91Ó°ÊÓ

Understanding the speed of a wave on a string is crucial in many physics problems. The wave speed, denoted by \( v \), is determined by the relationship between the wave's angular frequency \(\omega\) and wave number \(k\). Essentially, wave speed tells us how fast the wave is traveling along the string.

The formula to calculate wave speed is given by:
\[ v = \frac{\omega}{k} \]
From the wave equation \( y(x, t) = (2.0 \, \text{mm}) \sin [(20 \,\text{rad/m}) x - (600 \,\text{rad/s}) t] \), we can extract the values:

• Wave number \( k = 20 \, \text{rad/m} \)
• Angular frequency \( \omega = 600 \, \text{rad/s} \)

Plugging these values into our wave speed formula, we get:
\[ v = \frac{600 \, \text{rad/s}}{20 \, \text{rad/m}} = 30 \, \text{m/s} \] This tells us the wave on the string is traveling at a speed of 30 meters per second.

Linear density is another key concept in understanding waves on a string. It refers to the mass per unit length of the string, which affects how the wave propagates. The symbol for linear density is \( \mu \). Linear density can be found using the formula:
\[ \mu = \frac{T}{v^2} \] where \( T \) is the tension in the string and \( v \) is the wave speed.

Given in our problem, we have:

• Tension \( T = 15 \, \text{N} \)
• Wave speed \( v = 30 \, \text{m/s} \)

Using the formula, we calculate:
\[ \mu = \frac{15 \,\text{N}}{(30 \, \text{m/s})^2} = \frac{15}{900} \text{ kg/m} = 0.0167 \, \text{kg/m} \]
To convert this to grams per meter (g/m):
\[ 0.0167 \, \text{kg/m} = 16.7 \, \text{g/m} \] This means the string's linear density is 16.7 grams per meter.

The wave equation is the fundamental mathematical description of how waves move on a string. Generally, a transverse wave on a string can be described by:
\[ y(x, t) = A \sin(kx - \omega t) \]
Here, \( y(x, t) \) is the displacement at any point \( x \) and time \( t \). The parameters in the equation include:

• Amplitude \( A \): The maximum displacement from the rest position (2.0 mm in our problem).
• Wave number \( k \): Describes the number of wave cycles per unit distance (20 rad/m in our problem).
• Angular frequency \( \omega \): Describes how quickly the wave oscillates in time (600 rad/s in our problem).

Combining these parameters gives a complete description of the wave's behavior in space and time, allowing us to analyze and predict the movement of the wave as it propagates along the string.