Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

The Simple Harmonic Oscillator

(a) When you do a dimensional analysis for the simple harmonic oscillator, both the damped and undamped cases, you find there is no length parameter. Why? What sets the length scale for the actual oscillation? Is this realistic? Discuss.

(b) Using the exponential method to find a general solution for the undamped simple harmonic oscillator gives

This is complex in general and would be hard to interpret physically. Prove that if the initial conditions are

x(0) = x0
v(0) = v0

and both are real, that x(t) and v(t) remain real for all t. Does x and v being real require that α and β must be real? Explain.

(c) Consider the case where the damping for the simple harmonic oscillator is small. This means that in each oscillation, there will be a little bit of energy lost, but it will be hardly noticable until you have made many oscillations. To analyze this, let's make a "physicist's approximation." Assume that the damping can be ignored and find the position x(t) and the velocity v(t) as a function of time. (You can use any starting condition you like.) Then calculate the work done by the damping force over one oscillation. This is the energy lost in one period. Now calculate the fraction of the energy lost per oscillation, ΔE/E. By putting in the time appropriately, calculate an approximation for the time dependence of the energy of the oscillator, dE/dt.