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Physics · Transverse waves

JEE Advanced 2019 — Paper 2 — Question 34

A musical instrument is made using four different metal strings, 1,2,31,2,3 and 4 with mass per unit length μ\mu, 2μ,3μ2 \mu, 3 \mu and 4μ4 \mu respectively. The instrument is played by vibrating the strings by varying the free length in between the range L0L_{0} and 2L02 L_{0}. It is found that in string- 1(μ)1(\mu) at free length L0L_{0} and tension T0T_{0} the fundamental mode frequency is f0f_{0}. List -I gives the above four strings while list -II lists the magnitude of some quantity.

List-IList-II
(I) String-1 ( μ\mu )(P) 11
(II) String-2 ( 2μ2 \mu )(Q) 1/21 / 2
(III) String-3 ( 3μ3 \mu )(R) 1/21 / \sqrt{2}
(IV) String-4 ( 4μ4 \mu )(S) 1/31 / \sqrt{3}
(T) 3/163 / 16
(U) 1/161 / 16

If the tension in each string is T0T_{0}, the correct match for the highest fundamental frequency in f0f_{0} units will be,

  1. Option A:
    I → P, II → R, III → S, IV → Q\text{I } \rightarrow \text{ P, II } \rightarrow \text{ R, III } \rightarrow \text{ S, IV } \rightarrow \text{ Q}
    Correct
  2. Option B:
    I → Q, II → S, III → R, IV → P\text{I } \rightarrow \text{ Q, II } \rightarrow \text{ S, III } \rightarrow \text{ R, IV } \rightarrow \text{ P}
  3. Option C:
     I → P, II → Q, III → T, IV → S\text{ I } \rightarrow \text{ P, II } \rightarrow \text{ Q, III } \rightarrow \text{ T, IV } \rightarrow \text{ S}
  4. Option D:
     I → Q, II → P, III → R, IV → T\text{ I } \rightarrow \text{ Q, II } \rightarrow \text{ P, III } \rightarrow \text{ R, IV } \rightarrow \text{ T}

Answer: A

Step-by-step solution

f0=12L0T0μf_0 = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}} f1=12L0T02μ=f02f_1 = \frac{1}{2L_0} \sqrt{\frac{T_0}{2\mu}} = \frac{f_0}{\sqrt{2}} f2=12L0T03μ=f03f_2 = \frac{1}{2L_0} \sqrt{\frac{T_0}{3\mu}} = \frac{f_0}{\sqrt{3}} f4=12L0T04μ=f02f_4 = \frac{1}{2L_0} \sqrt{\frac{T_0}{4\mu}} = \frac{f_0}{2}

Answer key and solution verified before publishing.

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Exam
JEE Advanced 2019
Paper
Paper 2
Subject
Physics
Chapter
Transverse waves
Topic
Standing Waves on a String and Modes of Vibration

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