Chapter 11, Problem 39P

(a)

To determine

To Find: The speed of the longitudinal waves in water.

Answer to Problem 39P

The speed of the longitudinal waves in water is $1.4\times {10}^3\text{ m/s}$ .

Explanation of Solution

Given:

Elastic modulus of water, $E=2\times {10}^9{\text{N/m}}^{\text{2}}$

Density of water, $\rho =1000{\text{ kg/m}}^{\text{3}}$

Formula used:

The speed of the longitudinal waves in water is obtained by the following formula,

  $\begin{array}{c}v\text{ =}\sqrt{\frac{\text{Elastic modulus}}{\text{Density}}}\\ =\sqrt{\frac{E}{\rho }}\end{array}$

Here,

  $v$ is speed of longitudinal wave,

  $E$ is elastic modulus,

  $\rho$ is density.

Calculation:

Substitute the given values in the above equation,

  $v=\sqrt{\frac{2\times {10}^9}{1000}}=1.4\times {10}^3\text{ m/s}$

Conclusion:

Thus, the speed of the longitudinal waves in water is $1.4\times {10}^3\text{ m/s}$ .

(b)

To determine

To Find: The speed of the longitudinal waves in granite.

Answer to Problem 39P

The speed of the longitudinal waves in granite is $4.1\times {10}^3\text{ m/s}$ .

Explanation of Solution

Given:

Elastic modulus of granite, $E=45\times {10}^9{\text{N/m}}^{\text{2}}$

Density of granite, $\rho =2.7\times {10}^3{\text{ kg/m}}^{\text{3}}$

Formula used:

The speed of the longitudinal waves in granite is obtained by the following formula,

  $\begin{array}{c}v\text{ =}\sqrt{\frac{\text{Elastic modulus}}{\text{Density}}}\\ =\sqrt{\frac{E}{\rho }}\end{array}$

Here,

  $v$ is speed of longitudinal wave,

  $E$ is elastic modulus,

  $\rho$ is density.

Calculation:

Substitute the given values in the above equation,

  $v=\sqrt{\frac{45\times {10}^9}{2.7\times {10}^3}}=4.1\times {10}^3\text{ m/s}$

Conclusion:

Thus, the speed of the longitudinal waves in granite is $4.1\times {10}^3\text{ m/s}$ .

(c)

To determine

To Find: The speed of the longitudinal waves in steel.

Answer to Problem 39P

The speed of the longitudinal waves in steel is $5.1\times {10}^3\text{ m/s}$ .

Explanation of Solution

Given:

Elastic modulus of steel, $E=200\times {10}^9{\text{N/m}}^{\text{2}}$

Density of steel, $\rho =7.8\times {10}^3{\text{ kg/m}}^{\text{3}}$

Formula used:

The speed of the longitudinal waves in steel is obtained by the following formula,

  $\begin{array}{c}v\text{ =}\sqrt{\frac{\text{Elastic modulus}}{\text{Density}}}\\ =\sqrt{\frac{\text{E}}{\rho }}\end{array}$

Here,

  $v$ is speed of longitudinal wave,

  $E$ is elastic modulus,

  $\rho$ is density.

Calculation:

Substitute the given values in the above equation,

  $v=\sqrt{\frac{200\times {10}^9}{7.8\times {10}^3}}=5.1\times {10}^3\text{ m/s}$

Conclusion:

Thus, the speed of the longitudinal waves in steel is $5.1\times {10}^3\text{ m/s}$ .