Consider a metal bar of length L, cross-sectional area A, and Young's modulus Y. When a tension force F is applied to the bar, it causes an extension AL. Model the material as a cubic lattice where the atoms lie at the corners of the cubes and are connected by springs with equilibrium length x. Calculate the force constant k of the atomic springs by deriving expressions for (1) the number of atoms in any cross-sectional area, (2) the number of atoms in a single chain of length L, (3) the microscopic extension Ar between atoms, (4) the tensile force f between atoms, (5) write f = kAr and show that k=Yx, (6) calculate the value of k for a typical metal such as aluminum, for which Y = 70 GN/m² and 04pm

Consider a metal bar of length L, cross-sectional area A, and Young's modulus Y. When a tension force F is applied to the bar, it causes an extension AL. Model the material as a cubic lattice where the atoms lie at the corners of the cubes and are connected by springs with equilibrium length x. Calculate the force constant k of the atomic springs by deriving expressions for (1) the number of atoms in any cross-sectional area, (2) the number of atoms in a single chain of length L, (3) the microscopic extension Ar between atoms, (4) the tensile force f between atoms, (5) write f = kAr and show that k=Yx, (6) calculate the value of k for a typical metal such as aluminum, for which Y = 70 GN/m² and 04pm