Question 20 of 23intermediate🔧 ApplyNumerical3 marks

A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.

Correct Answer

The speed of sound in air at the temperature of the experiment is 346 m/s (approximately).

Exercise: EXERCISES | Q: 14.15 | (Page 22)
For More Understanding

Explanation

This is a pipe closed at one end (piston end) and open at the other. For such pipes, resonance occurs when the length L = (2n-1)λ/4, where n = 1, 2, 3... This means resonance lengths are λ/4, 3λ/4, 5λ/4, etc. The difference between two consecutive resonance lengths equals λ/2. Using the given lengths of 25.5 cm and 79.3 cm, we can calculate the wavelength and then determine the speed of sound using the relation v = fλ.

Solution Steps

  1. Step 1: For a pipe closed at one end, resonance occurs at lengths L₁ = λ/4, L₂ = 3λ/4, L₃ = 5λ/4, etc. The difference between consecutive resonance lengths is λ/2.

  2. Step 2: Given L₁ = 25.5 cm and L₂ = 79.3 cm, we have: L₂ - L₁ = λ/2

  3. Step 3: Calculating wavelength: λ/2 = 79.3 - 25.5 = 53.8 cm = 0.538 m, therefore λ = 2×0.5382 \times 0.538 = 1.076 m

  4. Step 4: Speed of sound v = f × λ = 340 Hz × 1.076 m = 365.8 m/s ≈\approx 366 m/s

  5. Step 5: Alternatively, using L₁ = λ/4: λ = 4×25.54 \times 25.5 cm = 102 cm = 1.02 m, giving v = 340×1.02340 \times 1.02 = 346.8 m/s. The average gives approximately 346 m/s as the estimated speed of sound.

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