Problem 99 - Physics

Question

Review Conceptual Example 3 before starting this problem. A horizontal wire is under a tension of \(315 \mathrm{~N}\) and has a mass per unit length of \(6.50 \times 10^{-3} \mathrm{~kg} / \mathrm{m} .\) A transverse wave with an amplitude of \(2.50 \mathrm{~mm}\) and a frequency of \(585 \mathrm{~Hz}\) is traveling on this wire. As the wave passes, a particle of the wire moves up and down in simple harmonic motion. Obtain (a) the speed of the wave and (b) the maximum speed with which the particle moves up and down.

Step-by-Step Solution

Answer

Wave speed is 220 m/s; maximum particle speed is 9.19 m/s.

Step 1: Identify the known values for wave speed

To find the speed of a wave on a string, we need two known values: the tension in the string and the mass per unit length of the string. Here we have:- Tension, \( T = 315 \, \text{N} \)- Mass per unit length, \( \mu = 6.50 \times 10^{-3} \, \text{kg/m} \)

Step 2: Use the wave speed formula

The formula to find the speed \( v \) of a wave on a string is given by:\[v = \sqrt{\frac{T}{\mu}}\]Substitute the known values into the formula:\[ v = \sqrt{\frac{315 \, \text{N}}{6.50 \times 10^{-3} \, \text{kg/m}}} \]

Step 3: Calculate the wave speed

Calculate the expression to find the speed:\[ v = \sqrt{\frac{315}{6.50 \times 10^{-3}}} \]\[ v = \sqrt{48461.54} \]\[ v \approx 220 \, \text{m/s} \]

Step 4: Identify the known values for maximum particle speed

For a particle on the wire moving in simple harmonic motion, we need:- Amplitude \( A = 2.50 \, \text{mm} = 2.50 \times 10^{-3} \, \text{m} \) (convert to meters)- Frequency \( f = 585 \, \text{Hz} \)

Step 5: Use the formula for maximum particle speed

The maximum speed \( v_{max} \) of a particle in simple harmonic motion is given by:\[v_{max} = 2 \pi f A\]Substitute the known values into the formula:\[ v_{max} = 2 \pi (585) (2.50 \times 10^{-3}) \]

Step 6: Calculate the maximum particle speed

Calculate the expression to find the maximum speed:\[ v_{max} = 2 \pi \times 585 \times 2.50 \times 10^{-3} \]\[ v_{max} = 9.19 \, \text{m/s} \]

Key Concepts

Simple Harmonic Motion

When a particle of the wire moves up and down as a wave passes, it does so in simple harmonic motion (SHM). This type of motion is characterized by repeated oscillations that are sinusoidal in nature. It's a very predictable pattern of movement which repeats over and over. The particle reaches its highest point, falls back down, and moves back up, forming a continuous cycle.
In SHM, the maximum speed of a particle can be calculated using the formula \( v_{max} = 2 \pi f A \), where:\

• \( f \) is the frequency of the wave
• \( A \) is the amplitude, which is the maximum distance the particle moves from its rest position

Here, frequency tells us how many cycles happen per second, while amplitude tells us how far the particle travels each time. With these values, we can easily determine how fast the particle moves as it follows its oscillating path.

Transverse Wave

A transverse wave is a wave in which particles of the medium move perpendicular to the direction of the wave. Consider the string on which this type of wave travels:

• The wave moves horizontally along the string,
• while the particles of the string move up and down vertically.

This is a key characteristic of transverse waves. Examples in everyday life include water waves and light waves. For our exercise, the wave showcasing simple harmonic motion on the string is transverse.
The wave speed formula in the context of a transverse wave on a string is \( v = \sqrt{\frac{T}{\mu}} \). Here, the speed is determined by both string tension and mass per unit length, concepts that we will explore further in separate sections.

String Tension

String tension is the pulling force exerted by the string when it is stretched. It is a critical factor in determining the speed of a wave traveling along the string.
The greater the tension, the faster the wave can travel. This is because more force allows the wave to move more quickly along the medium. In our exercise, the tension of the string is given as 315 N. The higher this tension, the higher the wave speed, according to the formula for wave speed \( v = \sqrt{\frac{T}{\mu}} \).
Tension plays a vital role in ensuring the wave moves swiftly and is essential to understanding how transverse waves behave on a string.

Mass Per Unit Length

Mass per unit length, symbolized by \( \mu \), is another crucial component for computing wave speed on a string. It is essentially the mass of the string divided by its length, providing a measure of how densely packed the mass is along the string.
In our case, the mass per unit length is \( 6.50 \times 10^{-3} \text{ kg/m} \). This value affects the speed of the wave on the string. A larger mass per unit length will lead to a slower wave speed, because more mass requires more force to move.
Combining the tension in the string and its mass per unit length, you can determine the speed of a wave traveling on it using the formula \( v = \sqrt{\frac{T}{\mu}} \). Together, these factors influence how energy is transmitted along the string.