1. A person is listening to a new instrument capable of sounds ranging from 10 dB to 150 dB and with frequencies from 12 Hz to 30,000 Hz. Comment on the suitability of this instrument for the musician or audience.

2. A guitar emits a 440 Hz tone. How far apart are the crests in room temperature air (20 degrees celsius)? How long does it take the string to vibrate from crest position to trough position (across twice the amplitude)?

Answer 1: This instrument is suitable for both the musician and the audience, as it has a wide range of sound levels and frequencies. The 10 dB to 150 dB range is suitable for both loud and soft music, while the 12 Hz to 30,000 Hz range covers a wide range of musical notes.

Answer 2: The crests of the 440 Hz tone in room temperature air (20 degrees celsius) are approximately 7.5 meters apart. It takes the string 0.0045 seconds to vibrate from crest position to trough position (across twice the amplitude).

1. The instrument described has a wide range of sound levels, from 10 dB to 150 dB. To put this into context, 10 dB is a relatively soft sound, similar to a whisper, while 150 dB is extremely loud, comparable to a jet engine or fireworks. This wide dynamic range allows the musician to explore a broad range of volume levels, providing them with versatility in expressing their musical ideas.

Additionally, the instrument has a frequency range from 12 Hz to 30,000 Hz. This covers a wide spectrum of audible frequencies, allowing for a diverse range of tones and musical possibilities. The lower frequency range goes down to 12 Hz, which is on the lower end of human hearing, providing a deep and resonant sound. The higher frequency range extends up to 30,000 Hz, capturing the full spectrum of human hearing capabilities.

Overall, the instrument's suitability for the musician or audience depends on their preferences and the intended musical context. The wide dynamic range and frequency capabilities provide ample opportunities for artistic expression, but the musician and audience may need to consider their tolerance for loud sound levels and their desired sound aesthetics before determining the instrument's suitability.

2. To determine the distance between the crests of a 440 Hz tone in room temperature air (20 degrees Celsius), we can use the speed of sound formula. The speed of sound in air at 20 degrees Celsius is approximately 343 meters per second.

The formula for the distance between crests, or wavelength (λ), is given by:

λ = v / f

Where:
λ = Wavelength (distance between crests)
v = Speed of sound in air (343 m/s)
f = Frequency of the sound wave (440 Hz)

Calculating the wavelength:

λ = 343 m/s / 440 Hz = 0.7795 meters

Therefore, the distance between the crests of a 440 Hz tone in room temperature air is approximately 0.7795 meters (or 77.95 centimeters).

To determine the time it takes for the string to vibrate from the crest position to the trough position (across twice the amplitude), we need to consider the frequency of the vibration.

The formula for the period (T) of a vibration is given by:

T = 1 / f

Where:
T = Period (time for one complete vibration)
f = Frequency of the vibration (440 Hz)

Calculating the period:

T = 1 / 440 Hz = 0.00227 seconds

Therefore, it takes approximately 0.00227 seconds (or 2.27 milliseconds) for the string to vibrate from the crest position to the trough position, completing one complete vibration cycle.

1. To comment on the suitability of this instrument for the musician or audience, we need to consider both the range of sound and the human perception of sound.

a) Sound Intensity (in dB): The instrument's sound range of 10 dB to 150 dB covers a wide range, from very soft to extremely loud sounds. This wide dynamic range allows for versatile expression but should be managed carefully to avoid potential damage to the listener's ears.

b) Frequency (in Hz): The instrument's frequency range covers 12 Hz to 30,000 Hz, which includes most of the audible range for humans (20 Hz to 20,000 Hz). This wide frequency range allows for the production of various tones and pitches throughout the audible spectrum.

Considering these factors, this instrument seems well-suited for musicians who need a versatile instrument capable of producing a wide range of sound intensities and frequencies. However, it is crucial for both the musician and the audience to be conscious of hearing health and the potential risks associated with high sound intensities.

2. To determine the distance between crests in room temperature air and the duration of the vibration, we can use some basic physics formulas.

a) Wavelength and Crest-to-Crest Distance: The speed of sound in air at room temperature is approximately 343 meters per second. To find the distance between crests, we can use the formula: wavelength = speed of sound / frequency.

Given that the guitar emits a 440 Hz tone, we can plug in the values:
wavelength = 343 m/s / 440 Hz ≈ 0.779 meters or 77.9 centimeters.

Therefore, the distance between crests is approximately 0.779 meters or 77.9 centimeters.

b) Time Period of Vibration: The frequency of a vibrating string is the number of cycles (vibrations) it completes per second. The time period is the inverse of frequency, expressed as T = 1/frequency.

For the guitar emitting a 440 Hz tone, the time period can be calculated as:
T = 1 / 440 Hz ≈ 0.0023 seconds or 2.3 milliseconds.

Therefore, it takes approximately 0.0023 seconds or 2.3 milliseconds for the string to vibrate from the crest position to the trough position (across twice the amplitude).