Problem 51 - University Physics with Modern Physics

Question

Three pieces of string, each of length \(L\), are joined together end to end, to make a combined string of length 3\(L\). The first piece of string has mass per unit length \(\mu_1\), the second piece has mass per unit length \(\mu2 = 4\mu1\), 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 pieces are joined together? Explain.

Step-by-Step Solution

Answer

(a) \( T = \frac{3.5L \sqrt{\mu_1}}{\sqrt{F}} \). (b) No, the order doesn't affect the time.

Step 1: Find the Wave Speed in Each Segment

The wave speed in a string segment can be calculated using the formula: \( v = \sqrt{\frac{F}{\mu}} \), where \( F \) is the tension and \( \mu \) is the mass per unit length. - For the first piece: \( v_1 = \sqrt{\frac{F}{\mu_1}} \).- For the second piece: \( v_2 = \sqrt{\frac{F}{4\mu_1}} \).- For the third piece: \( v_3 = \sqrt{\frac{F}{\mu_1/4}} = 2\sqrt{\frac{F}{\mu_1}} \).

Step 2: Calculate Time Taken for Wave to Travel Each Segment

The time taken for a wave to travel through each segment is given by \( t = \frac{L}{v} \). - For the first piece: \( t_1 = \frac{L}{v_1} = \frac{L}{\sqrt{\frac{F}{\mu_1}}} = \frac{L \sqrt{\mu_1}}{\sqrt{F}} \).- For the second piece: \( t_2 = \frac{L}{v_2} = \frac{L}{\sqrt{\frac{F}{4\mu_1}}} = \frac{2L \sqrt{\mu_1}}{\sqrt{F}} \).- For the third piece: \( t_3 = \frac{L}{v_3} = \frac{L}{2\sqrt{\frac{F}{\mu_1}}} = \frac{L \sqrt{\mu_1}}{2\sqrt{F}} \).

Step 3: Sum Up the Times for the Entire String

The total time \( T \) for the wave to travel the entire length of the string is the sum of the times for each segment:\[ T = t_1 + t_2 + t_3 = \frac{L \sqrt{\mu_1}}{\sqrt{F}} + \frac{2L \sqrt{\mu_1}}{\sqrt{F}} + \frac{L \sqrt{\mu_1}}{2\sqrt{F}} \].This simplifies to:\[ T = \frac{3.5L \sqrt{\mu_1}}{\sqrt{F}} \].

Step 4: Analyze Dependence on Order of Joining

The time calculation depends only on the sum of the individual times, each determined by tension \( F \) and the specific mass per unit length \( \mu \). As such, the order in which the different pieces are joined does not affect the total time. Each piece contributes a fixed amount to the total time regardless of its position.

Key Concepts

Transverse Waves

Transverse waves are a type of wave where the medium's displacement is perpendicular to the direction of wave propagation. Imagine them like waves on a string, where the string moves up and down while the wave travels horizontally. This concept is vital for understanding the behavior of waves in strings or wires, which are common in physics and engineering problems.

In the exercise, a transverse wave travels through three different segments of string that are connected end to end. Each segment has different mass characteristics, which influences how the wave travels through each part. This is because the wave speed in a medium is influenced by the tension in the string and the string's mass per unit length.

• Wave speed is determined by the tension and mass per unit length: the formula is \( v = \sqrt{\frac{F}{\mu}} \).
• This means that in sections of the string with less mass, waves can travel faster.
• It is crucial to understand that transverse waves are key to the behavior of this particular exercise's problem involving waves on a stretched string.

Mass per Unit Length

Mass per unit length is a concept used to describe how a material's mass is distributed along its length. It is typically denoted as \( \mu \) and plays a critical role in determining the speed at which waves can travel through the medium. In this exercise, understanding mass per unit length is essential.

The waves' velocity in a string is impacted by its mass per unit length, with lighter strings allowing faster wave travel under the same tension. This exercise features three segments of string with different mass densities:

• The first string segment has a mass per unit length of \( \mu_1 \).
• The second segment is heavier, with \( \mu_2 = 4\mu_1 \).
• The third is lighter with \( \mu_3 = \frac{\mu_1}{4} \).

The varying mass per unit length across these segments affects the wave speeds as follows:

• Heavier segments result in slower wave speeds.
• Lighter segments allow waves to travel more quickly.

The combined effects of these differences are important for calculating the total time a wave takes to traverse the entire length of the string.

Wave Travel Time

Wave travel time is the duration it takes for a wave to travel through a specific distance. In the context of strings, this time is affected by the wave speed, which in turn depends on tension and mass per unit length. Calculating wave travel time involves understanding both these elements and how they interact over the given length of the material.

For this exercise:

• The total travel time \( T \) is derived by summing up the times for each segment of the string:
• Each segment's time \( t \) can be calculated using the formula \( t = \frac{L}{v} \), where \( v \) is the wave speed and \( L \) is the segment length.
• As segments have different mass per unit lengths, they each contribute different times to the final result.

The wave travel time is calculated as:
\[ T = \frac{3.5L \sqrt{\mu_1}}{\sqrt{F}} \]
This expression shows that the order of segments does not affect the travel time, meaning each segment's length and mass are the primary time determinants. By understanding how to apply these principles, students can solve similar problems involving complex wave and string dynamics.