A transversee wave propagating on the string can be described by the equation` y= 2 sin (10 x + 300 t) ` , where x and y are in metres and t in second. If the vibrating string has linear density of ` 0.6 xx 10^(-3) g//cm` , then the tension in the string is

Options

A. 5.4N
B. 0.054 N
C. 54 N
D. 0.0054 N

Answer

To find the tension in the string described by the wave equation \( y = 2 \sin(10x + 300t) \), we can follow these steps:

### Step 1: Identify parameters from the wave equation
The given wave equation is:
\[ y = 2 \sin(10x + 300t) \]

From this equation, we can identify:
- Amplitude \( A = 2 \) m
- Wave number \( k = 10 \) rad/m
- Angular frequency \( \omega = 300 \) rad/s

### Step 2: Calculate the wave speed
The speed \( V \) of the wave on the string can be calculated using the relationship:
\[ V = \frac{\omega}{k} \]

Substituting the values of \( \omega \) and \( k \):
\[ V = \frac{300}{10} = 30 \text{ m/s} \]

### Step 3: Convert linear density to appropriate units
The linear density \( \mu \) is given as:
\[ \mu = 0.6 \times 10^{-3} \text{ g/cm} \]

To convert this to kg/m, we use the conversion factor \( 1 \text{ g/cm} = 0.01 \text{ kg/m} \):
\[ \mu = 0.6 \times 10^{-3} \text{ g/cm} \times 0.01 = 0.6 \times 10^{-4} \text{ kg/m} \]

### Step 4: Calculate the tension in the string
The tension \( T \) in the string can be calculated using the formula:
\[ T = \mu V^2 \]

Substituting the values of \( \mu \) and \( V \):
\[ T = (0.6 \times 10^{-4}) \times (30)^2 \]
\[ T = (0.6 \times 10^{-4}) \times 900 \]
\[ T = 540 \times 10^{-4} \text{ N} \]

### Step 5: Convert tension to a more standard form
To express the tension in a more standard form:
\[ T = 0.0540 \text{ N} \]

### Final Answer
The tension in the string is:
\[ T = 0.054 \text{ N} \]

---