Question

A metal bar has a Young's modulus of \(112.1 \cdot 10^{9} \mathrm{~N} / \mathrm{m}^{2} .\) The speed of sound in the bar is \(5628 \mathrm{~m} / \mathrm{s}\). What is the mass density of this bar?

Short answer

Answer: The mass density of the metal bar is approximately \(3537.38 \mathrm{~kg} / \mathrm{m}^3\).

Step-by-step solution

Rearrange the formula to make mass density the subject

From the formula \(v=\sqrt{\dfrac{E}{\rho}}\), we want to isolate \(\rho\). First, square both sides of the equation to remove the square root: \(v^2 = \dfrac{E}{\rho} \) Now, multiply by \(\rho\) and divide by \(v^2\): \(\rho = \dfrac{E}{v^2}\) Now we have a formula for mass density in terms of Young's modulus and speed of sound.

Substitute the given values and calculate mass density

We are given the Young's modulus \(E = 112.1 \times 10^{9} \mathrm{~N} / \mathrm{m}^2\) and the speed of sound \(v = 5628 \mathrm{~m} / \mathrm{s}\). Substitute these values into the formula we derived in Step 1: \(\rho = \dfrac{112.1 \times 10^{9} \mathrm{~N} / \mathrm{m}^2}{(5628 \mathrm{~m} / \mathrm{s})^2} \) Square the speed of sound value and divide to find mass density: \(\rho = \dfrac{112.1 \times 10^{9} \mathrm{~N} / \mathrm{m}^2}{(5628 \mathrm{~m} / \mathrm{s})^2} = \dfrac{112.1 \times 10^{9} \mathrm{~N} / \mathrm{m}^2}{31.673 \times 10^{6} \mathrm{m}^2 / \mathrm{s}^2} \) Simplify the final expression: \(\rho = \dfrac{112.1 \times 10^{9} \mathrm{~N} / \mathrm{m}^2}{31.673 \times 10^{6} \mathrm{m}^2 / \mathrm{s}^2} = 3537.38 \mathrm{kg} / \mathrm{m}^3 \) So the mass density of the metal bar is approximately \(3537.38 \mathrm{kg} / \mathrm{m}^3\).

Key concepts

Understanding Young's Modulus

Young's modulus, denoted by the symbol \(E\), is a measure of the stiffness of a material. It defines the relationship between stress (force per unit area) and strain (deformation) in a material under tension or compression. The value of Young's modulus is constant for a given material and is one of the fundamental properties used in engineering and material sciences.

In the exercise given, Young's modulus is a critical factor for calculating the mass density as it represents the ability of the metal bar to resist deformation. The higher the value of Young's modulus, the stiffer the material. Imagine stretching a rubber band versus a metal wire; the wire (having a higher Young's modulus) is much more resistant to stretching under the same force. This resistance due to material's stiffness is the cornerstone for understanding Young's modulus.

When solving problems involving Young's modulus, one should ensure that the units used for stress and strain are consistent. Forces are often measured in newtons (\(N\)) and area in square meters (\(m^2\)), resulting in a modulus measured in pascals (\(Pa\)) or newtons per square meter (\(N/m^2\)).

Deciphering the Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium. It’s an essential concept when discussing mechanical waves and their properties. The speed of sound varies depending on the medium—being faster in solids than in gases because the particles are closer together and can transmit vibrations more quickly.

In the context of the given problem, knowing the speed of sound in the metal bar (\(5628 \frac{m}{s}\)) provides information about how quickly mechanical waves can travel through the material. This value is determined by both the material's stiffness and its density, following the equation \(v = \sqrt{\frac{E}{\rho}}\), where \(\rho\) is the mass density and \(E\) is Young's modulus. A higher speed of sound in a material typically suggests a greater stiffness to density ratio. In practical terms, this means that messages or signals (like tapping one end of the rail) can travel rapidly through the material without much loss of energy.

Mechanical Waves Fundamentals

Mechanical waves are disturbances that travel through a medium, such as solids, liquids, or gases, transferring energy from one place to another. The most familiar mechanical wave is sound. These waves require a medium to propagate, which means they cannot travel through a vacuum (like space).

There are two main types of mechanical waves: transverse and longitudinal. Transverse waves have oscillations perpendicular to the direction of propagation, like waves on a string. Longitudinal waves, on the other hand, oscillate in the direction of propagation, which is how sound travels in a medium.

When solving problems that involve mechanical waves, like calculating the speed of sound in a solid, it is crucial to grasp how they propagate and how the medium's properties affect wave speed. The relationship between a medium's Young's modulus and its density directly determines how mechanical waves behave in that medium. Understanding this interplay provides a comprehensive picture that goes beyond mere calculations to grasp the physics behind sound propagation and the nature of mechanical waves.