Question

The fundamental frequency of a string stretched with a weight \(M\) kg is \(n\) hertz. Keeping the vibrating length constant, the weight required to produce its octave is

• A. \(M\)
• B. \(8M\)
• C. \(2M\)
• D. \(4M\)

Hint:

For a stretched string, doubling the frequency requires four times the tension if length and density remain constant.

Answer 1

The Correct Option is D

Step 1: Write the formula for fundamental frequency of a stretched string.
The fundamental frequency of a stretched string is given by:
\[ n = \frac{1}{2L}\sqrt{\frac{T}{\mu}} \] where \(T\) is the tension and \(\mu\) is the linear density.

Step 2: Relation between tension and frequency.
Keeping the length and linear density constant,
\[ n \propto \sqrt{T} \]
Step 3: Condition for octave.
An octave means the frequency is doubled:
\[ n' = 2n \]
Step 4: Find new tension.
\[ \frac{n'}{n} = \sqrt{\frac{T'}{T}} \Rightarrow 2 = \sqrt{\frac{T'}{T}} \] \[ T' = 4T \]
Step 5: Relation between tension and weight.
Since tension is proportional to the weight suspended,
\[ M' = 4M \]
Step 6: Conclusion.
The weight required to produce an octave is \(4M\).