A musical instrument is made using four different metal strings, 1,2,3 and 4 with mass per unit length μ , 2 μ, 3 μ and 4 μ respectively. The instrument is played by vibrating the strings by varying the free length in between the range L 0 and 2 L 0 . It is found that in string- 1(μ) at free length L 0 and tension T 0 the fundamental mode frequency is f 0 . List -I gives the above four strings while list -II lists the magnitude of some quantity. List-I List-II --- --- (I) String-1 ( μ ) (P) 1 (II) String-2 ( 2 μ ) (Q) 1 / 2 (III) String-3 ( 3 μ ) (R) 1 / √(2) (IV) String-4 ( 4 μ ) (S) 1 / √(3) (T) 3 / 16 (U) 1 / 16 The length of the string 1, 2, 3 and 4 are kept fixed at L 0, 3L 0/2, 5L 0/4 and 7L 0/4, respectively. Strings 1, 2, 3 and 4 are vibrated at their 1 st , 3 rd , 5 th and 14 th harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of T 0 will be:

I rightarrow P, II rightarrow Q, III rightarrow T, IV rightarrow U

I rightarrow P, II rightarrow Q, III rightarrow R, IV rightarrow T

I rightarrow P, II rightarrow R, III rightarrow T, IV rightarrow U

I rightarrow T, II rightarrow Q, III rightarrow R, IV rightarrow U

Physics · Transverse waves

JEE Advanced 2019 — Paper 2 — Question 35

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

List-I	List-II
(I) String-1 ( $\mu$ )	(P) $1$
(II) String-2 ( $2 \mu$ )	(Q) $1 / 2$
(III) String-3 ( $3 \mu$ )	(R) $1 / \sqrt{2}$
(IV) String-4 ( $4 \mu$ )	(S) $1 / \sqrt{3}$
	(T) $3 / 16$
	(U) $1 / 16$

\text{The length of the string 1, 2, 3 and 4 are kept fixed at } L_0, \frac{3L_0}{2}, \frac{5L_0}{4} \text{ and } \frac{7L_0}{4}, \text{ respectively.}

\text{Strings 1, 2, 3 and 4 are vibrated at their } 1^{st}, 3^{rd}, 5^{th} \text{ and } 14^{th} \text{ harmonics, respectively such that all the strings}

\text{have same frequency. The correct match for the tension in the four strings in the units of } T_0 \text{ will be:}

Option A:
$\text{I } \rightarrow \text{ P, II } \rightarrow \text{ Q, III } \rightarrow \text{ T, IV } \rightarrow \text{ U}$
Correct
Option B:
$\text{ I } \rightarrow \text{ P, II } \rightarrow \text{ Q, III } \rightarrow \text{ R, IV } \rightarrow \text{ T}$
Option C:
$\text{ I } \rightarrow \text{ P, II } \rightarrow \text{ R, III } \rightarrow \text{ T, IV } \rightarrow \text{ U}$
Option D:
$\text{ I } \rightarrow \text{ T, II } \rightarrow \text{ Q, III } \rightarrow \text{ R, IV } \rightarrow \text{ U}$

Answer: A

Step-by-step solution

\text{For string -1}

f_0 = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\text{For string -2}

\frac{3}{2 \times \frac{3L_0}{2}} \sqrt{\frac{T_2}{\mu}} = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\frac{3}{3L_0} \sqrt{\frac{T_2}{\mu}} = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\frac{1}{L_0} \sqrt{\frac{T_2}{\mu}} = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\sqrt{T_2} = \frac{1}{2} \sqrt{T_0}

T_2 = \frac{T_0}{4}

\text{For string -3}

\frac{5}{2 \times \frac{5L_0}{4}} \sqrt{\frac{T_3}{\mu}} = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\frac{5}{\frac{5L_0}{2}} \sqrt{\frac{T_3}{\mu}} = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\frac{2}{L_0} \sqrt{\frac{T_3}{\mu}} = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\sqrt{T_3} = \frac{1}{4} \sqrt{T_0}

T_3 = \frac{T_0}{16}

\text{For string -4}

\frac{14}{2 \times \frac{7L_0}{4}} \sqrt{\frac{T_4}{\mu}} = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\frac{14}{\frac{7L_0}{2}} \sqrt{\frac{T_4}{\mu}} = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\frac{4}{L_0} \sqrt{\frac{T_4}{\mu}} = \frac{1}{2L_0} \sqrt{\frac{T_0}{\mu}}

\sqrt{T_4} = \frac{1}{8} \sqrt{T_0}

T_4 = \frac{T_0}{64}

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

A musical instrument is made using four different metal strings, 1,2,3 and 4 with mass per unit length mu , 2 mu, 3 mu and 4 mu…

Answer: A

Step-by-step solution

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