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A composite string is made by joining a thin and a thick uniform string joined end to end (series) and it is held fixed horizontally between two rigid walls with some tension. V is the velocity of a transverse wave in thin string. Lengths, radii of cross section and densities of materials for the two strings are 2l, r, $ρ$ and l, 2r, $4 ρ$ respectively. If a small transverse wave pulse is generated at the fixed end of the thin string, the time it takes to reach the other end of composite string is

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$\frac{2 l}{3 V}$

$\frac{6 l}{V}$

$\frac{3 l}{4 V}$

$\frac{4 l}{3 V}$

answer is A.

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Detailed Solution

velocity of a wave in a string is V = $\sqrt{\frac{T}{{πr}^{2} ρ}}$

so $\frac{V_{1}}{V_{2}} = \frac{r_{2}}{r_{1}} \sqrt{\frac{ρ_{2}}{ρ_{1}}} = \frac{2 r}{r} \frac{2}{1} = 4$

$V_{1} = V and V_{2} = \frac{V}{4}$

time taken $\frac{2 l}{V} + \frac{l}{V / 4} = \frac{6 l}{V}$

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