Answer :
The frequency at which the violin string will vibrate when fingered one-third of the way down is [tex]\({661.5} \)[/tex] Hz.
To find the frequency of the violin string when fingered one-third of the way down, we use the fact that only two-thirds of the string vibrates.
When a string is fingered one-third of the way down, the vibrating length of the string [tex]\( L' \)[/tex] is two-thirds of the original length [tex]\( L \)[/tex].
Given:
- Original frequency [tex]\( f = 441 \)[/tex] Hz
The frequency f when the string is fingered can be found using the relation for frequency and length:
[tex]\[ f' = \left( \frac{L}{L'} \right) f \][/tex]
Since[tex]\( L' = \frac{2}{3} L \)[/tex]:
[tex]\[ f' = \left( \frac{L}{\frac{2}{3} L} \right) \cdot 441 \text{ Hz} \][/tex]
[tex]\[ f' = \left( \frac{3}{2} \right) \cdot 441 \text{ Hz} \][/tex]
[tex]\[ f' = 661.5 \text{ Hz} \][/tex]
Final answer:
The frequency of the unfingered violin string is 441 Hz. When fingered one-third of the way down, the new frequency will be 661.5 Hz.
Explanation:
To find the new frequency of the violin string when it is fingered one-third of the way down, we can use the relationship between the length and frequency of a vibrating string. If the length of the string is reduced to two-thirds, its new length would be (2/3) * original length. Since frequency is inversely proportional to length, we can set up the equation:
original frequency * (original length / new length) = new frequency
Substituting the given values, we have:
441 Hz * (1 / (2/3)) = new frequency
Simplifying the equation, we find that the violin string will vibrate at a frequency of 661.5 Hz when fingered one-third of the way down.
