Changes

It should be noted that in reality, Hooke's law is merely an approximation, and no physical spring actually has precisely this behavior.  However, for most materials a version of Hooke's law holds for reasonable ranges of <math>\vec x</math>.  This is called the ''elastic range'' of the material.

It should be noted that in reality, Hooke's law is merely an approximation, and no physical spring actually has precisely this behavior.  However, for most materials a version of Hooke's law holds for reasonable ranges of <math>\vec x</math>.  This is called the ''elastic range'' of the material.

If a particle moves only under the influence of the force exerted by a spring, then Newton's second law (<math>\vec F = m \vec a</math>) implies that its displacement satisfies the second-order [[differential equation]] <math>m\ddot{\vec x} = - k \vec x</math>. This is solved by <math>\vec x(t) = \vec x_0 \cos(\omega t)</math>, where <math>\omega = \sqrt{k/m}</math> is the frequency of oscillation: thus the particle moves in a "[[sine|sine wave]]" shape. This is an example of simple harmonic motion.

If a particle moves only under the influence of the force exerted by a spring, then Newton's second law (<math>\vec F = m \vec a</math>) implies that its displacement satisfies the second-order [[differential equation]]

 

<math>m\ddot{\vec x} = - k \vec x</math>.  

 

This is solved by:

 

<math>\vec x(t) = \vec x_0 \cos(\omega t + \phi)</math>

 

where <math>\phi</math> is a phase shift and <math>\omega</math> is the angular frequency and is:

 

<math>\omega = \sqrt{\frac{k}{m}}</math>

 

Thus the particle moves in a [[sine|sinusoidal]] manner. This is an example of simple harmonic motion.

[[Category:Physics]]

[[Category:Physics]]