The lengths of the strings 1, 2, 3 and 4 are kept fixed at , , , and , respectively. Strings 1, 2, 3, and 4 are vibrated at their , , , and harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the f

Question

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. [2019]

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

The lengths of the strings 1, 2, 3 and 4 are kept fixed at $L_{0}$ , $\frac{3 L_{0}}{2}$ , $\frac{5 L_{0}}{4}$ , and $\frac{7 L_{0}}{4}$ , 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

Smart Achievers · Accepted Answer

(2)

$As ν = \frac{p}{2 ℓ} \sqrt{\frac{T}{m}}$

$∴$ $T = \frac{ν^{2} ℓ^{2} m}{p^{2}}$

$String-1 T_{0} = \frac{f_{0}^{2} 4 L_{0}^{2} μ}{ℓ^{2}}$

$String-2 T_{2} = \frac{f_{0}^{2} 4 {(\frac{3}{2})}^{2} L_{0}^{2} (2 μ)}{{(3)}^{2}} = \frac{T_{0}}{2}$

$String-3 T_{3} = \frac{f_{0}^{2} 4 {(\frac{5}{2})}^{2} L_{0}^{2} (3 μ)}{5^{2}} = \frac{3}{16} T_{0}$

$String-4 T_{4} = \frac{f_{0}^{2} 4 {(\frac{7}{4})}^{2} L_{0}^{2} (4 μ)}{{(14)}^{2}} = \frac{T_{0}}{16}$

Solution

1 I → T, II → Q, III → R, IV → U

2 I → P, II → Q, III → T, IV → U

3 I → P, II → Q, III → R, IV → T

4 I → P, II → R, III → T, IV → U

Want Us to Email you About Special Offers & Updates?