Problem
Problem
A wire having a linear density of is stretched between two rigid supports with a tension of 450 N. It is observed that the wire resonates at a frequency of 420 Hz. The next higher frequency at which the same wire resonates is 490 Hz. Find the length of the wire.Intuitive explanation
The key insight is recognizing that consecutive resonant frequencies in a stretched wire correspond to consecutive harmonic modes—each differing by exactly one half-wavelength fitting along the wire's length. Since the difference between any two consecutive resonant frequencies depends only on the wire's length and the wave speed (which is fixed by tension and linear density), you can use that frequency gap to find the length directly, without needing to identify which specific harmonics 420 Hz and 490 Hz represent.
Intuitive explanation
The key insight is recognizing that consecutive resonant frequencies in a stretched wire correspond to consecutive harmonic modes—each differing by exactly one half-wavelength fitting along the wire's length. Since the difference between any two consecutive resonant frequencies depends only on the wire's length and the wave speed (which is fixed by tension and linear density), you can use that frequency gap to find the length directly, without needing to identify which specific harmonics 420 Hz and 490 Hz represent.