1. The problem statement, all variables and given/known data According to an old model due to JJ Thompson, an atom consists of a cloud of positive charge within which electrons sit like plums in a pudding. THe electrons are supposed to emit light when they vibrate about their equilibrium positions in this cloud. Assume that in the case of the hydrogen atom the positive cloud is a sphere of radius R = .050nm with a charge of e uniformly distributed over the volume of this sphere. The (point-like) electron is held at the center of this charge distribution by the electrostatic attraction. a) By deriving its equation of motion, show that when the electron is displaced from its equilibrium position by a distance r, it will execute S.H.M. 2. Relevant equations 3. The attempt at a solution ∫∫ E ⋅ dA = e(r^3)/(R^3)ε by Gauss' Law (Not sure if enclosed charge is e or 2e) Solving for E we have ke^2/R^3 * (1/r) Fe = m(d^2x/dt^2) plug in E and point charge q and we have a second order differential equation that should allow to prove SHM. Unfortunately I'm in high school and we haven't learned how to solve second order differential equations and I'm assuming we're not expected to. I can use a = v dv/dx and solve for v in terms of but I'm not sure how that will help me prove SHM.