A string of length L consists of two distinct sections. The left half has linear mass density $μ_{1} = μ_{0} / 2$ , while the right half has linear mass density $μ_{2} = 3 μ_{0}$ . Tension in the string is F₀. The time required for a transverse wave pulse to travel from one end of the string to the other is

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$\frac{L}{2} \sqrt{\frac{μ_{o}}{2 F_{o}}} (\sqrt{2} + \sqrt{6})$

$\frac{L}{2} \sqrt{\frac{2 μ_{o}}{F_{o}}} (1 + \sqrt{3})$

$\frac{L}{4} \sqrt{\frac{μ_{o}}{F_{o}}} (\sqrt{2} + \sqrt{6})$

$\frac{L}{2} \sqrt{\frac{μ_{o}}{2 F_{o}}} (1 + \sqrt{6})$

answer is D.

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Detailed Solution

Total time taken $t = t_{1} + t_{2} \Rightarrow t = \frac{L}{2 V_{1}} + \frac{L}{2 V_{2}}$
$\begin{array}{l} \Rightarrow t = \frac{L}{2} [\sqrt{\frac{μ_{0}}{2 F_{0}}} + \sqrt{\frac{3 μ_{0}}{F_{0}}}] \\ \Rightarrow t = \frac{L}{2} \sqrt{\frac{μ_{0}}{2 F_{0}}} [1 + \sqrt{6}] \end{array}$