Solution8. Worked Examples
📐 Worked Examples
Example: Node Measurement
On a 1 m string in 4th harmonic, nodes are observed at x = 0, 0.25, 0.50, 0.75, 1.00 m. Find λ.
1. Learning Objectives2. Language Objectives3. Key Terms3.1. Study Cards4. Glossary5. Theory6. Memorization Exercises7. Video Lesson
🎯 Learning Objectives
- Determine wavelength from node or antinode spacing in a stationary wave
- Relate distance between adjacent nodes (or antinodes) to half a wavelength
- Apply node/antinode measurements to experimental setups (strings, air columns)
- Explain how to measure λ in different harmonic modes
🗣️ Language Objectives
- Use terms “node spacing,” “antinode spacing,” “half-wavelength,” “harmonic” accurately
- Describe measurement procedures clearly in English
- Interpret instructions and report results with correct terminology
- Discuss experimental uncertainty and precision in academic language
📚 Key Terms and Translations
| English Term | Russian | Kazakh |
|---|---|---|
| Wavelength (λ) | Длина волны (λ) | Толқын ұзындығы (λ) |
| Node spacing | Расстояние между узлами | Тоқтау нүктелері арасындағы қашықтық |
| Antinode spacing | Расстояние между пучностями | Максимум нүктелері арасындағы қашықтық |
| Harmonic | Гармоника | Гармоника |
| Fundamental mode | Основной режим | Негізгі режим |
🃏 Vocabulary Study Cards
Node Spacing
Definition: Distance between two adjacent nodes = λ/2
Use: Measure to find λ
Antinode Spacing
Definition: Distance between adjacent antinodes = λ/2
Harmonic
Definition: A resonant mode with n half‐wavelengths fitting in the medium
📖 Glossary of Terms
Wavelength (λ)
Distance over which the wave’s shape repeats; in stationary wave, λ = 2·(node spacing).
Russian: Расстояние, через которое форма волны повторяется; в стоячей волне λ = 2·расстояние между узлами.Kazakh: Толқын формасы қайталанатын қашықтық; тұрақты толқында λ = 2·тоқтау нүктелері арасындағы қашықтық.
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Harmonic
An integer multiple of the fundamental frequency; n-th harmonic has n·(λ/2) segments.
Russian: Целочисленное кратное основной частоте; n-ая гармоника содержит n·(λ/2) сегментов.Kazakh: Негізгі жиіліктің бүтін еселігі; n-ші гармоника n·(λ/2) сегменттен тұрады.
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🔬 Theory: Determining Wavelength from Nodes/Antinodes
In a stationary wave, adjacent nodes (or antinodes) are separated by <su_tooltip text="половина длины волны/толқын ұзындығының жартысы" behavior="click"]λ/2[/su_tooltip]. Thus:
- Measure distance Δx between two adjacent nodes → λ = 2·Δx
- For n nodes spanning distance L: L = (n–1)·(λ/2) → λ = 2L/(n–1)
- In the n-th harmonic on a string of length L: λ = 2L/n
Medium: A string of length 0.8 m shows 5 nodes (including ends). Calculate λ.
Medium: In 3rd harmonic on a 1.2 m wire, what is λ?
Hard (Critical Thinking): Discuss sources of error when measuring node spacing in a real experiment.
💪 Memorization Exercises
Fill in the Blanks
- Distance between adjacent nodes = ______.
- λ in terms of node spacing Δx: λ = ______.
- For n intervals in length L, λ = ______.
- In the fundamental mode (n=1), λ = ______.
🎥 Video Lesson