Three pieces of string, each of length L, are joined together end-to-end, to make a combined string of length 3L. The first piece of string has mass per unit length `mu_1`, the second piece has mass per unit length `mu_2 = 4mu_1` and the third piece has mass per unit length `mu_3 = 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 and `mu_1` .
(b) Does your answer to part (a) depend on the order in which the three piece are joined together? Explain.

Explanation

To solve the problem step-by-step, we will analyze the situation of the three pieces of string joined together and calculate the time taken for a transverse wave to travel the entire length of the combined string.

### Step-by-Step Solution

**Given:**
- Length of each string piece = \( L \)
- Total length of combined string = \( 3L \)
- Mass per unit length of the first piece = \( \mu_1 \)
- Mass per unit length of the second piece = \( \mu_2 = 4\mu_1 \)
- Mass per unit length of the third piece = \( \mu_3 = \frac{\mu_1}{4} \)
- Tension in the string = \( F \)

### Part (a): Time taken for a transverse wave to travel the entire length \( 3L \)

1. **Wave Speed Formula:**
The speed of a transverse wave on a string is given by:
\[ v = \sqrt{\frac{F}{\mu}} \]
where \( F \) is the tension and \( \mu \) is the mass per unit length.

2. **Calculate Wave Speeds for Each Piece:**
- For the first piece:
\[ v_1 = \sqrt{\frac{F}{\mu_1}} \]
- For the second piece:
\[ v_2 = \sqrt{\frac{F}{4\mu_1}} = \frac{1}{2} \sqrt{\frac{F}{\mu_1}} = \frac{v_1}{2} \]
- For the third piece:
\[ v_3 = \sqrt{\frac{F}{\frac{\mu_1}{4}}} = 2\sqrt{\frac{F}{\mu_1}} = 2v_1 \]

3. **Calculate Time Taken for Each Piece:**
The time taken for a wave to travel a distance \( d \) is given by:
\[ t = \frac{d}{v} \]
- For the first piece (length \( L \)):
\[ t_1 = \frac{L}{v_1} \]
- For the second piece (length \( L \)):
\[ t_2 = \frac{L}{v_2} = \frac{L}{\frac{v_1}{2}} = \frac{2L}{v_1} \]
- For the third piece (length \( L \)):
\[ t_3 = \frac{L}{v_3} = \frac{L}{2v_1} = \frac{L}{2v_1} \]

4. **Total Time Taken:**
The total time taken for the wave to travel the entire length \( 3L \) is:
\[ T_{\text{total}} = t_1 + t_2 + t_3 = \frac{L}{v_1} + \frac{2L}{v_1} + \frac{L}{2v_1} \]
To simplify:
\[ T_{\text{total}} = \left(1 + 2 + \frac{1}{2}\right) \frac{L}{v_1} = \frac{7L}{2v_1} \]
Substituting \( v_1 = \sqrt{\frac{F}{\mu_1}} \):
\[ T_{\text{total}} = \frac{7L}{2\sqrt{\frac{F}{\mu_1}}} = \frac{7L \sqrt{\mu_1}}{2\sqrt{F}} \]

### Part (b): Dependence on the order of joining the pieces

The time taken for the wave to travel the entire length of the string does not depend on the order in which the pieces are joined. This is because the total time is a function of the individual lengths and their respective mass densities. The wave speed is determined by the tension and mass per unit length of each segment, and since these properties are intrinsic to each piece, the total time remains the same regardless of the arrangement.

### Final Answers:
(a) The time taken for a transverse wave to travel the entire length \( 3L \) is:
\[ T_{\text{total}} = \frac{7L \sqrt{\mu_1}}{2\sqrt{F}} \]

(b) The answer does not depend on the order in which the three pieces are joined together.