Serway & Jewett — Oscillatory Motion: Problem 30

You take on a research assistantship with a molecular physicist. She is studying the vibrations of diatomic molecules. In these vibrations, the two atoms in the molecule move back and forth along the line connecting them (see Figure 20.5c). As an introduction to her research, she asks you to familiarize yourself with the Lennard–Jones potential (see Example 7.9), which describes the potential energy function for a diatomic molecule. She asks you to determine the effective spring constant, in terms of the parameters \(\sigma\) and \(\epsilon\), for the bond holding the atoms together in the molecule for small vibrations around the equilibrium separation \(r_{eq}\). After being stumped for a while, you ask her for a hint. She responds, “Example 7.9 provides the derivative of the potential energy function. Compare that to Equation 7.29 to find the force between the atoms. You want to show that F is of the form \(-kx\), and find \(k\). Let the separation distance \(r = r_{eq} + x\), where \(x\) is small and take advantage of the series approximations in Appendix Section B.5.” Wow, that’s several hints! You sit down and get to work.