Question

In a liquid with density 1300 kg/m3, longitudinal waves with frequency 400 Hz are found to have wavelength 8.00 m. Calculate the bulk modulus of the liquid. (b) A metal bar with a length of 1.50 m has density 6400 kg/m3. Longitudinal sound waves take 3.90 \(\times\) 10\(^{-4}\) s to travel from one end of the bar to the other. What is Young's modulus for this metal?

Short answer

Bulk modulus of the liquid: \(1.3312 \times 10^7 \) Pa. Young's modulus of the metal: \(9.436 \times 10^{10} \) Pa.

Step-by-step solution

Determine the velocity of sound in the liquid

The velocity of sound in a medium is given by the formula \( v = f \cdot \lambda \), where \( f \) is the frequency and \( \lambda \) is the wavelength. For the liquid, the frequency \( f = 400 \) Hz and the wavelength \( \lambda = 8.00 \) m. Substitute these values into the equation:\[v = 400 \text{ Hz} \times 8.00 \text{ m} = 3200 \text{ m/s}\]

Calculate the bulk modulus of the liquid

The bulk modulus \( B \) can be expressed using the formula:\[ B = \rho \cdot v^2 \]where \( \rho \) is the density of the liquid and \( v \) is the velocity of sound in the liquid. We've calculated \( v = 3200 \) m/s and the given density \( \rho = 1300 \) kg/m\(^3\). Now, substitute these values:\[B = 1300 \text{ kg/m}^3 \times (3200 \text{ m/s})^2 = 1.3312 \times 10^7 \text{ Pa}\]

Determine the velocity of sound in the metal bar

The velocity of sound in an object is given by the formula \( v = \frac{d}{t} \), where \( d \) is the distance the sound travels (the length of the bar) and \( t \) is the time taken. Here, \( d = 1.50 \) m and \( t = 3.90 \times 10^{-4} \) s. Substitute these values:\[v = \frac{1.50 \text{ m}}{3.90 \times 10^{-4} \text{ s}} = 3846.15 \text{ m/s}\]

Calculate Young's modulus for the metal

Young's modulus \( E \) is related to the velocity of sound in a solid by the formula:\[ E = \rho \cdot v^2 \]where \( \rho \) is the density of the metal and \( v \) is the velocity of the sound in the metal. We've calculated \( v = 3846.15 \) m/s and the given density \( \rho = 6400 \) kg/m\(^3\). Substitute these values:\[E = 6400 \text{ kg/m}^3 \times (3846.15 \text{ m/s})^2 = 9.436 \times 10^{10} \text{ Pa}\]

Key concepts

Longitudinal Waves

Longitudinal waves are a type of wave where the particle displacement is parallel to the direction of wave propagation. Such waves consist of compressions and rarefactions, much like a slinky moving back and forth as it expands and contracts. A common example of longitudinal waves is sound traveling through air, liquids, or solids.
This type of wave is crucial for understanding the behavior of sound in different media, like how sound travels through a metal bar or a liquid in the context of the given problem. When you know the frequency and the wavelength of a longitudinal wave, you can determine the velocity of sound in that medium using the formula: \( v = f \cdot \lambda \), where \( v \) is the velocity, \( f \) is the frequency, and \( \lambda \) is the wavelength. This relationship helps in evaluating the characteristics of the medium, such as density and compressibility, further connecting to concepts like the Bulk Modulus.

Young's Modulus

Young's Modulus is a measure of the stiffness of a solid material. This elastic modulus is crucial as it quantifies the ability of a material to resist deformation under tensile stress. In the context of the problem, calculating Young's Modulus involves understanding how sound speed and material density relate in a solid object, such as the metal bar.
To find Young's Modulus \( E \), we use the formula:\[ E = \rho \cdot v^2 \]Here, \( \rho \) is the density of the material, and \( v \) is the velocity of sound in the material. Young's Modulus not only reveals the inherent material rigidity but also connects to other physical properties like Poisson’s ratio, which further describes the behavior of materials under various stresses.

• It helps engineers and designers choose materials with appropriate stiffness.
• It is fundamental for calculations in mechanical and structural engineering.

Sound Velocity in Medium

The velocity of sound in a medium plays a critical role in understanding how quickly sound energy is transmitted through different materials. This velocity depends on both the medium's elastic properties and its density.
In the problem, sound velocity was calculated using known parameters of frequency and wavelength for a liquid and considering the length and time it takes for sound to travel through a metal bar. This reinforces the formula for wave speed, \(v = f \cdot \lambda \) for liquid, and \( v = \frac{d}{t} \) for solids.

• The velocity of sound is higher in solids than in liquids and gases, due to tighter molecule packing.
• Understanding sound velocity aids in determining crucial properties like Bulk Modulus and Young's Modulus.
• It is essential for applications in acoustics and materials science, contributing to advances in technology and industry.