Question

A string of length \(L\) and linear density \(m\) has a fundamental frequency \(n\) when stretched by tension \(T\). The fundamental frequency of another string having double the length and double linear density, when the same tension is applied, is

• A. \(\dfrac{n}{2\sqrt{2}}\)
• B. \(2n\)
• C. \(\dfrac{n}{2}\)
• D. \(\dfrac{n}{\sqrt{2}}\)

Hint:

For strings, frequency varies inversely with length and inversely with the square root of linear density.

Answer 1

The Correct Option is A

Step 1: Write the formula for fundamental frequency of a stretched string.
\[ n = \frac{1}{2L}\sqrt{\frac{T}{m}} \]
Step 2: Parameters of the second string.
Length \(L' = 2L\), linear density \(m' = 2m\), tension \(T' = T\).

Step 3: Write the frequency of the second string.
\[ n' = \frac{1}{2(2L)}\sqrt{\frac{T}{2m}} \]
Step 4: Simplify the expression.
\[ n' = \frac{1}{4L}\cdot\frac{1}{\sqrt{2}}\sqrt{\frac{T}{m}} \]
Step 5: Express in terms of \(n\).
\[ n' = \frac{1}{2\sqrt{2}}\left(\frac{1}{2L}\sqrt{\frac{T}{m}}\right) = \frac{n}{2\sqrt{2}} \]
Step 6: Conclusion.
The fundamental frequency of the second string is \(\dfrac{n}{2\sqrt{2}}\).