Dynamics of Simple Harmonic Motion
* Many systems that are in stable equilibrium will oscillate with simple harmonic motion when displaced by from equilibrium by a small amount.
* Near equilibrium the force acting to restore the system can be approximated by the Hooke's law no matter how complex the "actual" force. For example, the force holding atoms together in a solid can be approximated by spring forces when the vibrations of atoms in a solid are small.
* For a one-dimensional system to undergo Simple Harmonic Motion, the equation of motion of the "Force Law" - Newton's Second Law - must take the form,
* This is a second order differential equation where
| q(t) |
= Any spacial or angular coordinate of a system such as x or q
|
| K |
= Constant which is a function of system parameters
|
* Any time you can reduce the "Force Law" equation to the above structural form, you know that the system will undergo Simple Harmonic Motion whose General Solution will look very similar to that of a mass-on-a-spring.
* One must examine the initial conditions to determine the actual value of the phase angle f since most calculator only returns values between ±p/2. See trigonometric inverse .