Question

Two uniform strings X and Y made of same material made to vibrate under same tension. If the second overtone of string X is equal to the third overtone of string Y and the radius of string X is thrice that of string Y. The ratio of length of string X to that of Y is

• A. \(1:8\)
• B. \(1:6\)
• C. \(1:3\)
• D. \(1:4\)

Hint:

Frequency of the p-th harmonic is proportional to p divided by (length times radius) for the same tension and material.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
For a string, \(f_p=\dfrac{p}{2L}\sqrt{\dfrac{T}{\mu}}\) with \(\mu=\rho\pi r^{2}\). For the same tension and material, \(f_p\propto\dfrac{p}{Lr}\).

Step 2: Overtones to harmonics
The second overtone of X is its 3rd harmonic. The third overtone of Y is its 4th harmonic.

Step 3: Equate frequencies
\(\dfrac{3}{L_Xr_X}=\dfrac{4}{L_Yr_Y}\Rightarrow\dfrac{L_X}{L_Y}=\dfrac{3r_Y}{4r_X}\)

Step 4: Substitute
With \(r_X=3r_Y\): \(\dfrac{L_X}{L_Y}=\dfrac{3}{4\times3}=\dfrac{1}{4}\).

Final Answer:
The ratio of lengths of X to Y is 1 : 4. \[ \boxed{\text{(D) }1:4} \]