The linear density of a vibrating string is `10^(-4) kg//m`. A transverse wave is propagating on the string, which is described by the equation `y=0.02 sin (x+30t)`, where x and y are in metres and time t in seconds. Then tension in the string is

Options

A. 0.09 N
B. 0.36 N
C. 0.9 N
D. 3.6 N

Explanation

To find the tension in the string, we will follow these steps:

### Step 1: Identify the given parameters
- Linear density (μ) of the string: \( \mu = 10^{-4} \, \text{kg/m} \)
- Wave equation: \( y = 0.02 \sin(x + 30t) \)

### Step 2: Determine the angular frequency (ω) and wave number (k)
From the wave equation \( y = A \sin(kx + \omega t) \):
- Amplitude (A) = 0.02 m
- Angular frequency (ω) = 30 rad/s
- Wave number (k) can be identified from the equation. Since the argument of sine is \( (x + 30t) \), we can see that \( k = 1 \, \text{rad/m} \).

### Step 3: Calculate the wave velocity (v)
The wave velocity (v) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{30 \, \text{rad/s}}{1 \, \text{rad/m}} = 30 \, \text{m/s} \]

### Step 4: Calculate the tension (T) in the string
The tension in the string can be calculated using the formula:
\[ T = \mu v^2 \]
Substituting the values of μ and v:
\[ T = (10^{-4} \, \text{kg/m}) \cdot (30 \, \text{m/s})^2 \]
Calculating \( (30 \, \text{m/s})^2 \):
\[ (30 \, \text{m/s})^2 = 900 \, \text{m}^2/\text{s}^2 \]
Now substituting this back into the tension formula:
\[ T = 10^{-4} \cdot 900 = 0.09 \, \text{N} \]

### Final Answer
The tension in the string is \( T = 0.09 \, \text{N} \).

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