Part a
Using the fact that at maximum displacement all energy is stored as potential energy, derive an expression for the total energy \(E_{total}\) of the system in terms of \(k\) and \(x_0\).
Part b
Using energy conservation, derive an expression for the kinetic energy (KE) of the mass at an arbitrary displacement \(x\) (where \(0 \le |x| \le x_0\)) from equilibrium, given that the potential energy at \(x\) is \(\frac{1}{2}kx^2\).
Part c
Given that kinetic energy is also expressed as \(\frac{1}{2}mv^2\), derive an expression for the speed \(v\) of the mass at displacement \(x\) in terms of \(k\), \(m\), \(x_0\), and \(x\).