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A string in musical instrument is 50 cm long and its fundamental frequency is 800 Hz. If a frequency of 1000 Hz is to be produced, then required length of string is
- a)62.5 cm
- b)40 cm
- c)50 cm
- d)37.5 cm
Correct answer is option 'B'. Can you explain this answer?
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Given data:
- Length of the string = 50 cm
- Fundamental frequency = 800 Hz
Calculating the speed of the wave:
The speed of the wave on a string is given by the formula:
\[v = f \times \lambda\]
where:
- \(v\) = speed of the wave
- \(f\) = frequency
- \(\lambda\) = wavelength
Calculating the wavelength:
The wavelength of the wave on a string is given by the formula:
\[\lambda = 2L/n\]
where:
- \(L\) = length of the string
- \(n\) = harmonic number (for fundamental frequency, \(n = 1\))
For the given fundamental frequency:
\[800 = v / \lambda\]
Substitute \(\lambda = 2L\) and solve for \(v\):
\[800 = v / 2L\]
\[v = 1600L\]
Calculating the length for 1000 Hz frequency:
For the desired frequency of 1000 Hz:
\[1000 = v / \lambda\]
Substitute the value of \(v\) obtained earlier:
\[1000 = 1600L / \lambda\]
Since the speed of the wave remains constant, we can equate the two expressions for \(v\):
\[1600L = 1000 \times 2L\]
Solve for \(L\):
\[1600L = 2000L\]
\[L = 40 cm\]
Therefore, the required length of the string to produce a frequency of 1000 Hz is 40 cm.
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A string in musical instrument is 50 cm long and its fundamental frequency is 800 Hz. If a frequency of 1000 Hz is to be produced, then required length of string isa)62.5 cmb)40 cmc)50 cmd)37.5 cmCorrect answer is option 'B'. Can you explain this answer?
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