Answer :
The beat frequency is approximately 3.94 Hz.
The frequency f of a vibrating string is given by the formula:
[tex]\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \][/tex]
where:
- L is the length of the string,
- T is the tension in the string,
- [tex]\( \mu \)[/tex] is the linear mass density of the string.
When the tension is decreased by 2.5%, the new tension T' can be expressed as:
[tex]\[ T' = T - 0.025T = 0.975T \][/tex]
The new frequency ( f' ) of the string with decreased tension can be calculated using the relationship between tension and frequency. Since frequency is proportional to the square root of the tension, we have:
[tex]\[ f' = \frac{1}{2L} \sqrt{\frac{T'}{\mu}} = \frac{1}{2L} \sqrt{\frac{0.975T}{\mu}} \][/tex]
This simplifies to:
[tex]\[ f' = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \sqrt{0.975} \]\[ f' = f \sqrt{0.975} \][/tex]
Given the original frequency ( f = 294 ) Hz:
[tex]\[ f' = 294 \sqrt{0.975} \][/tex]
Now, we can calculate [tex]\( \sqrt{0.975} \)[/tex]:
[tex]\[ \sqrt{0.975} \approx 0.987 \][/tex]
So:
[tex]\[ f' = 294 \times 0.987 \approx 290.058 \text{ Hz} \][/tex]
The beat frequency is the absolute difference between the two frequencies:
[tex]\[ \text{Beat frequency} = |f - f'| = |294 - 290.058| = 3.942 \text{ Hz} \][/tex]
Rounding to a reasonable precision, the beat frequency heard when the two strings are played together is approximately:
[tex]\[ \text{Beat frequency} \approx 3.94 \text{ Hz} \][/tex]
Final answer:
The beat frequency heard when one of the two violin strings tuned to 294 Hz has its tension decreased by 2.5% and is played together with the other will be approximately 7.35 Hz.
Explanation:
When two violin strings tuned to the same frequency are played together, and one string's tension is decreased, the frequency of that string will also decrease. This change in tension and hence the frequency will lead to a phenomenon known as beats. The beat frequency is the absolute difference between the frequencies of the two strings. Since the original frequency is 294 Hz and the tension is decreased by 2.5%, we can calculate the change in frequency (which will be very small) introduced by that change in tension. However, in a real-world scenario, simply calculating the frequency change from the percentage decrease in tension is non-trivial, as it involves understanding the physical properties of the string and using the wave equation.
Since we're ignoring complex calculations and precise physical properties, let's assume for the sake of this example that the new frequency will be 2.5% less than the original, which would give a new frequency of approximately 286.65 Hz (since 2.5% of 294 is 7.35 Hz). Now both strings, when played together, will produce beats at the frequency of the absolute difference between them, 294 Hz - 286.65 Hz, which is 7.35 Hz. Therefore, the beat frequency heard will be approximately 7.35 Hz.
