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Question
A particular guitar wire is 30⋅0 cm long and vibrates at a frequency of 196 Hz when no finger is placed on it. The next higher notes on the scale are 220 Hz, 247 Hz, 262 Hz and 294 Hz. How far from the end of the string must the finger be placed to play these notes?
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Solution
Given:
Length of the guitar wire (L1) = 30.0 cm = 0.30 m
Frequency, when no finger is placed on it, (f1) =196 Hz
And (f2) =220 Hz, (f3) = 247 Hz, (f4) = 262 Hz and (f5) = 294 Hz
The velocity is constant for a medium.
We have:
\[f \propto \left( \frac{1}{L} \right)\]
\[\Rightarrow \frac{f_1}{f_2} = \frac{L_2}{L_1}\]
\[ \Rightarrow \frac{196}{220} = \frac{L_2}{0 . 3}\]
\[ \Rightarrow L_2 = \frac{196 \times 0 . 3}{220} = 0 . 267 m\]
\[ \Rightarrow L_2 = 26 . 7 cm\]
Again,
\[f_3 = 247 Hz\]
\[\Rightarrow \frac{f_3}{f_1} = \frac{L_1}{L_3}\]
\[ \Rightarrow \frac{247}{196} = \frac{0 . 3}{L_3}\]
\[ \Rightarrow L_3 = 196 \times \frac{0 . 3}{247} = 0 . 238 m\]
\[ \Rightarrow L_3 = 23 . 8 cm\]
\[Similarly, L_4 = 196 \times \frac{0 . 3}{262} = 0 . 224 m\]
\[ \Rightarrow L_4 = 22 . 4 cm\]
\[And, L_5 = 20 \text{ cm }\]
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