Problem 6 - University Physics with Modern Physics

Question

(a) In a liquid with density \(1300 \mathrm{kg} / \mathrm{m}^{3},\) longitudinal waves with frequency 400 \(\mathrm{Hz}\) are found to have wavelength 8.00 \(\mathrm{m}\) . Calculate the bulk modulus of the liquid. (b) A metal bar with a length of 1.50 \(\mathrm{m}\) has density 6400 \(\mathrm{kg} / \mathrm{m}^{3}\) . Longitudinal sound waves take \(3.90 \times 10^{-4} \mathrm{s}\) to travel from one end of the bar to the other. What is Young's modulus for this metal?

Step-by-Step Solution

Answer

(a) Bulk modulus = \(1.3312 \times 10^{10} \text{ Pa}\); (b) Young's modulus = \(9.44 \times 10^{10} \text{ Pa}\).

Step 1: Write down known values for the liquid

For the liquid, we know:- Density \( \rho = 1300 \text{ kg/m}^3 \)- Frequency \( f = 400 \text{ Hz} \)- Wavelength \( \lambda = 8.00 \text{ m} \)

Step 2: Calculate the speed of sound in the liquid

Use the relationship between speed \( v \), frequency \( f \), and wavelength \( \lambda \):\[ v = f \cdot \lambda \]Substitute in the values:\[ v = 400 \text{ Hz} \times 8.00 \text{ m} = 3200 \text{ m/s} \]

Step 3: Calculate the bulk modulus of the liquid

The speed of sound in a liquid is given by:\[ v = \sqrt{\frac{B}{\rho}} \]Solve for the bulk modulus \( B \):\[ B = \rho \cdot v^2 \]Substitute the known values:\[ B = 1300 \text{ kg/m}^3 \times (3200 \text{ m/s})^2 = 1.3312 \times 10^{10} \text{ Pa} \]

Step 4: Write down known values for the metal bar

For the metal bar, we know:- Density \( \rho = 6400 \text{ kg/m}^3 \)- Length \( L = 1.50 \text{ m} \)- Time \( t = 3.90 \times 10^{-4} \text{ s} \)

Step 5: Calculate the speed of sound in the metal bar

The speed \( v \) is given by the formula:\[ v = \frac{L}{t} \]Substitute in the values:\[ v = \frac{1.50 \text{ m}}{3.90 \times 10^{-4} \text{ s}} = 3846.15 \text{ m/s} \]

Step 6: Calculate Young's modulus for the metal

Speed \( v \) of sound in a metal bar is given by:\[ v = \sqrt{\frac{E}{\rho}} \]Solve for Young's modulus \( E \):\[ E = \rho \cdot v^2 \]Substitute the known values:\[ E = 6400 \text{ kg/m}^3 \times (3846.15 \text{ m/s})^2 = 9.44 \times 10^{10} \text{ Pa} \]

Key Concepts

Understanding Bulk Modulus

Bulk modulus is a measure of a material's resistance to uniform compression. It is defined as the ratio of the applied pressure increase to the relative decrease in volume. The higher the bulk modulus, the more incompressible the material is. It is commonly denoted by 'B'.

For liquids, the speed of sound is related to the bulk modulus and the density of the liquid. This relationship can be expressed with the formula:

• \[ v = \sqrt{\frac{B}{\rho}} \]

This formula tells us that the speed of sound in a liquid depends directly on the bulk modulus and inversely on the density. By rearranging this equation, we can solve for the bulk modulus:

• \[ B = \rho \cdot v^2 \]

In this calculation,

• \( \rho \) represents the density of the liquid,
• \( v \) the speed of sound.

By knowing the speed of sound and the density, one can easily determine the bulk modulus and understand the material's compressibility characteristics.

Young's Modulus Explored

Young's modulus is a fundamental mechanical property that measures a material's stiffness. It is defined as the ratio of tensile stress to tensile strain. Essentially, it tells us how much a material will stretch or compress under a given stress.

Young's modulus is typically applied to solids and is especially useful when analyzing longitudinal wave propagation in materials like metal bars.

• For a metal rod, the speed of sound \( v \) can be related to Young's modulus \( E \) and the density \( \rho \) by:
• \[ v = \sqrt{\frac{E}{\rho}} \]

Here, the direct relationship between the speed of sound and Young's modulus indicates that materials with a high Young's modulus will have faster sound wave propagation, meaning they are stiff. Conversely, a lower Young's modulus suggests a more flexible material.

• Rearranging the formula, you can solve for Young's modulus:
• \[ E = \rho \cdot v^2 \]

This equation highlights how Young's modulus can be calculated when the speed of sound and the density of the material are known. It's a practical way to assess a material's mechanical rigidity.

Speed of Sound in Materials

The speed of sound in any medium is the rate at which a mechanical wave propagates through it. It depends on both the elastic properties and the density of the medium.

Different materials transmit sound at different speeds, determined largely by their physical characteristics. The speed can be calculated using the relation between frequency and wavelength:

• \[ v = f \cdot \lambda \]

Here,

• \( v \) is the speed of sound,
• \( f \) the frequency,
• \( \lambda \) the wavelength.

The above equation shows that for a fixed wavelength, an increase in frequency results in an increase in speed. In materials like liquids and metals, the speed of sound links directly to parameters like the bulk modulus (as in liquids) or Young's modulus (as in solids).

Understanding the speed of sound in a medium is crucial. It helps in fields such as acoustics, material science, and engineering, where sound-related properties are pivotal in designing and analyzing structures and systems.