JEE Advanced · Physics · 16. Waves & Sound

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 $- 4 (4 μ)$ $(s)$ $\frac{1}{\sqrt{3}}$

$(t)$ $\frac{3}{16}$

$(u)$ $\frac{1}{16}$

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

	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 $- 4 (4 μ)$	$(s)$	$\frac{1}{\sqrt{3}}$
		$(t)$	$\frac{3}{16}$
		$(u)$	$\frac{1}{16}$

A $a - p, b - r, c - s, d - q$
B $a - p, b - q, c - t, d - s$
C $a - q, b - s, c - r, d - p$
D $a - q, b - p, c - r, d - t$

Verified Solution

Answer & Solution

Correct Answer

(A) $a - p, b - r, c - s, d - q$

Step-by-step Solution

Detailed explanation

Fundamental frequency is maximum when length is minimum i.e.

L_{0},

Case

1 . L = L_{0}, T = T_{0}, f = f_{0}; f_{1} = \frac{1}{2 L_{0}} \sqrt{\frac{T_{0}}{μ}} = f_{0}

(fundamental frequency of vibration in string fixed at both ends)
Case

2 . f_{2} = \frac{1}{L_{0}} \sqrt{\frac{T_{2}}{2 μ}} = \frac{f_{0}}{\sqrt{2}}

Case

3 . f_{3} = \frac{1}{L_{0}} \sqrt{\frac{T_{2}}{3 μ}} = \frac{f_{0}}{\sqrt{3}}

Case

4 . f_{4} = \frac{1}{L_{0}} \sqrt{\frac{T_{2}}{4 μ}} = \frac{f_{0}}{2}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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