Question

A stretched string of length '\(L\)' is fixed at both the ends, has '\(n\)' nodes (\(n>2\)). The length of the string in terms of \(n\) and wavelength \(\lambda\) is

• A. \((n+1)\frac{\lambda }{2}\)
• B. \(n\lambda\)
• C. \(\frac{n\lambda }{2}\)
• D. \((n-1)\frac{\lambda }{2}\)

Hint:

With both ends as nodes, n nodes leave n minus 1 gaps between them, each half a wavelength.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
In a stretched string with both ends fixed, the ends are nodes, and the distance between consecutive nodes is \(\dfrac{\lambda}{2}\).

Step 2: Count the Gaps
With \(n\) nodes (including the ends) there are \((n-1)\) gaps between them.

Step 3: Calculation
\(L=(n-1)\dfrac{\lambda}{2}\)

Step 4: Compare the options
This is the last option.

Final Answer:
The length of the string is (n - 1) times lambda by 2. \[ \boxed{\text{(D) }(n-1)\dfrac{\lambda}{2}} \]