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Question

A musical instrument is made using four different metal strings. 1,2,3 and 4 with mass per unit length μ,2μ,3μ and 4μ respectively. The instrument is played by vibrating the strings by varying the free length in between the range L0 and 2L0. It is found that in string- 1 (μ) at free length L0 and tension T0 the fundamental mode frequency is f0. 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(2μ) (Q) 1/2
(iii) String- 3(3μ) (R) 1/2
(iv) String-4(4μ) (S) 1/3
(T) 3/16
(U) 1/16

The length of the strings 1,2,3 and 4 are kept fixed at L0,3L02,5L04 and 7L04 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 T0 will be,


A

IP,IIR,IIIT,IVU

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B

IP,IIQ,IIIT,IVU

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C

IP,IIQ,IIIR,IVT

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D

IT,IIQ,IIIR,IVU

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Solution

The correct option is A

IP,IIR,IIIT,IVU



λ=2L
f=Vλ=12LTμ
For string (1)
f0=12L0×T1μ=12LT0μ

For string (2)
f0=3×22×3L0T22μT2=T02

For string (3)
f0=5×25L0T33μT3=3T016

For string (4)
f0=14×42×7L0T44μT4=T016

IP,IIQ,IIIT,IVU

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