E2.6 · Predict and test conditions for resonance

Ontario Grade 11 Physics

Waves and Sound

Ontario Grade 11 Physics — E2.6

What you will learn

• Describe resonance using a system’s natural frequency and the frequency of a repeated input.
• Predict when resonance is likely and describe the system’s response.
• Plan a fair test for resonance and distinguish predictions from measured results.

1. Prerequisite bridge: repeating motion

• Period is measured in seconds (s); frequency is measured in hertz (Hz).
• Amplitude is the maximum distance from equilibrium and is measured in metres (m) for a length.
• Frequency and amplitude are scalars. Displacement has a direction as well as a size.

2. Physical situation and model: matching frequencies

• Natural frequency describes the oscillating system.
• Driving frequency describes how often the repeated input acts.
• Resonance is expected when the two frequencies are equal or close.
• A large amplitude can show a strong response, but compare frequencies to predict resonance.

3. Predicting and testing resonance

• Identify or measure both frequencies before comparing them.
• Vary the driving frequency systematically and keep other test conditions steady.
• Measure amplitude as the response. Record observations separately from predictions.
• A largest measured response near the natural frequency supports the resonance prediction.

Worked example

1. Predicting a swing’s response

1. Define the system and direction
   The system is the swing and rider. Take forward displacement as positive and backward displacement as negative. Frequency is a scalar, so it has no direction. The unknown is whether the driving frequency is close to the swing’s natural frequency.
2. Find the natural frequency
   The given time is for 12 cycles. Divide the number of cycles by the total time. The result is in cycles per second, or hertz.
   $$f_{\text{natural}}=\frac{12\ \text{cycles}}{20.0\ \text{s}}=0.600\ \text{Hz}$$
3. Compare the frequencies
   The driving frequency is 0.60 Hz, which matches the calculated natural frequency to the stated precision. A strong response is likely. This is reasonable because the repeated pushes occur at the swing’s natural rate.
   $$f_{\text{driving}}=0.60\ \text{Hz}\approx f_{\text{natural}}=0.600\ \text{Hz}$$

Worked example

2. Choosing a driving rate

1. Define the system and unknown
   The system is the test oscillator. Define displacement to the right of equilibrium as positive. The unknown is which proposed driving frequency is closest to the natural frequency.
2. Calculate the natural frequency
   The period is the time for one cycle. Use the reciprocal relationship to convert the given period into frequency, keeping seconds in the calculation.
   $$f_{\text{natural}}=\frac{1}{0.80\ \text{s}}=1.25\ \text{Hz}$$
3. Select the closest match
   The 1.25 Hz choice equals the calculated natural frequency. It is the best first choice for testing resonance. This predicts where to look for the greatest amplitude; it does not claim that a test has already been performed.
   $$f_{\text{driving}}=1.25\ \text{Hz}$$

Worked example

3. Interpreting a proposed frequency sweep

1. Define the system and direction
   The system is the oscillator being tested. Define displacement to the right of equilibrium as positive. Frequency and amplitude are scalars; the direction convention applies to displacement. The unknown is which trial should produce the strongest response.
2. Compare the frequencies
   Resonance is expected at the driving frequency closest to the natural frequency. Of the three proposed rates, 2.0 Hz matches the stated natural frequency.
   $$2.0\ \text{Hz}=f_{\text{natural}}$$
3. State a testable prediction
   Predict that the measured amplitude will be greatest in the 2.0 Hz trial if the force and other conditions are kept comparable. Measure the maximum displacement from equilibrium for each frequency. These are predicted results until the trials are actually performed.
   $$A_{2.0\ \text{Hz}}>A_{1.5\ \text{Hz}},\quad A_{2.0\ \text{Hz}}>A_{2.5\ \text{Hz}}$$

Common mistakes and how to avoid them

Lesson summary

• An oscillator repeats motion around an equilibrium position.
• Natural frequency describes how quickly a system tends to oscillate freely; driving frequency describes how quickly a repeated input acts.
• Resonance is expected when the driving frequency equals or nearly equals the natural frequency.
• A fair test varies driving frequency and measures amplitude while keeping other conditions as steady as possible.
• Predictions are not measured evidence. Compare actual measurements with the prediction after testing.

Check your understanding

Key terms

Amplitude

The greatest displacement of an oscillation from its equilibrium position.

Driving force

A repeated push or pull that acts on an oscillating system.

Driving frequency

The number of times per second that a repeated force acts.

Frequency

The number of complete cycles per second, measured in hertz (Hz).

Natural frequency

The frequency at which a system tends to oscillate when disturbed and then allowed to move without repeated driving.

Oscillation

A repeated back-and-forth motion around an equilibrium position.

Period

The time taken for one complete cycle, measured in seconds (s).

Resonance

A strong response when the driving frequency equals or is close to the natural frequency.

Continue through SPH3U

View the complete SPH3U Ontario Grade 11 Physics curriculum and lessons

• E2.5 · Analyse the Doppler effect for a moving sound source
• E2.7 · Analyse resonance conditions and applications
• E1.1 · Analyse how wave properties influence structures and devices
• E1.2 · Assess wave and noise impacts and technologies that reduce them
• E2.1 · Use terminology for waves, interference, standing waves, and resonance
• E2.2 · Investigate mechanical waves and interference

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Physics (SPH3U), expectation E2.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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