In the figure below two strings of different linear mass density μ 1 and μ 2 and length ℓ 1 and ℓ 2 are connected to each other. The strings pass over ideal pulley and connected to weight "W".

Question

In the figure below two strings of different linear mass density $μ_{1}$ and $μ_{2}$ and length $ℓ_{1}$ and $ℓ_{2}$ are connected to each other. The strings pass over ideal pulley and connected to weight "W". When excited with a tuning fork the string vibrates such that the joint between the two wires does not move at all. If $N_{1}$ and $N_{2}$ are the order of harmonic in the first and second wire respectively, match the entries in List-I and List-II.

List - I	List - II
(I) $\frac{μ _{2}}{μ _{1}} = 4 : 1$	(P) $ℓ_{1} = 3 ℓ_{2}, N_{1} : N_{2} = 1 : 1$
(II) $\frac{μ _{1}}{μ _{2}} = 1 : 9$	(Q) $ℓ_{1} = ℓ_{2}, N_{1} : N_{2} = 1 : 2$
(III) $\frac{μ _{2}}{μ _{1}} = 1 : 4$	(R) $6 ℓ_{1} = ℓ_{2}, N_{1} : N_{2} = 1 : 3$
(IV) $\frac{μ _{1}}{μ _{2}} = 4 : 9$	(S) $ℓ_{1} = 3 ℓ_{2}, N_{1} : N_{2} = 3 : 2$
	(T) $2 ℓ_{1} = ℓ_{2}, N_{1} : N_{2} = 1 : 3$

TestHub · Accepted Answer

​ y ∝ v ∝ μ ​ 1 ​ ​ λ 2 ​ λ 1 ​ ​ = μ 1 ​ μ 2 ​ ​ ​ ​ N 1 ​ 2 λ 1 ​ ​ = ℓ 1 ​ , N 2 ​ 2 λ 2 ​ ​ = ℓ 2 ​ ​ N 2 ​ N 1 ​ ​ × λ 2 ​ λ 1 ​ ​ = ℓ 2 ​ ℓ 1 ​ ​ ​ For (I) N 2 ​ N 1 ​ ​ × 1 2 ​ = ℓ 2 ​ ℓ 1 ​ ​ N 2 ​ N 1 ​ ​ = 2 ℓ 2 ​ ℓ 1 ​ ​ So, QS For (II) N 2 ​ N 1 ​ ​ × 1 3 ​ = ℓ 2 ​ ℓ 1 ​ ​ N 2 ​ N 1 ​ ​ = 3 ℓ 2 ​ ℓ 1 ​ ​ So, P For (III) N 2 ​ N 1 ​ ​ × 2 1 ​ = ℓ 2 ​ ℓ 1 ​ ​ N 2 ​ N 1 ​ ​ = ℓ 2 ​ 2 ℓ 1 ​ ​ So, R For (IV) N 2 ​ N 1 ​ ​ × 2 3 ​ = ℓ 2 ​ ℓ 1 ​ ​ N 2 ​ N 1 ​ ​ = 3 ℓ 2 ​ 2 ℓ 1 ​ ​ So, T

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