The tension in a string with a linear density of 0.0010 kg/m iş 0.40 N. A 100 Hz sinusoidal wave on this string has a wavelength of

Options

A. 0.05cm
B. 2.0cm
C. 5.0cm
D. 20cm

Answer

To solve the problem, we need to find the wavelength of a sinusoidal wave on a string given its tension and linear density. Here’s the step-by-step solution:

### Step 1: Identify the given values
- Linear density (μ) = 0.0010 kg/m
- Tension (T) = 0.40 N
- Frequency (f) = 100 Hz

### Step 2: Use the wave speed formula
The speed (v) of a wave on a string can be calculated using the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where:
- \( T \) is the tension in the string,
- \( \mu \) is the linear density of the string.

### Step 3: Calculate the wave speed (v)
Substituting the values into the formula:
\[ v = \sqrt{\frac{0.40 \, \text{N}}{0.0010 \, \text{kg/m}}} \]
Calculating the denominator:
\[ v = \sqrt{400} = 20 \, \text{m/s} \]

### Step 4: Relate wave speed to frequency and wavelength
The relationship between wave speed (v), frequency (f), and wavelength (λ) is given by:
\[ v = f \cdot \lambda \]
We can rearrange this to find the wavelength:
\[ \lambda = \frac{v}{f} \]

### Step 5: Substitute the values to find wavelength (λ)
Substituting the values of v and f:
\[ \lambda = \frac{20 \, \text{m/s}}{100 \, \text{Hz}} = 0.20 \, \text{m} \]

### Step 6: Convert the wavelength to centimeters
To convert meters to centimeters, we multiply by 100:
\[ \lambda = 0.20 \, \text{m} \times 100 = 20 \, \text{cm} \]

### Final Answer
The wavelength of the sinusoidal wave on the string is **20 cm**.

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