Question

A student is performing the experiment of Resonance Column. The diameter of the column tube is \(4 \mathrm{~cm}\). The frequency of the tuning fork is \(512 \mathrm{~Hz}\). The air temperature is \(38^{\circ} \mathrm{C}\) in which the speed of sound is \(336 \mathrm{~m} / \mathrm{s}\). The zero of the meter scale coincides with the top end of the Resonance Column tube. When the first resonance occurs, the reading of the water level in the column is (A) \(14.0 \mathrm{~cm}\) (B) \(15.2 \mathrm{~cm}\) (C) \(16.4 \mathrm{~cm}\) (D) \(17.6 \mathrm{~cm}\)

Short answer

The length of the air column at the first resonance is 16.4 cm, so the correct answer is (C) 16.4 cm.

Step-by-step solution

Understanding the Experiment

In a Resonance Column experiment, a tube filled with air is used to determine the speed of sound in the air. A tuning fork is struck and held over the tube, and the length of the air column is changed until the first resonance condition is met, which occurs when the length of the air column is equal to one-quarter of the wavelength of the sound.

Calculate the Wavelength

Using the speed of sound and the frequency provided, calculate the wavelength of the sound wave using the formula \( v = f \lambda \), where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength. Solving for \( \lambda \), we have \( \lambda = \frac{v}{f} \).

Plug in the Known Values

Now, plug in the values for the speed of sound (\( v = 336 \text{ m/s} \)) and the frequency of the tuning fork (\( f = 512 \text{ Hz} \)) into the wavelength formula to find \( \lambda \).

Compute the Length for the First Resonance

Since the first resonance occurs at one-quarter of the wavelength, divide the calculated wavelength by 4 to find the length of the first resonance, \( L = \frac{\lambda}{4} \).

Select the Correct Answer

Compare the calculated length for the first resonance to the given options (A) 14.0 cm, (B) 15.2 cm, (C) 16.4 cm, (D) 17.6 cm, to find the correct answer.

Key concepts

Speed of Sound Calculation

Understanding the speed of sound is crucial when performing experiments involving sound waves, such as the Resonance Column experiment. The speed of sound can be affected by factors such as air temperature, humidity, and the medium through which it travels. In our specific experiment, at a temperature of 38°C, the speed of sound in air is given as 336 m/s.

To calculate the speed of sound in different conditions, you can use the formula: \( v = f \lambda \), where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength of the sound wave. By rearranging the formula, you can solve for any of the variables if the other two are known. For instance, if you have the frequency and want to find the speed of sound, you could rearrange to \( v = f \cdot \lambda \) when wavelength is given.

Sound Wavelength Determination

The wavelength of a sound wave is a key element in understanding sound propagation and is directly related to the frequency of the sound and the medium it travels through. In our exercise, we determine the wavelength by using the formula \( \lambda = \frac{v}{f} \) given the speed of sound (\( v \)) and the frequency (\( f \)) of the tuning fork.

After finding the speed of sound at the experiment's air temperature, we insert the frequency of the tuning fork to get the wavelength. This is crucial for the Resonance Column experiment because the wavelength helps us find at what length of the air column the first resonance, an enhancement of sound, occurs. The calculated wavelength represents the full cycle of the sound wave, and specific fractions of this wavelength will correspond to resonance points in the tube.

First Resonance in Sound Waves

The concept of resonance in sound waves is integral to many acoustical applications and experiments. Resonance occurs when a system is able to naturally vibrate at larger amplitudes at certain frequencies, known as the system's natural frequencies or resonant frequencies.

In the context of our experiment with the Resonance Column, the first resonance happens when the length of the air column inside the tube is one-quarter the wavelength of the sound emitted by the tuning fork. It's at this point that the sound waves are effectively amplified, creating a louder and clearer sound. The first resonance is achieved because the reflected wave from the water surface adds constructively to the incoming wave from the tuning fork at that specific length, leading to increased amplitude – essentially, the waves are in phase. By dividing the calculated wavelength by 4, we find the length of air column which corresponds to the first resonant point, which helps us determine which option (A, B, C or D) correctly represents the first resonance length as detected in the experiment.