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?

검증된 단계별 안내

Understand the relationship between the speed of longitudinal waves (v_l) and transverse waves (v_t) in a material. The problem states that v_l = 30 * v_t.

Recall the formula for the speed of longitudinal waves in a material: v_l = sqrt(Y / ρ), where Y is Young's modulus and ρ is the density of the material.

Recall the formula for the speed of transverse waves in a stretched wire: v_t = sqrt(T / μ), where T is the tension in the wire and μ is the linear mass density (mass per unit length) of the wire.

Since v_l = 30 * v_t, substitute the expressions for v_l and v_t into this equation: sqrt(Y / ρ) = 30 * sqrt(T / μ).

Square both sides of the equation to eliminate the square roots, and solve for the tension T in terms of Young's modulus Y, density ρ, and linear mass density μ. Then, use the relationship between tension and stress (T = F/A) to find the stress in the wire.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Young's Modulus

Young's modulus (Y) is a measure of the stiffness of a material, defined as the ratio of stress (force per unit area) to strain (deformation per unit length). It quantifies how much a material will deform under a given load, and is crucial for understanding the relationship between stress and wave speed in a material.

Longitudinal Waves

Longitudinal waves are waves in which the particle displacement is parallel to the direction of wave propagation. In solids, these waves travel through the material by compressing and expanding it, and their speed is influenced by the material's density and elastic properties, such as Young's modulus.

Transverse Waves

Transverse waves are waves where the particle displacement is perpendicular to the direction of wave propagation. In solids, these waves move through the material by shearing it, and their speed is determined by the material's shear modulus and density. Understanding the speed difference between longitudinal and transverse waves is key to solving the problem.