Sound pendulum

This activity allows students to understand the Doppler effect by analyzing the frequency variations of a sound emitted by a moving pendulum. It develops the ability to interpret a complex acoustic phenomenon.

The Doppler effect is most commonly encountered when a siren-equipped vehicle passes by, producing the characteristic drop in pitch. But what if the source of sound were swinging back and forth on a pendulum instead of passing by in a straight line? The result is a beautifully periodic version of the Doppler effect: as the pendulum swings toward the listener, the pitch rises; as it swings away, the pitch drops. This oscillating frequency shift produces a distinctive warbling sound that encodes information about the pendulum's speed at every moment of its swing. Since the pendulum speed varies sinusoidally (fastest at the bottom, zero at the turning points), the frequency shift also varies sinusoidally, creating a natural connection between the Doppler effect and harmonic oscillation. By analyzing the frequency pattern of a sound-emitting pendulum with FizziQ's spectral tools, students can simultaneously investigate wave physics and pendulum mechanics in a single elegant experiment.

Learning objectives:

The student studies the frequency variations of a sound emitted by a smartphone suspended in a pendulum movement or uses the 'Doppler Pendulum Effect' recording from the FizziQ library. By carefully analyzing the characteristics of the sound, the student determines the frequency emitted, the oscillation period and the maximum speed of the pendulum, then reflects on the asymmetry observed in the frequency curve.

Level:

High school

FizziQ

Author:

Duration (minutes) :

40

What students will do :

- Analyze the frequency variations of a sound emitted by a pendulum using spectral analysis
- Identify the Doppler-shifted frequencies during approach and recession phases
- Determine the pendulum's oscillation period and maximum speed from the frequency data
- Apply the Doppler formula to relate frequency shift to pendulum velocity
- Understand how periodic motion creates a characteristic pattern of frequency modulation

Scientific concepts:

- Doppler effect
- Frequency and period
- Pendulum movement
- Variable speed
- Spectral analysis

Sensors:

- Microphone (sound recording and frequency analysis)
- FizziQ spectrum analyzer (FFT)

What is required:

- Smartphone with the FizziQ application
- Wire or string for hanging a smartphone (option 1)
- Recording 'Doppler Pendulum Effect' from the sound library (option 2)
- FizziQ experience notebook

Experimental procedure:

Option A: Suspend a smartphone emitting a pure tone (e.g., 1000 Hz from FizziQ's synthesizer) on a string to create a sound pendulum. Option B: Use the 'Doppler Pendulum Effect' recording from the FizziQ sound library.
If using Option A, set the pendulum length to at least 1 meter and place a second smartphone (for recording) at the same height as the pendulum's lowest point, about 1.5 meters away.
Open FizziQ on the recording phone and select the Frequency Meter or Spectrum Analyzer.
Start recording. Release the pendulum from an angle of about 20-30° and let it swing.
Record for at least 15-20 seconds to capture multiple complete oscillations.
Examine the frequency versus time graph. You should see periodic oscillations: the frequency rises above the emitted frequency during approach and drops below it during recession.
Identify the maximum frequency (f_max, during fastest approach) and the minimum frequency (f_min, during fastest recession).
Calculate the emitted frequency: f₀ = (f_max + f_min) / 2.
Calculate the maximum Doppler shift: Δf = (f_max - f_min) / 2.
Use the Doppler formula to calculate the maximum pendulum speed: v_max = c × Δf / f₀, where c = 343 m/s.
Independently calculate v_max from the pendulum parameters: v_max = √(2gL(1 - cos θ)), where L is the length and θ is the release angle.
Compare the two values of v_max. They should agree within 10-20%.
Measure the oscillation period from the frequency graph (time between consecutive maxima) and compare with T = 2π√(L/g).

Expected results:

For a 1-meter pendulum released at 30° and emitting at 1000 Hz, the maximum speed at the bottom is approximately 1.7 m/s. The Doppler shift should be about ±5 Hz (f_max ≈ 1005 Hz, f_min ≈ 995 Hz). This small shift requires good frequency resolution to detect. The frequency should oscillate sinusoidally at twice the pendulum frequency (because the maximum speed occurs twice per full oscillation: once approaching, once receding). The oscillation period should be about 2.0 seconds, matching the theoretical prediction. Measurement precision depends on the spectrum analyzer's frequency resolution and the background noise level.

Scientific questions:

- Why does the perceived frequency increase when the pendulum swings toward the microphone?
- At what point in the swing is the Doppler shift zero? Why?
- Why does the frequency oscillate at twice the pendulum frequency?
- How would increasing the pendulum length affect the maximum frequency shift?
- What is the relationship between the Doppler shift and the instantaneous velocity of the pendulum?
- How is this experiment related to the Doppler effect used in medical ultrasound?

Scientific explanations:

The Doppler effect, discovered by Christian Doppler in 1842, is the apparent variation in frequency of a wave perceived by an observer when the source and the observer are in relative motion. For a sound source approaching the observer, the perceived frequency is higher than the emitted frequency; when it moves away, the perceived frequency is lower.

In the case of a sound pendulum, the source (smartphone) oscillates back and forth from a point of balance, creating a periodic movement of approach and distance. The apparent frequency f' is linked to the emitted frequency f by the relation: f' = f × (1 + v/c), where v is the component of the speed of the source in the direction of the observer (positive when approaching, negative when moving away) and c is the speed of sound (approximately 343 m/s at 20°C).

For a simple pendulum, the speed varies continuously: zero at the ends of the oscillation and maximum at the equilibrium point. This variation in speed results in a modulation of the perceived frequency, creating a characteristic "rise-fall" pattern of the frequency.

The asymmetry observed in the frequency curve is explained by two main factors: 1) The projection of the pendulum's speed towards the observer is not sinusoidal but depends on the position of the observer; 2) The relationship between speed and frequency is not linear but fractional, which accentuates the variations. Knowing the base frequency f and the maximum amplitude of the shift Δf, we can calculate the maximum speed of the pendulum: v_max = c × Δf/f.

If the transmitted frequency is 1000 Hz and the maximum variation observed is ±3 Hz, the maximum speed of the pendulum is approximately 1 m/s. This experiment illustrates a phenomenon also observed in many contexts: passing ambulance sirens, astronomical Doppler effect, and even medical imaging by Doppler ultrasound.

Extension activities:

Frequently asked questions:

Q: The frequency variations are too small to see clearly. How can I improve the signal?
R: Use a higher emitted frequency (2000-3000 Hz gives larger absolute shifts), increase the pendulum amplitude, and use a shorter distance between the pendulum and microphone. Also ensure the microphone is in the plane of the swing.

Q: The frequency graph shows random fluctuations rather than a smooth oscillation. Why?
R: Background noise can overwhelm the Doppler signal. Perform the experiment in a quiet room. Also ensure the emitted tone is loud enough to dominate the spectrum. The FizziQ library recording may give cleaner results than a live experiment.

Q: Why does the frequency modulation appear at twice the pendulum frequency?
R: In each complete oscillation, the pendulum passes the microphone moving in the same direction twice: once swinging left-to-right and once right-to-left. Each passage produces a maximum frequency (approach) and minimum frequency (recession), so the frequency oscillates twice per pendulum period.

Q: My calculated maximum speed from the Doppler formula does not match the pendulum calculation. Why?
R: Check that the microphone is in the plane of the swing. If the microphone is off-axis, it sees a reduced component of the velocity, giving a smaller apparent Doppler shift. Also verify the pendulum length and release angle measurements.