Three pieces of string, each of length \(L\), are joined together end to end, to make a combined string of length \(3L\). The has mass per unit length \({\mu _2} = 4{\mu _1}\), and the third piece has mass per unit length \({\mu _3} = \frac{{{\mu _1}}}{4}\). (a) If the combined string is under tension \(F\), how much time does it take a transverse wave to travel the entire length \(3L\)? Give your answer in terms of \(L,F,\;{\rm{and}}\;{\mu _1}\).

(b) Does your answer to part (a) depend on the order in which the three pieces are joined together? Explain.

Length of each string \(L\).

The mass per unit length \({\mu _2} = 4{\mu _1}\).

The third piece mass per unit length \({\mu _3} = \frac{{{\mu _1}}}{4}\).

The speed of transverse waves on a string depends on the tension and mass per unit length. It is given by:

\(v = \sqrt {\frac{F}{\mu }} \)

Here, \(F\) is tension, \(\mu \) is linear density and \(v = xt\).

According to the question,

\(\begin{array}{c}\frac{x}{t} = \sqrt {\frac{F}{\mu }} \\t = x\sqrt {\frac{\mu }{F}} \end{array}\)

This is the time it takes for the wave to travel on string of length \(x\).

Use the above formula and solve as follows:

Time takes solve as:

\({t_1} = L\sqrt {\frac{{{\mu _1}}}{F}} \)

Time takes solve as:

\(\begin{array}{c}{t_2} = L\sqrt {\frac{{{\mu _2}}}{F}} \\ = L\sqrt {\frac{{4{\mu _1}}}{F}} \\ = 2L\sqrt {\frac{{{\mu _1}}}{F}} \end{array}\)

And time takes solve as:

\(\begin{array}{c}{t_3} = L\sqrt {\frac{{{\mu _2}}}{F}} \\ = L\sqrt {\frac{{\frac{{{\mu _1}}}{4}}}{F}} \\ = \frac{L}{2}\sqrt {\frac{{{\mu _1}}}{F}} \end{array}\)

Total time takes is calculated as:

\(\begin{array}{c}t = {t_1} + {t_2} + {t_3}\\ = L\sqrt {\frac{{{\mu _1}}}{F}} + 2L\sqrt {\frac{{{\mu _1}}}{F}} + \frac{L}{2}\sqrt {\frac{{{\mu _1}}}{F}} \\ = \frac{{7L}}{2}\sqrt {\frac{{{\mu _1}}}{F}} \end{array}\)

Hence, the total time takes is \(\frac{{7L}}{2}\sqrt {\frac{{{\mu _1}}}{F}} \).