Problem 73 - Pearson Physics

Question

(a) When you blow across the opening of a partially filled 2-L plastic bottle, you hear a tone. After pouring some water out of the bottle, you blow across the opening again. Does this tone have a higher frequency, a lower frequency, or the same frequency as the first tone? (b) Choose the best explanation from among the following: A. The same bottle produces the same frequency. B. The greater distance from the top of the bottle to the level of the water results in a higher frequency. C. A lower level of water results in a longer column of air and hence a lower frequency.

Step-by-Step Solution

Answer

The tone has a lower frequency; explanation C is correct.

Step 1: Understanding the Problem

When you blow across the top of the bottle, you create a resonant frequency that depends on the column of air inside the bottle. We need to determine how changes in the air column length affect the frequency of the sound produced.

Step 2: Identifying Key Concepts

The frequency of the sound is determined by the length of the air column. When the air column is longer, the frequency is lower; when the air column is shorter, the frequency is higher. This is because frequency and wavelength are inversely related to the length of the air column that vibrates when you blow over it.

Step 3: Analyzing the Steps

Initially, the bottle is partially filled, and you get a certain frequency when blowing over it. By pouring water out, you increase the air column's length inside the bottle, which should be more significant than before. This lengthening of the air column decreases the frequency of the resonant tone produced.

Step 4: Choosing the Best Explanation

Option C accurately describes the situation. Pouring out water results in a longer column of air, which lowers the frequency. Therefore, option C - 'A lower level of water results in a longer column of air and hence a lower frequency' is correct.

Key Concepts

Sound Waves

Sound waves are disturbances that travel through a medium, such as air, water, or solids, in the form of oscillations or vibrations. When you blow across the opening of a bottle, for example, you initiate sound waves that resonate inside. These waves carry energy from one location to another, producing the perception of sound.

Sound waves are longitudinal waves, meaning the vibrations occur in the same direction as the wave direction. This is unlike transverse waves, where vibrations are perpendicular to the direction of wave travel.

• Wavelength is the distance between two consecutive points in the same phase, like crest to crest or trough to trough.
• Amplitude measures the wave's maximum displacement from its rest position and relates to the sound's volume.


• Frequency measures the number of oscillations per second, determining pitch.

By understanding these essential characteristics of sound waves, we can better grasp the concept of resonant frequency when dealing with air columns, like in our bottle experiment.

Air Column

An air column is a column of air, and in our context, it refers to the air inside a bottle or similar container. The length of the air column is crucial in determining the sound's resonant frequency.

When you blow across the top of a bottle, you cause the air column inside to vibrate, creating sound waves. The bottle's structure and the length of the air column define the tone's pitch.

• A longer air column leads to a lower resonant frequency, producing a deeper tone.
• A shorter air column results in a higher resonant frequency, producing a higher-pitched tone.

Changes in the column length cause variations in sound frequency. This principle is observed in musical instruments like organs and wind instruments and explains why pouring water out of a bottle lowers the frequency.

Frequency and Wavelength

Frequency and wavelength are interconnected properties crucial to understanding sound in air columns.

Frequency (\( f \)) is defined as the number of cycles per second of a wave and is measured in hertz (Hz). Higher frequencies correspond to higher pitches in sound perception.
Conversely, wavelength (\( \lambda \)) is the length of one complete wave cycle and is inversely related to frequency. When the air column inside a bottle changes, it affects these properties.

• As the air column lengthens, the wavelength increases, resulting in a lower frequency.
• Shortening the air column reduces the wavelength, resulting in a higher frequency.

The formula describing this relationship is \( v = f \lambda \), where \( v \) represents the speed of sound in the medium. Keeping this relationship in mind helps us understand why longer columns emit lower frequencies, as seen in the changes within the bottle's air column.