Problem 17 - Master Resource Book in JEE Main Physics

Question

In the experiment of measuring speed of sound by resonance tube, it is observed that for tuning fork of frequency \(v=480 \mathrm{~Hz}\), length of air column \(\mathrm{cm}\), \(l_{1}=30 \mathrm{~cm}, l_{2}=70 \mathrm{~cm}\), then \(v_{1}\) is equal to (a) \(338 \mathrm{~ms}^{-1}\) (b) \(379 \mathrm{~ms}^{-1}\) (c) \(384 \mathrm{~ms}^{-1}\) (d) \(332 \mathrm{~ms}^{-1}\)

Step-by-Step Solution

Answer

The speed of sound, \( v_1 \), is 384 m/s (option c).

Step 1: Understanding the Problem

The problem involves determining the speed of sound using a resonance tube with a tuning fork of known frequency and measured lengths of the air column where resonance occurs. The key here is to use the relation between resonating lengths and the wavelength of the sound in the tube.

Step 2: Identifying the Formula

In resonance, the difference between the resonating lengths of the air column gives half of the wavelength of the sound wave. Therefore, the formula to calculate the speed of sound is given by:\[ v = 2f(l_2 - l_1) \]where \( v \) is the speed of sound, \( f \) is the frequency of the tuning fork, and \( l_1, l_2 \) are the resonating lengths.

Step 3: Plug in the Values

Substitute the given values into the formula: the frequency \( f = 480 \) Hz, and the lengths \( l_1 = 30 \) cm, \( l_2 = 70 \) cm. Calculate the difference in lengths:\[ l_2 - l_1 = 70 \, \text{cm} - 30 \, \text{cm} = 40 \, \text{cm} \]Convert this to meters: \( 40 \, \text{cm} = 0.4 \, \text{m} \).

Step 4: Calculate the Speed of Sound

Use the formula to calculate the speed:\[ v = 2 \times 480 \, \text{Hz} \times 0.4 \, \text{m} \]Calculate step-by-step:\[ v = 2 \times 480 \times 0.4 = 384 \, \text{m/s} \]

Step 5: Conclusion

After calculating, the speed of sound is found to be 384 m/s, which matches with one of the options provided.

Key Concepts

Resonance Tube

A resonance tube is a simple but effective way to measure the speed of sound in air. It involves a tube partially filled with water and a column of air above the water. By striking a tuning fork and placing it near the opening of the tube, you can produce standing waves in the air column. When resonance occurs, the sound waves inside the tube have a distinct and strong amplitude, making the sound notably louder.

This phenomenon happens at specific air column lengths where the conditions are ripe for standing waves. By carefully adjusting the water level, you can find these lengths quite precisely. This makes the resonance tube an efficient experiment to demonstrate wave interference and resonance in physics classrooms.

• Simple way to observe standing waves.
• Helps to identify wavelengths through resonance.
• Gives a practical application of wave physics.

Tuning Fork Frequency

A tuning fork is a useful tool in acoustics experiments due to its precise frequency. When struck, it produces a pure musical tone, defined by its frequency, which is measured in Hertz (Hz). In resonance tube experiments, the frequency of the tuning fork is crucial, as it directly affects the formation of standing waves inside the tube.

For instance, in our exercise, we use a tuning fork with a frequency of 480 Hz. This means that 480 complete cycles of sound waves occur every second. The frequency sets the stage for how the air molecules vibrate within the tube and plays a key role in determining the speed of sound along with the resonance lengths.

• The precise frequency helps in accurate measurements.
• Produces a consistent tone for resonance experiments.
• The frequency directly impacts wavelength calculations.

Wavelength Calculation

Calculating the wavelength is an important step in finding the speed of sound. When using the resonance tube, the relationship between the resonant lengths and the wavelength is crucial. The distance between two consecutive resonating lengths corresponds to half of the wave's full length—or, simply, half the wavelength.

Using the known frequencies and these lengths, like in the example: if two resonating lengths are 30 cm and 70 cm, their difference gives half the wavelength. Here, it is 40 cm in the example where the wavelength itself will be two times this difference, or 80 cm (0.8 m).

• Key to linking physical phenomena with wave theory.
• Essential for calculating speed of sound.
• Bridges observational data with theoretical formulas.

Resonating Lengths

Resonating lengths in a resonance tube experiment are the lengths at which the air column naturally vibrates in response to the tuning fork's frequency. These measurements are crucial for calculating the speed of sound because they relate directly to the wavelength of the sound wave inside the tube.

For example, if the first resonating length is 30 cm and the next one is 70 cm, these two points of resonance indicate where conditions support the standing wave pattern. The difference between these lengths (here, 40 cm) is vital because it is half of the respective wavelength.

• Directly measured from experiment setup.
• Gives half of the wavelength for standing waves.
• Critical for determining the speed of sound accurately.