1. If a steel rope and a nylon rope of equal lengths are to stretch by equal amounts when subjected to equal tensions, what must be the ratio of their distances?
2.A simple hand operated hydraulic press can generate a pressure of 6.0 X 10^9 N/m^2 . If the system is used to compress a small volume, what fraction of the original volume does the final volume of steel occupy?
1. What distances are you talking about? The rope lengths? Are the cross sectional areas the same?
2. What material is being comoressed?
To solve these questions, we need to understand the concepts of stress, strain, and elastic constants.
1. The ratio of the distances can be determined by considering the relation between stress and strain. Stress is defined as the force applied per unit area, and strain is the measure of deformation caused by that stress. The equation that relates stress (σ), force (F), and area (A) is σ = F/A.
Now, to compare the elongation of the two ropes, we can recall Hooke's Law. According to Hooke's Law, the strain (ε) produced in a material is directly proportional to the stress (σ) applied to it. Mathematically, ε = σ/E, where E is the modulus of elasticity, or Young's modulus.
Since both ropes are subjected to equal tensions, we can assume the applied stress is the same. Therefore, the ratio of their distances (d₁ and d₂) can be calculated using the ratio of their moduli of elasticity (E₁ and E₂):
d₁/d₂ = (E₁/E₂)
The ratio of the distances will be equal to the ratio of the moduli of elasticity of the two materials.
2. To determine the fraction of the original volume occupied by the final volume of steel, we need to consider the concept of bulk modulus. Bulk modulus (K) measures the resistance of a material to compressibility under pressure. It quantifies how much a substance will deform when subjected to a given amount of pressure.
Using the formula P = K * ΔV/V, where P is the pressure applied, ΔV is the change in volume, and V is the original volume, we can calculate the change in volume of the steel.
ΔV = (P * V) / K
To find the fraction of the original volume occupied by the final volume, we can divide the change in volume by the original volume:
Fraction = ΔV / V
So, by substituting the given values of pressure and bulk modulus into the equation, we can find the fraction of the original volume occupied by the steel.