For a solid rod,the Young's modulus of elasticity is `3.2times10^(11)Nm^(-2)` and density is `8times10^(3) kg m^(-3)`. The velocity of longitudinal wave in the rod will be

Options

A. `3.65times10^(3)ms^(-1)`
B. `18.96times10^(3)ms^(-1)`
C. `145.75times10^(3)ms^(-1)`
D. `6.32times10^(3)ms^(-1)`

Answer

To find the velocity of longitudinal waves in a solid rod, we can use the formula:

\[ V = \sqrt{\frac{Y}{\rho}} \]

where:
- \( V \) is the velocity of the longitudinal wave,
- \( Y \) is the Young's modulus of elasticity,
- \( \rho \) is the density of the material.

### Step 1: Identify the given values
From the problem statement, we have:
- Young's modulus \( Y = 3.2 \times 10^{11} \, \text{N/m}^2 \)
- Density \( \rho = 8 \times 10^{3} \, \text{kg/m}^3 \)

### Step 2: Substitute the values into the formula
Substituting the values into the formula for the velocity of longitudinal waves:

\[ V = \sqrt{\frac{3.2 \times 10^{11}}{8 \times 10^{3}}} \]

### Step 3: Calculate the fraction inside the square root
First, calculate the fraction:

\[ \frac{3.2 \times 10^{11}}{8 \times 10^{3}} = \frac{3.2}{8} \times 10^{11 - 3} = 0.4 \times 10^{8} = 4 \times 10^{7} \]

### Step 4: Take the square root
Now, take the square root of the result:

\[ V = \sqrt{4 \times 10^{7}} = 2 \times 10^{3} \, \text{m/s} \]

### Final Result
Thus, the velocity of the longitudinal wave in the rod is:

\[ V = 2000 \, \text{m/s} \]