attached mass and is ignored. Since acceleration is simply the second derivative of x with respect to time, This is a second order linear differential equation for the displacement as a function of time. Rearranging: the solution of which is the sum of a sine and cosine: and are arbitrary constants that may be found by considering the initial displacement and velocity of the mass. The graph of this function with (zero initial position with some positive initial velocity) is displayed in the image on the right. Energy dynamics In simple harmonic motion of a spring-mass system, energy will fluctuate between kinetic energy and potential energy, but the total energy of the system remains the same. A spring that obeys Hooke's Law with spring constant k will have a total system energy E of: Here, A is the amplitude of the wave-like motion that is produced by the oscillating behavior of the spring. The potential energy U of such a system can be determined through the spring constant k and its displacement x: The kinetic energy K of an object in simple harmonic motion can be found using the mass of the attached object m and the velocity at which the object oscillates v: Since there is no energy loss in such a system, energy is always conserved and thus: Frequency & period The angular frequency ω of an object in simple harmonic motion, given in radians per second, is found using the spring constant k and the mass of the oscillating object m: The period T, the amount of time for the spring-mass system to complete one full cycle, of such harmonic motion is given by: The frequency f, the number of oscillations per unit time, of something in simple harmonic motion is found by taking the inverse of the period: Theory In classical physics, a spring can be seen as a device that stores potential energy, specifically elastic potential energy, by straining the bonds between the atoms of an elastic material. Hooke's law of elasticity states that the extension of an elastic rod (its distended length minus its relaxed length) is linearly proportional to its tension, the force used to stretch it. Similarly, the contraction (negative extension) is proportional to the compression (negative tension). This law actually holds only approximately, and only when the deformation (extension or contraction) is small compared to the rod's overall length. For deformations