A violin A-string, fixed at both ends, has a length and a linear mass density. One of the resonance frequencies of the string is 1320 Hz, and the next resonance frequency is 1760 Hz, with no frequencies between these two.

(a) What is the lowest resonance frequency (fundamental) of the A-string?

(b) Which harmonic is the frequency?

(c) What is the tension in the string?

The fundamental frequency of the given A-string with resonant frequencies is 440 Hz, and it represents the first harmonic. Using harmonics and the physical properties of the string, we can also calculate the tension required to produce a specific frequency on a string instrument.

Explanation:

Fundamental Frequency and Harmonics

The concepts of the fundamental frequency, harmonics, and the calculation of tension in a vibrating string are fixed at both ends in physics. To find the fundamental frequency or the first harmonic (f1), we can use the knowledge that the given frequencies are successive harmonics without any other frequencies in between. Given the two resonant frequencies (1320 Hz and 1760 Hz), we know that they are consecutive harmonics, making them the third and fourth harmonics respectively. Hence, the fundamental frequency is 1320 Hz / 3 = 440 Hz.

For the tension (T), we use the formula for the speed of a wave on a string (v = √(T/μ)) and the relationships between speed, frequency (f), and wavelength (λ). With the knowledge that λ can be calculated using L (length of the string) and the number of antinodes (n), we can find the tension by rearranging the formula to T = μ x (f x λ)2. This would give us the tension needed to produce a specific harmonic on the string when it is vibrating with a known frequency.

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