To solve the problem step by step, we will analyze the given information and apply the relevant physics concepts.
### Given Data:
- Speed of the wave, \( V = 40 \, \text{cm/s} \)
- Displacement equation: \( y = (5.0 \, \text{cm}) \sin(1.0 - 4.0 \, \text{s}^{-1} \cdot t) \)
- Linear density of the string, \( \mu = 4.0 \, \text{g/cm} \)
### Step 1: Identify Parameters from the Wave Equation
The general form of a sinusoidal wave is given by:
\[
y = y_m \sin(kx - \omega t)
\]
From the given equation, we can identify:
- Amplitude, \( y_m = 5.0 \, \text{cm} \)
- Angular frequency, \( \omega = 4.0 \, \text{s}^{-1} \)
### Step 2: Calculate the Wavenumber \( k \)
The displacement equation can be rewritten as:
\[
y = 5.0 \sin(1.0 - 4.0 t)
\]
From this, we can see that \( kx = 1.0 \) when \( t = 0 \). Therefore, we can set \( k = 1.0 \, \text{cm}^{-1} \).
### Step 3: Calculate the Frequency \( f \)
The frequency \( f \) can be calculated using the formula:
\[
f = \frac{\omega}{2\pi}
\]
Substituting the value of \( \omega \):
\[
f = \frac{4.0}{2\pi} \approx 0.64 \, \text{Hz}
\]
### Step 4: Calculate the Wavelength \( \lambda \)
The relationship between wavenumber \( k \) and wavelength \( \lambda \) is given by:
\[
k = \frac{2\pi}{\lambda}
\]
Rearranging gives:
\[
\lambda = \frac{2\pi}{k}
\]
Substituting \( k = 1.0 \, \text{cm}^{-1} \):
\[
\lambda = \frac{2\pi}{1.0} \approx 6.28 \, \text{cm} \approx 63 \, \text{cm}
\]
### Step 5: Calculate the Tension \( T \) in the String
Using the wave speed \( V \) and the linear density \( \mu \), we can find the tension \( T \) using the formula:
\[
V = \sqrt{\frac{T}{\mu}}
\]
Rearranging gives:
\[
T = \mu V^2
\]
First, convert \( \mu \) from grams to kilograms:
\[
\mu = 4.0 \, \text{g/cm} = 0.004 \, \text{kg/cm} = 0.004 \times 100 = 0.4 \, \text{kg/m}
\]
Now substituting \( V = 0.4 \, \text{m/s} \):
\[
T = 0.004 \, \text{kg/cm} \cdot (0.4 \, \text{m/s})^2 = 0.004 \cdot 0.16 = 0.00064 \, \text{N} = 0.064 \, \text{N}
\]
### Step 6: Verify the Wave Equation
The wave equation can also be expressed as:
\[
y = y_m \sin(kx - \omega t)
\]
Substituting \( y_m = 5.0 \, \text{cm} \), \( k = 1.0 \, \text{cm}^{-1} \), and \( \omega = 4.0 \, \text{s}^{-1} \):
\[
y = 5.0 \sin(1.0 x - 4.0 t)
\]
### Summary of Results:
1. Frequency \( f \approx 0.64 \, \text{Hz} \)
2. Wavelength \( \lambda \approx 63 \, \text{cm} \)
3. Tension \( T \approx 0.064 \, \text{N} \)
4. Wave equation \( y = 5.0 \sin(1.0 x - 4.0 t) \)
