A wire vibrates in fundamental mode with frequency of 50 Hz when stretched between two rigid supports. The linear mass density of wire is `3 xx 10^(-2)` kg/m and mass of wire is `4.5 xx 10^(-2)` kg. The tension developed in wire is `x xx 135` N. Find value of x.
`{:(0,1,2,3,4,5,6,7,8,9),(,,,,,,,,,):}`

Explanation

To solve the problem step by step, we will use the given data and relevant formulas.

### Step 1: Identify the given data
- Frequency (f) = 50 Hz
- Linear mass density (μ) = \(3 \times 10^{-2}\) kg/m
- Mass of the wire (m) = \(4.5 \times 10^{-2}\) kg

### Step 2: Calculate the length of the wire
Using the formula for linear mass density:
\[ \mu = \frac{m}{L} \]
We can rearrange this to find the length (L):
\[ L = \frac{m}{\mu} \]
Substituting the values:
\[ L = \frac{4.5 \times 10^{-2}}{3 \times 10^{-2}} = 1.5 \text{ m} \]

### Step 3: Use the formula for frequency in terms of tension
The formula for frequency in the fundamental mode is given by:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Squaring both sides gives:
\[ f^2 = \frac{1}{4L^2} \frac{T}{\mu} \]
Rearranging to find tension (T):
\[ T = 4L^2 \mu f^2 \]

### Step 4: Substitute the known values into the tension formula
Substituting \(L = 1.5\) m, \(\mu = 3 \times 10^{-2}\) kg/m, and \(f = 50\) Hz:
\[ T = 4 \times (1.5)^2 \times (3 \times 10^{-2}) \times (50)^2 \]

### Step 5: Calculate the components
Calculating \( (1.5)^2 \):
\[ (1.5)^2 = 2.25 \]
Calculating \( (50)^2 \):
\[ (50)^2 = 2500 \]
Now substituting these values back into the tension formula:
\[ T = 4 \times 2.25 \times (3 \times 10^{-2}) \times 2500 \]

### Step 6: Calculate the tension
Calculating the product:
\[ T = 4 \times 2.25 \times 3 \times 10^{-2} \times 2500 \]
Calculating \(4 \times 2.25 = 9\):
\[ T = 9 \times (3 \times 10^{-2}) \times 2500 \]
Calculating \(3 \times 2500 = 7500\):
\[ T = 9 \times 7500 \times 10^{-2} \]
Calculating \(9 \times 7500 = 67500\):
\[ T = 67500 \times 10^{-2} = 675 \text{ N} \]

### Step 7: Express tension in the required format
We need to express \(T\) in the form \(x \times 135\) N:
\[ 675 = x \times 135 \]
To find \(x\):
\[ x = \frac{675}{135} = 5 \]

### Final Answer
The value of \(x\) is \(5\).

---