__Classical Mechanics Unit W1 The Harmonic Oscillator __

** Overview:** This unit treats the mechanics of the
harmonic oscillator, one example of which is a mass on a spring. The
mathematics here has many applications in physics. Almost any system
which is in stable equilibrium will undergo simple harmonic motion if
displaced slightly from equilibrium. The electromagnetic field in a
cavity is equivalent to an infinite collection of harmonic oscillators
of different frequencies. The mathematics of AC electric
circuits is essentially identical to that used in this unit.

__Read: ____ __

D. Kleppner and R. Kolenkow, *An Introduction to Mechanics
* , Chapt. 10 - The Harmonic Oscillator

__COMMENT:__

If you start with this unit before doing unit CM1 you may need to review the concepts of work and energy (Chapt. 4). You should work through notes 10.1 and 10.2 which describe the use of complex variables in the analysis of periodic motion. The mathematical level of this discussion is not all that sophisticated and you will find this technique useful in periodic motion analysis and again in the unit AC (AC circuits) should you choose to do it.

__After completing this unit you should understand:__

- The undamped, unforced harmonic oscillator with the equation of motion
d
^{2}x/dt^{2}+ ω_{0}^{2}x = 0, where ω_{0}is the natural angular frequency of the motion. This equation has the simple solution x(t) = A cos(ω_{0}t + φ), where A and φ depend on the initial conditions. - The damped unforced harmonic oscillator with the equation of motion
d
^{2}x/dt^{2}+ γdx/dt ω_{0}^{2}x = 0, where γ is the damping constant. This equation has the simple solution x(t) = A exp(-γ/2t) cos(ω_{1}t + φ), where ω_{1}^{2}= ω_{0}^{2}-γ^{2}/4 and A and φ depend on the initial conditions. - The damped forced harmonic oscillator with the equation of motion
d
^{2}x/dt^{2}+ γdx/dt + ω_{0}^{2}x = ( F_{0}/m) cos(ω t), where ω is the driving frequency. After a long time has passed this equation has the steady-state solution x(t) = A cos(ω t + φ), where A and φ are determined by the values of F_{0}, m, γ, ω and ω_{0}and do*not*depend on the initial conditions . - Why the solution of the damped unforced harmonic oscillator can be added to the solution in (c) to obtain the general solution for the damped forced harmonic oscillator.

__Problems:__

Back to the Physics 323/324 Home Page.