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Two successive resonant frequencies of a stretched wire: find its length or mass per unit length

This type of question, asked 2 times: 2021 Paper 22, 27 Jul 2022. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 22

    25
    A wire having a linear mass density 9.0×10−4 kg/m9.0 \times10^{- 4}\text{ }kg/m is stretched between two rigid supports with a tension of 900 N900\text{ }N. The wire resonates at a frequency of 500 Hz500\text{ }Hz. The next higher frequency at which the same wire resonates is 550 Hz550\text{ }Hz. The length of the wire is___ m.

  2. Question 2

    27 Jul 2022

    117
    A wire of length 30 cm30\text{ }cm, stretched between rigid supports, has it's nth n^{\text{th~}} and (n+1)th (n + 1)^{\text{th~}} harmonics at 400 Hz400\text{ }Hz and 450 Hz450\text{ }Hz, respectively. If tension in the string is 2700 N2700\text{ }N, it's linear mass density is........ kg/mkg/m