Question:
Two uniform wires of same material are vibrating under the same tension. If the first overtone of the first wire is equal to the second overtone of the second wire and radius of first wire is twice the radius of second wire then the ratio of length of first to second wire is
Two uniform wires of same material are vibrating under the same tension. If the first overtone of the first wire is equal to the second overtone of the second wire and radius of first wire is twice the radius of second wire then the ratio of length of first to second wire is
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Wires with same tension and material have v proportional to 1/r; equate the overtone frequencies.
Updated On: Oct 1, 2026
- \(\frac{1}{4}\)
- \(\frac{1}{3}\)
- \(\frac{2}{3}\)
- \(\frac{1}{2}\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
For a string fixed at both ends, \(f_n=\dfrac{nv}{2L}\) with \(v=\sqrt{T/\mu}\). For the same material, \(\mu\propto r^2\), so \(v\propto\dfrac1r\) at the same tension.
Step 2: Name the overtones:
First overtone is the second harmonic (\(n=2\)): \(f=\dfrac vL\). Second overtone is the third harmonic (\(n=3\)): \(f=\dfrac{3v}{2L}\).
Step 3: Equate the frequencies:
\(\dfrac{v_1}{L_1}=\dfrac{3v_2}{2L_2}\), so \(\dfrac{L_1}{L_2}=\dfrac23\cdot\dfrac{v_1}{v_2}\).
Step 4: Insert the radius ratio:
\(r_1=2r_2\) gives \(\dfrac{v_1}{v_2}=\dfrac{r_2}{r_1}=\dfrac12\). So \(\dfrac{L_1}{L_2}=\dfrac23\times\dfrac12=\dfrac13\). Option B.
Step 5: Why the other options are wrong.
1/4, 2/3 and 1/2 come from using \(v\propto\dfrac1{r^2}\), ignoring the speed difference, or using the wrong harmonic numbers.
Final Answer:
The ratio of lengths is 1/3. \[ \boxed{\text{(B) }\dfrac13} \]
For a string fixed at both ends, \(f_n=\dfrac{nv}{2L}\) with \(v=\sqrt{T/\mu}\). For the same material, \(\mu\propto r^2\), so \(v\propto\dfrac1r\) at the same tension.
Step 2: Name the overtones:
First overtone is the second harmonic (\(n=2\)): \(f=\dfrac vL\). Second overtone is the third harmonic (\(n=3\)): \(f=\dfrac{3v}{2L}\).
Step 3: Equate the frequencies:
\(\dfrac{v_1}{L_1}=\dfrac{3v_2}{2L_2}\), so \(\dfrac{L_1}{L_2}=\dfrac23\cdot\dfrac{v_1}{v_2}\).
Step 4: Insert the radius ratio:
\(r_1=2r_2\) gives \(\dfrac{v_1}{v_2}=\dfrac{r_2}{r_1}=\dfrac12\). So \(\dfrac{L_1}{L_2}=\dfrac23\times\dfrac12=\dfrac13\). Option B.
Step 5: Why the other options are wrong.
1/4, 2/3 and 1/2 come from using \(v\propto\dfrac1{r^2}\), ignoring the speed difference, or using the wrong harmonic numbers.
Final Answer:
The ratio of lengths is 1/3. \[ \boxed{\text{(B) }\dfrac13} \]
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