Answer
(a) $T = \frac{7L}{2}~\sqrt{\frac{\mu_1}{F}}$
(b) The total time does not depend on the order in which the pieces are joined together.
Work Step by Step
(a) We can find the time it takes for the wave to travel along the first piece of string.
$t_1 = \frac{L}{v_1}$
$t_1 = \frac{L}{\sqrt{\frac{F}{\mu_1}}}$
$t_1 = \sqrt{\frac{L^2~\mu_1}{F}}$
We can find the time it takes for the wave to travel along the second piece of string.
$t_2 = \frac{L}{v_2}$
$t_2 = \frac{L}{\sqrt{\frac{F}{\mu_2}}}$
$t_2 = \sqrt{\frac{L^2~\mu_2}{F}}$
$t_2 = \sqrt{\frac{L^2~(4\mu_1)}{F}}$
$t_2 = 2\sqrt{\frac{L^2~\mu_1}{F}}$
We can find the time it takes for the wave to travel along the third piece of string.
$t_3 = \frac{L}{v_3}$
$t_3 = \frac{L}{\sqrt{\frac{F}{\mu_3}}}$
$t_3 = \sqrt{\frac{L^2~\mu_3}{F}}$
$t_3 = \sqrt{\frac{L^2~(\mu_1/4)}{F}}$
$t_3 = \frac{1}{2}~\sqrt{\frac{L^2~\mu_1}{F}}$
We can find the total time $T$.
$T = t_1+t_2+t_3$
$T = \sqrt{\frac{L^2~\mu_1}{F}}+2~\sqrt{\frac{L^2~\mu_1}{F}}+\frac{1}{2}~\sqrt{\frac{L^2~\mu_1}{F}}$
$T = \frac{7}{2}~\sqrt{\frac{L^2~\mu_1}{F}}$
$T = \frac{7L}{2}~\sqrt{\frac{\mu_1}{F}}$
(b) The total time does not depend on the order that the pieces are joined together. To find the total time, we simply add the time for the wave to travel along each of the three sections. The order that three numbers are added does not change the final sum.
