Question

A metal bar has a mass density of \(3579 \mathrm{~kg} / \mathrm{m}^{3}\). The speed of sound in this bar is \(6642 \mathrm{~m} / \mathrm{s}\). What is Young's modulus for this bar?

Short answer

Answer: The approximate value of Young's modulus for the metal bar is \(157.9 \times 10^{9} \mathrm{~N}/\mathrm{m}^2\).

Step-by-step solution

Write the formula for Young's Modulus

The formula to find Young's modulus is: \(Y = \rho v^2\).

Plug in the given values for mass density and speed of sound

Replace \(\rho\) with its given value, \(3579 \mathrm{~kg}/\mathrm{m}^3\), and \(v\) with its given value, \(6642 \mathrm{~m}/\mathrm{s}\), in the formula: \(Y = (3579 \mathrm{~kg}/\mathrm{m}^3) \times (6642 \mathrm{~m}/\mathrm{s})^2\).

Perform the calculation

Multiply \((6642 \mathrm{~m}/\mathrm{s})^2\) to get \(44,119,364 \mathrm{~m^2}/\mathrm{s^2}\) and then multiply this result by \(3579 \mathrm{~kg}/\mathrm{m}^3\) to find the Young's modulus: \(Y = 3579 \times 44,119,364 \mathrm{~N}/\mathrm{m}^2\).

Calculate the final value of Young's modulus

Perform the multiplication to get the final value: \(Y = 157,898,388,446 \mathrm{~N}/\mathrm{m}^2\). Young's modulus for the metal bar is approximately \(157.9 \times 10^{9} \mathrm{~N}/\mathrm{m}^2\).

Key concepts

Mass Density

Mass density is a fundamental concept in physics that relates to the distribution of mass within a given volume. Commonly denoted by the Greek letter \(\rho\) (rho), it is defined as the mass per unit volume and expressed in units such as kilograms per cubic meter (\(\mathrm{kg/m^3}\)).

Understanding the concept of mass density is crucial when studying material properties because it directly affects how substances behave under various conditions. For example, a material with a high mass density means that its mass is packed more tightly, which tends to make it stronger or more durable.

In the context of our problem, the mass density of the metal bar, \(3579 \mathrm{kg/m^3}\), provides an insight into how densely packed the metal atoms are within the bar. This information is essential when calculating material properties such as Young's modulus.

Speed of Sound

The speed of sound is the velocity at which sound waves travel through a medium. It varies depending on the medium's physical properties such as its elasticity, temperature, and density. In solids, sound tends to travel faster than in liquids, and in liquids faster than in gases, due to the closer proximity of the molecules in solids providing a more efficient transfer of kinetic energy.

In our problem, the speed of sound in the metal bar is given as \(6642 \mathrm{m/s}\). This not only refers to how swiftly sound waves propagate through the metal but also reflects the material's stiffness or elasticity to some extent. Higher speeds of sound often correlate with stiffer materials which, as a result, can better resist deformation.

Elasticity of Materials

Elasticity describes a material's ability to resist deformation and return to its original shape when external forces are removed. The measure of this property is encapsulated in Young's modulus, often symbolized as \(Y\). It quantifies the stiffness of a material and is expressed in pascals (\(\mathrm{N/m^2}\)). A higher Young's modulus indicates a stiffer material.

Young's modulus is dependent on both the material's ability to resist change in shape (elasticity) and its volume-specific mass (density), as the equation \(Y = \rho v^2\) incorporates both mass density (\(\rho\)) and the speed of sound (\(v\)), which is related to the material's elastic properties.

When solving for Young's modulus as in our exercise, we consider how the intrinsic property of elasticity and the measured speed of sound interact to define the material's stiffness and its behavior under stress.

Physics Problem Solving

Physics problems can often seem daunting, but breaking them down into manageable steps can help in understanding and solving them efficiently. In our exercise, the steps begin with establishing the appropriate formula, which connects mass density and the speed of sound to Young's modulus. Then, the given values are substituted into the equation.

Performing the calculation is where understanding the units and how they interact is critical—ensuring the correct manipulation of numbers and units to avoid common mistakes. Finally, obtaining the final value displays the solution's physical meaning—how the calculated modulus reflects the metal's resistance to deformation.

Improving clarity and comprehension in physics problems can be achieved by visually demonstrating each step, discussing the significance of the units used, and connecting each part of the problem back to the foundational physics concepts.