
| 5.5 | 3.2 | 1.3 |
Identify which mass has the lower natural frequency.
[1]
Use the data to estimate the resonant frequency for each mass.
[1]
Explain whether the data are consistent with (f_0=(1/2\pi)\sqrt{k/m}).
[1]
Question 42
A string is fixed at both ends and is driven by a vibration generator.
Explain how the boundary conditions determine the first harmonic.
[1]
The string length is (1.20, ext{m}) and the wave speed is (96, ext{m s}^{-1}). Calculate the frequencies of the first three harmonics and explain the phase relation between adjacent loops in the third harmonic.
[1]
Question 43
A singer can cause a glass to vibrate strongly by singing a note close to one of the glass's natural frequencies.
Define natural frequency and forced vibration.
[1]
Discuss how resonance can produce a large vibration amplitude while remaining consistent with conservation of energy. Include the role of damping.
[1]
Question 44
A vehicle suspension system is designed to reduce oscillations after the wheel passes over a bump.
Distinguish between light damping, critical damping and heavy damping using displacement-time behaviour.
[1]
Evaluate which damping regime is most suitable for a vehicle suspension and justify your answer.
[1]
Question 45
A rotating machine is mounted on a platform. The vibration amplitude of the platform is measured as the rotation frequency is increased. Data are collected before and after rubber dampers are fitted.
Identify the dangerous operating frequency range before dampers are fitted.
[1]
Describe the effect of the dampers on the resonance curve.
[1]
Suggest one additional engineering method to reduce the risk of destructive resonance.
[1]
Question 46
Two pipes have the same length and contain air at the same temperature. Pipe A is open at both ends. Pipe B is closed at one end and open at the other.
State the displacement boundary condition at an open end and at a closed end.
[1]
Compare and contrast the allowed harmonics and first-harmonic frequencies of the two pipes.
[1]
Question 47
A bridge has a natural vibration mode that may be excited by periodic forces from traffic or wind.
Outline how resonance could occur in the bridge.
[1]
Evaluate three engineering strategies for reducing the risk of destructive resonance. Your answer should refer to frequency response and energy dissipation.
[1]
Question 48
A musical instrument contains a pipe that is closed at one end and open at the other. The pipe length is (0.40, ext{m}), and the speed of sound is (320, ext{m s}^{-1}).
Explain the displacement node and antinode positions for the first harmonic.
[1]
Determine the first three allowed frequencies and explain why the missing harmonics are not produced.
[1]
Question 49
A travelling wave and a standing wave are observed on the same string.
State two conditions needed for the standing wave to form from travelling waves.
[1]
Compare and contrast the energy transfer, amplitude variation and phase relationships in the travelling wave and the standing wave.
[1]
Question 50
Infrared absorption by atmospheric molecules can be modelled using resonance.
Outline the resonance condition for a molecule interacting with infrared radiation.
[1]
Discuss the usefulness and limitations of this resonance model for explaining the greenhouse effect.
[1]
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