User:Awsamb1/Spring (device)

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 :

$$E = \left ( \frac{1}{2} \right )kA^2$$

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 the attached mass m :

$$U = \left ( \frac{1}{2} \right )kx^2 $$

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 :

$$K = \left ( \frac{1}{2} \right )mv^2$$

Since there is no energy loss in such a system, energy is always conserved and thus :

$$E = K + U$$

Frequency & Period
The angular frequency ω of an object in simple harmonic, given in radians per second, is found using the spring constant k and the mass of the oscillating object m :

$$\omega=\sqrt{\frac{k}{m}}$$

The period T, the amount of time for the spring-mass system to complete one full cycle, of such harmonic motion is given by :

$$T = \frac{2\pi}{\omega}=2\pi\sqrt{\frac{k}{m}}$$

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 :

$$f = \frac{1}{T} = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$