(b) A metal bar with a length of 1.50 m has density 6400 kg/m3 . Longitudinal sound waves take 3.90 * 10-4 s to travel from one end of the bar to the other. What is Young's modulus for this metal?

Verified step by step guidance

1. Calculate the speed of sound in the metal bar using the formula: \( v = \frac{L}{t} \), where \( L \) is the length of the bar and \( t \) is the time taken for sound to travel through the bar.
2. Determine the mass of the metal bar using the formula: \( m = \rho \times V \), where \( \rho \) is the density of the metal and \( V \) is the volume of the bar. Since the bar is a cylinder, its volume can be calculated as \( V = A \times L \), where \( A \) is the cross-sectional area.
3. Calculate the cross-sectional area of the bar, \( A \), by rearranging the mass formula: \( A = \frac{m}{\rho \times L} \).
4. Use the formula for Young's modulus in terms of sound speed, density, and cross-sectional area: \( Y = \rho \times v^2 \), where \( v \) is the speed of sound calculated in step 1 and \( \rho \) is the density of the metal.
5. Substitute the values obtained from the previous steps into the Young's modulus formula to find the modulus of the metal.

Key Concepts

Young's Modulus

Young's modulus is a measure of the stiffness of a material, defined as the ratio of tensile stress to tensile strain. It quantifies how much a material will deform under a given load, providing insight into its elastic properties. A higher Young's modulus indicates a stiffer material that deforms less under stress.

Density

Density is the mass per unit volume of a substance, typically expressed in kilograms per cubic meter (kg/m³). It is a critical property that influences how materials respond to forces and can affect the speed of sound in solids. In this context, the density of the metal bar helps in calculating its mass, which is necessary for determining stress.

Speed of Sound in Solids

The speed of sound in solids is the rate at which sound waves propagate through a material, influenced by the material's density and elasticity. It can be calculated using the formula v = √(E/ρ), where E is Young's modulus and ρ is density. Understanding this relationship is essential for solving the problem, as it connects the time taken for sound to travel through the bar to its mechanical properties.