Question

Two strings of the same material and same length are given equal tension. If they are vibrating with fundamental frequencies \( 1600 \text{ Hz} \) and \( 900 \text{ Hz} \), then the ratio of their respective diameters is

• A. \( 16 : 9 \)
• B. \( 4 : 3 \)
• C. \( 81 : 256 \)
• D. \( 3 : 4 \)
• E. \( 9 : 16 \)

Hint:

Remember the general proportionality \( f \propto \frac{1}{L \cdot d} \sqrt{\frac{T}{\rho}} \) for vibrating strings. Recognizing what parameters remain constant allows you to quickly set up ratios without doing full calculations.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The fundamental frequency of a stretched string vibrating transversely depends on three factors: its length, the tension applied, and its linear mass density (mass per unit length).
The linear mass density itself is determined by the material's density and the string's cross-sectional area, which directly relates to its diameter.

Step 2: Key Formula or Approach:
The fundamental frequency \( f \) is given by Mersenne's law:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \] where \( L \) is length, \( T \) is tension, and \( \mu \) is linear mass density.
Express \( \mu \) in terms of diameter \( d \) and material density \( \rho \):
\[ \mu = \text{Volume density} \times \text{Cross-sectional area} = \rho \cdot \pi \left(\frac{d}{2}\right)^2 = \frac{\pi \rho d^2}{4} \]

Step 3: Detailed Explanation:
Substitute the expression for \( \mu \) into the frequency formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\frac{\pi \rho d^2}{4}}} \] \[ f = \frac{1}{2L} \sqrt{\frac{4T}{\pi \rho d^2}} \] \[ f = \frac{1}{2L} \cdot \frac{2}{d} \sqrt{\frac{T}{\pi \rho}} \] \[ f = \frac{1}{L \cdot d} \sqrt{\frac{T}{\pi \rho}} \] The problem states that both strings have the same material (same density \( \rho \)), same length (\( L \)), and are under equal tension (\( T \)).
Therefore, all terms except \( f \) and \( d \) are constant.
This establishes an inverse proportionality between frequency and diameter:
\[ f \propto \frac{1}{d} \implies d \propto \frac{1}{f} \] We can write this as a ratio for the two strings:
\[ \frac{d_1}{d_2} = \frac{f_2}{f_1} \] Given the fundamental frequencies are \( f_1 = 1600 \text{ Hz} \) and \( f_2 = 900 \text{ Hz} \), we plug these in to find the ratio of their respective diameters:
\[ \frac{d_1}{d_2} = \frac{900}{1600} \] \[ \frac{d_1}{d_2} = \frac{9}{16} \] Thus, the ratio \( d_1 : d_2 \) is \( 9 : 16 \).

Step 4: Final Answer:
The correct ratio matches option (E).