Question:
A wire of length 'L' and linear density 'm' is stretched between two rigid supports with tension 'T'. It is observed that wire resonates in the \(P^{th}\) harmonic at a frequency of 320 Hz and resonates again at next higher frequency of 400 Hz in two successive modes. The value of 'P' is
A wire of length 'L' and linear density 'm' is stretched between two rigid supports with tension 'T'. It is observed that wire resonates in the \(P^{th}\) harmonic at a frequency of 320 Hz and resonates again at next higher frequency of 400 Hz in two successive modes. The value of 'P' is
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Successive harmonics of a stretched wire differ by the fundamental frequency. So f1 = 80 Hz and P = 320/80.
Updated On: Oct 1, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
A wire fixed at both ends resonates at \(f_p = p\,f_1\), where \(f_1 = \dfrac{1}{2L}\sqrt{\dfrac Tm}\) is the fundamental and \(p\) is a whole number.
Step 2: Key Formula or Approach:
The gap between two successive resonances is \(f_{p+1} - f_p = f_1\).
Step 3: Detailed Explanation:
\[ f_1 = 400 - 320 = 80 \text{ Hz} \]
For the lower resonance:
\[ p = \frac{320}{80} = 4 \]
Check: the next harmonic is \(5\times80 = 400\) Hz. This agrees with the given frequency.
Option (A) 2 would give a fundamental of 160 Hz and a next resonance at 480 Hz. Options (C) and (D) give fundamental frequencies of 40 Hz and 32 Hz, which do not give a gap of 80 Hz.
Final Answer:
The wire resonates in the 4th harmonic at 320 Hz, option (B). \[ \boxed{P=4 \text{ (B)}} \]
A wire fixed at both ends resonates at \(f_p = p\,f_1\), where \(f_1 = \dfrac{1}{2L}\sqrt{\dfrac Tm}\) is the fundamental and \(p\) is a whole number.
Step 2: Key Formula or Approach:
The gap between two successive resonances is \(f_{p+1} - f_p = f_1\).
Step 3: Detailed Explanation:
\[ f_1 = 400 - 320 = 80 \text{ Hz} \]
For the lower resonance:
\[ p = \frac{320}{80} = 4 \]
Check: the next harmonic is \(5\times80 = 400\) Hz. This agrees with the given frequency.
Option (A) 2 would give a fundamental of 160 Hz and a next resonance at 480 Hz. Options (C) and (D) give fundamental frequencies of 40 Hz and 32 Hz, which do not give a gap of 80 Hz.
Final Answer:
The wire resonates in the 4th harmonic at 320 Hz, option (B). \[ \boxed{P=4 \text{ (B)}} \]
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