Question

A metal bar has a Young's modulus of \(266.3 \cdot 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and a mass density of \(3497 \mathrm{~kg} / \mathrm{m}^{3}\). What is the speed of sound in this bar?

Short answer

Answer: The speed of sound in the metal bar is approximately 8723.13 m/s.

Step-by-step solution

Write down the given values

The given values are: Young's modulus, \(Y = 266.3 \cdot 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) Mass density, \(ρ = 3497 \mathrm{~kg} / \mathrm{m}^{3}\)

Rewrite the speed of sound formula

We rewrite the formula for the speed of sound in a solid material: \(v = \sqrt{\frac{Y}{ρ}}\)

Substitute the values into the formula

Now, substitute the given values of Young's modulus (\(Y\)) and mass density (\(ρ\)) into the formula: \(v = \sqrt{\frac{266.3 \cdot 10^{9} \mathrm{~N} / \mathrm{m}^{2}}{3497 \mathrm{~kg} / \mathrm{m}^{3}}}\)

Solve for the speed of sound

Solve the expression inside the square root to get the speed of sound: \(v = \sqrt{\frac{266.3 \cdot 10^{9}}{3497}}\) Nm/kg Calculate the value: \(v = \sqrt{7.61 \cdot 10^{7}}\) m/s

Find the final value for the speed of sound

Calculate the square root to get the final value for the speed of sound: \(v \approx 8723.13\) m/s The speed of sound in the metal bar is approximately 8723.13 m/s.

Key concepts

Young's Modulus

Understanding Young's modulus is crucial when exploring the properties of materials. It's a measure of a material's ability to withstand changes in length when under lengthwise tension or compression. Think of it as a score for stiffness: higher values mean the material is stiffer.

In technical terms, Young's modulus (\( E \) or \( Y \) in some textbooks) is defined as the ratio of stress (force per unit area) to strain (proportional deformation) in a material. Essentially, \[ Y = \frac{\text{Stress}}{\text{Strain}} \]
where stress is \[ \text{Stress} = \frac{\text{Force}}{\text{Area}} \] and strain is \[ \text{Strain} = \frac{\text{Change in Length}}{\text{Original Length}}. \]
We use this concept to understand how much a material compresses or extends when subjected to a certain load.

Mass Density

Mass density, often represented by the Greek letter \( \rho \), is the measure of mass per unit volume. It illustrates how much matter is packed into a given space. For example, lead has a higher mass density than styrofoam, meaning it packs more mass in the same volume. The formula to calculate mass density is: \[ \rho = \frac{m}{V} \],
where \( m \) is the mass and \( V \) is the volume of the material. This property is vital because it impacts how sound waves travel through different materials—denser materials can transmit sound faster because their particles are more closely packed and can transmit vibrations more efficiently.

Speed of Sound Formula

The speed of sound formula in solids relates to how quickly sound waves can travel through a material. Different materials will transmit sound at different speeds. The formula ties together Young's modulus and mass density, illustrating their role in this speed: \[ v = \sqrt{\frac{Y}{\rho}} \].
Here \( v \) stands for the speed of sound, \( Y \) is Young's modulus, and \( \rho \) is the mass density. The square root indicates that the speed of sound is the square root of the quotient of Young's modulus and mass density. When using this formula, ensure that you're using consistent units for all the values to get an accurate result.

Physics Problem Solving

Physics problem solving is an essential skill that requires a systematic approach to apply concepts like Young's modulus and mass density to practical scenarios. The key is to break down a problem into manageable steps, like identifying given values, understanding the relationships between those values as represented by equations, and carefully substituting and simplifying until you arrive at a solution.

Visualization often helps, as does double-checking each step for unit consistency and mathematical accuracy. Remembering that physics problems represent real-world phenomena can provide context that makes the solving process more intuitive and meaningful.