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(A) Let the mass of the string be $m$ and its length be $l$.The linear mass density of the string is $\mu = \frac{m}{l}$.The tension at any point in a

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$\sqrt{3}v_A$

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(A) Let the mass of the string be $m$ and its length be $l$.The linear mass density of the string is $\mu = \frac{m}{l}$.The tension at any point in a hanging string is equal to the weight of the portion of the string below that point.At point $A$,the length of the string below it is $\frac{l}{4}$. Thus,the mass below $A$ is $m_A = \mu 	imes \frac{l}{4} = \frac{m}{4}$.The tension at $A$ is $T_A = m_A g = \frac{mg}{4}$.The speed of the wave at $A$ is $v_A = \sqrt{\frac{T_A}{\mu}} = \sqrt{\frac{mg/4}{m/l}} = \sqrt{\frac{gl}{4}}$.At point $B$,the length of the string below it is $\frac{3l}{4}$. Thus,the mass below $B$ is $m_B = \mu 	imes \frac{3l}{4} = \frac{3m}{4}$.The tension at $B$ is $T_B = m_B g = \frac{3mg}{4}$.The speed of the wave at $B$ is $v_B = \sqrt{\frac{T_B}{\mu}} = \sqrt{\frac{3mg/4}{m/l}} = \sqrt{\frac{3gl}{4}}$.Comparing $v_B$ and $v_A$,we get $v_B = \sqrt{3} 	imes \sqrt{\frac{gl}{4}} = \sqrt{3} v_A$.

List-$I$	List-$II$
$(1)$ Planck's constant	$(i)$ $[ML^{-1} T^{-2}]$
$(2)$ Gravitational constant	(ii) $[ML^{-1} T^{-1}]$
$(3)$ Bulk modulus	(iii) $[ML^2 T^{-1}]$
$(4)$ Coefficient of viscosity	(iv) $[M^{-1} L^3 T^{-2}]$

$A$ string of mass $m$ and length $l$ hangs from the ceiling as shown in the figure. $A$ wave in the string moves upward. $v_A$ and $v_B$ are the speeds of the wave at points $A$ and $B$ respectively. Then $v_B$ is

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