What must be the stress (F/A) in a stretched wire of a material whose Young's modulus is Y for the speed of longitudinal waves to equal 30 times the speed of transverse waves?

Verified step by step guidance

1. Identify the relationship between the speed of longitudinal waves (v_L) and Young's modulus (Y), density (\rho), and stress (\sigma). The formula is v_L = \sqrt{\frac{Y + \sigma}{\rho}}.
2. Identify the relationship between the speed of transverse waves (v_T) and the density (\rho) and stress (\sigma). The formula is v_T = \sqrt{\frac{\sigma}{\rho}}.
3. Set up the equation given in the problem where the speed of longitudinal waves is 30 times the speed of transverse waves: v_L = 30v_T.
4. Substitute the expressions for v_L and v_T from steps 1 and 2 into the equation from step 3: \sqrt{\frac{Y + \sigma}{\rho}} = 30\sqrt{\frac{\sigma}{\rho}}.
5. Solve the equation for \sigma in terms of Y. Square both sides to eliminate the square roots, then isolate \sigma on one side of the equation to find the expression for stress in terms of Young's modulus.

Key Concepts

Young's Modulus

Young's modulus (Y) is a measure of the stiffness of a material, defined as the ratio of tensile stress (force per unit area) to tensile strain (proportional deformation in length). It quantifies how much a material will stretch or compress under a given load, and is crucial for understanding the mechanical properties of materials in engineering and physics.

Wave Speed in Stretched Wire

The speed of waves in a stretched wire is determined by the material's properties and the tension in the wire. For longitudinal waves, the speed (v_L) is given by the formula v_L = sqrt(Y/ρ), where ρ is the density of the material. Understanding this relationship is essential for analyzing how stress affects wave propagation in different types of waves.

Transverse vs. Longitudinal Waves

Transverse waves are waves where the particle displacement is perpendicular to the direction of wave propagation, while longitudinal waves involve particle displacement parallel to the wave direction. The speeds of these waves in a material depend on its elastic properties and density, and knowing the relationship between them is key to solving problems involving wave mechanics in solids.