Answer the following by appropriately matching the lists based on the information given in the paragraph. A musical instrument is made using four different metal strings, 1 , 2 , 3 and 4 with mass per

Question

Answer the following by appropriately matching the lists based on the information given in the paragraph. 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.

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

Column I	Column II
(I) String-1( $μ$ )	(P) 1
(II) String-2 (2 $μ$ )	(Q) $\frac{1}{2}$
(III) String-3 (3 $μ$ )	(R) $\frac{1}{2}$
(IV) String-4 (4 $μ$ )	(S) $\frac{1}{3}$
	(T) $\frac{3}{16}$
	(U) $\frac{1}{16}$

Options:

Answer:

B

Solution:

For the fundamental mode:

$\frac{λ}{2} = L \Rightarrow λ = 2 L f = \frac{V}{λ} = \frac{1}{2 L} \frac{T}{μ}$

For string (1): $f_{0} = \frac{1}{2 L} \frac{T}{μ} \Rightarrow (P)$

For string (2): $f = \frac{1}{2 L} \frac{T}{2 μ} = \frac{f _{0}}{2} \Rightarrow (R)$

For string (3): $f = \frac{1}{2 L} \frac{T}{3 μ} = \frac{f _{0}}{3} \Rightarrow (S)$

For string (4): $f = \frac{1}{2 L} \frac{T}{4 μ} = \frac{f _{0}}{2} \Rightarrow (Q)$

TestHub · Accepted Answer

For the fundamental mode: ​ 2 λ ​ = L ⇒ λ = 2 L f = λ V ​ = 2 L 1 ​ μ T ​ ​ ​ For string (1): f 0 ​ = 2 L 1 ​ μ T ​ ​ ⇒ ( P ) For string (2): f = 2 L 1 ​ 2 μ T ​ ​ = 2 ​ f 0 ​ ​ ⇒ ( R ) For string (3): f = 2 L 1 ​ 3 μ T ​ ​ = 3 ​ f 0 ​ ​ ⇒ ( S ) For string (4): f = 2 L 1 ​ 4 μ T ​ ​ = 2 f 0 ​ ​ ⇒ ( Q )

Physics - Wave on String Question with Solution | TestHub

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