A travelling wave is given by `y = - 0.21 sin (x + 3t)` where x is in m, t is in seconds, y is in mm. (mass per unit length `0.135 g/cm`). Find the tension in the wire.

Options

A. `0.1215 N`
B. `20 N`
C. `15.35 N`
D. `5 N`

Answer

To find the tension in the wire given the wave equation \( y = -0.21 \sin(x + 3t) \) and the mass per unit length, we can follow these steps:

### Step-by-Step Solution

1. **Identify Parameters from the Wave Equation**:
The wave equation is given as \( y = -0.21 \sin(x + 3t) \).
- Here, the angular frequency \( \omega \) can be identified as \( 3 \, \text{rad/s} \) (coefficient of \( t \)).
- The wave number \( k \) can be identified as \( 1 \, \text{rad/m} \) (coefficient of \( x \)).

2. **Calculate Wave Speed**:
The wave speed \( v \) is given by the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{3}{1} = 3 \, \text{m/s} \]

3. **Convert Mass per Unit Length**:
The mass per unit length \( \mu \) is given as \( 0.135 \, \text{g/cm} \).
To convert this to \( \text{kg/m} \):
\[ \mu = 0.135 \, \text{g/cm} = 0.135 \times \frac{1 \, \text{kg}}{1000 \, \text{g}} \times \frac{100 \, \text{cm}}{1 \, \text{m}} = 0.135 \times 0.01 = 0.00135 \, \text{kg/m} \]

4. **Use the Wave Speed Formula**:
The relationship between wave speed, tension \( T \), and mass per unit length \( \mu \) is given by:
\[ v = \sqrt{\frac{T}{\mu}} \]
Squaring both sides gives:
\[ v^2 = \frac{T}{\mu} \]
Rearranging for tension \( T \):
\[ T = v^2 \cdot \mu \]

5. **Substituting Values**:
Now substituting the values of \( v \) and \( \mu \):
\[ T = (3 \, \text{m/s})^2 \cdot 0.00135 \, \text{kg/m} \]
\[ T = 9 \cdot 0.00135 = 0.01215 \, \text{N} \]

6. **Final Answer**:
Therefore, the tension in the wire is:
\[ T = 0.01215 \, \text{N} \]