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.

	List I		List II
$(a)$	String $- 1 (μ)$	$(p)$	$1$
$(b)$	String $- 2 (2 μ)$	$(q)$	$\frac{1}{2}$
$(c)$	String $- 3 (3 μ)$	$(r)$	$\frac{1}{\sqrt{2}}$
$(d)$	String $- 4 (4 μ)$	$(s)$	$\frac{1}{\sqrt{3}}$
		$(t)$	$\frac{3}{16}$
		$(u)$	$\frac{1}{16}$

The length 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 $1^{s t}, 3^{r d}, 5^{t h}$ and $14^{t h}$ 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.

TestHub · Accepted Answer

Case 1 . L = L 0 , T = T 0 , f = f 0 f 1 = 1 2 L 0 T 0 μ = f 0 (fundamental frequency of vibration in string fixed at both ends) Case 2 . L = 3 L 0 2 , T = T 2 , f 2 = f 0 f 2 = 3 2 × 3 L 0 2 T 2 2 μ = f 0 ⇒ f 0 = 1 2 L 0 T 2 μ ⇒ T 2 = T 0 2 Case 3 . L = 5 L 0 4 , T = T 3 , f 3 = f 0 f 3 = 5 2 × 5 L 0 4 T 3 3 μ = f 0 ⇒ f 0 = 2 3 L 0 T 3 μ ⇒ T 3 = 3 T 0 16 Case 4 . L = 7 L 0 4 ⇒ f 4 = 14 2 × 7 L 0 4 T 4 4 μ = f 0 ⇒ f 0 = 2 L 0 T 4 μ ⇒ T 4 = T 0 16

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