91Ó°ÊÓ

Linear density, often symbolized as \( \mu \), is a measure of mass per unit length. It is fundamental in wave mechanics, especially for waves traveling along strings or ropes. Calculating linear density involves dividing the mass of an object by its length. For instance, if we have a string that is 2.70 meters long and has a mass of 260 grams, we first convert the mass to kilograms (0.260 kg), then use the formula:
\( \mu = \frac{m}{L} = \frac{0.260 \text{ kg}}{2.70 \text{ m}} = 0.0963 \text{ kg/m} \).
This calculation sets the stage for understanding wave speed and related concepts.

Wave speed, denoted by \( v \), describes how fast a wave travels through a medium. For a string under tension, the wave speed depends on both the tension and the linear density of the string. This is expressed with the formula:
\( v = \sqrt{\frac{T}{\mu}} \)
where \( T \) is the tension and \( \mu \) is the linear density. For a string with tension 36.0 N and linear density 0.0963 kg/m, we can find the wave speed by calculating:
\( v = \sqrt{\frac{36.0 \text{ N}}{0.0963 \text{ kg/m}}} = 19.3 \text{ m/s} \).
Understanding wave speed is essential for determining other wave properties such as frequency and power.

Angular frequency, represented by \( \omega \), measures how rapidly the wave oscillates in radians per second. It is related to the frequency of the wave but gives a different perspective. Angular frequency can be found using the average power formula for sinusoidal waves:
\[ P_{avg} = \frac{1}{2} \mu \omega^2 A^2 v \]
Rearranging to solve for \( \omega \), we get:
\( \omega = \sqrt{ \frac{2 P_{avg}}{ \mu A^2 v }} \). Substituting known values (\( P_{avg} = 85.0 \text{ W} \), \( \mu = 0.0963 \text{ kg/m} \), \( A = 7.70 \times 10^{-3} \text{ m} \), and \( v = 19.3 \text{ m/s} \)):
\( \omega = \sqrt{ \frac{2 \times 85.0 \text{ W}}{0.0963 \text{ kg/m} \times (7.70 \times 10^{-3} \text{ m})^2 \times 19.3 \text{ m/s}}} = 1047.2 \text{ rad/s} \).
This concept bridges frequency and energy in wave dynamics.

The average power of a wave traveling along a string indicates how much energy is transported per second. For sinusoidal waves, the average power \( P_{avg} \) is given by:
\[ P_{avg} = \frac{1}{2} \mu \omega^2 A^2 v \]
This formula encompasses several key wave properties: linear density \( \mu \), angular frequency \( \omega \), amplitude \( A \), and wave speed \( v \). Each term plays a crucial role:

• \( \mu \): dictates mass effect
• \( \omega \): relates to oscillation speed
• \( A \): represents wave height
• \( v \): measures how quickly the wave propagates

Rearranging the formula allows solving for unknown quantities like angular frequency or average power.

Frequency, denoted by \( f \), represents the number of oscillations per second. To relate angular frequency \( \omega \) to frequency, use the formula:
\( f = \frac{\omega}{2\pi} \)
Given \( \omega = 1047.2 \text{ rad/s} \), the frequency is calculated as:
\( f = \frac{1047.2 \text{ rad/s}}{2\pi} \approx 166.8 \text{ Hz} \).
This conversion is essential for understanding how often waves pass through a point per second, thereby giving a complete picture of wave dynamics.