The amplitude of a transverse wave on a string is 4.5 cm. The ratio of the maximum particle speed to the speed of the wave is 3:1. What is the wavelengtlı (in cm) of the wave?

Options

A. 4.5 cm
B. 70 cm
C. 9.1 cm
D. 14 cm

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between maximum particle speed and wave speed
The problem states that the ratio of the maximum particle speed (\(V_{max}\)) to the speed of the wave (\(V_w\)) is 3:1. This can be expressed mathematically as:
\[ \frac{V_{max}}{V_w} = 3 \]
This implies:
\[ V_{max} = 3 V_w \]

### Step 2: Relate maximum particle speed to amplitude and angular frequency
The maximum particle speed for a transverse wave is given by:
\[ V_{max} = A \cdot \omega \]
where \(A\) is the amplitude and \(\omega\) is the angular frequency.

Given that the amplitude \(A = 4.5 \, \text{cm} = 0.045 \, \text{m}\), we can substitute this into the equation:
\[ V_{max} = 0.045 \cdot \omega \]

### Step 3: Relate wave speed to frequency and wavelength
The speed of the wave can be expressed as:
\[ V_w = f \cdot \lambda \]
where \(f\) is the frequency and \(\lambda\) is the wavelength.

### Step 4: Substitute \(V_w\) in terms of \(V_{max}\)
From the relationship \(V_{max} = 3 V_w\), we can express \(V_w\) as:
\[ V_w = \frac{V_{max}}{3} \]

### Step 5: Substitute \(V_w\) in terms of \(A\) and \(\omega\)
Now substituting \(V_w\) into the wave speed equation:
\[ \frac{V_{max}}{3} = f \cdot \lambda \]
Substituting \(V_{max} = A \cdot \omega\):
\[ \frac{A \cdot \omega}{3} = f \cdot \lambda \]

### Step 6: Relate frequency to angular frequency
We know that:
\[ f = \frac{\omega}{2\pi} \]
Substituting this into the equation gives:
\[ \frac{A \cdot \omega}{3} = \left(\frac{\omega}{2\pi}\right) \cdot \lambda \]

### Step 7: Cancel \(\omega\) and solve for \(\lambda\)
Assuming \(\omega \neq 0\), we can cancel \(\omega\) from both sides:
\[ \frac{A}{3} = \frac{\lambda}{2\pi} \]
Now, rearranging gives:
\[ \lambda = \frac{A \cdot 2\pi}{3} \]

### Step 8: Substitute the value of amplitude
Substituting \(A = 4.5 \, \text{cm}\):
\[ \lambda = \frac{4.5 \cdot 2\pi}{3} \]
Calculating this gives:
\[ \lambda = 3\pi \, \text{cm} \]

### Step 9: Calculate \(3\pi\)
Using \(\pi \approx 3.14\):
\[ \lambda \approx 3 \cdot 3.14 = 9.42 \, \text{cm} \]

### Final Answer
Thus, the wavelength of the wave is approximately:
\[ \lambda \approx 9.42 \, \text{cm} \]