**Simple harmonic motion** is that described by a mass moving under a restorative force **proportional to its displacement**.

The most common example of a harmonic motion is that of a mass attached to a spring within its elastic limit: when the string is stretched or compressed it does not undergo permanent deformations and it goes back to its original state. In this case, the force acting upon the mass is given by Hooke’s law:

Where *x* is the mass displacement (as well as the spring deformation) and *k* is a positive real number characteristic of the spring (its stiffness).

A mass moving under this force undergoes a periodic motion, going back and forth from positions *x = A* and *x = -A*.

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The force exerted by the spring is conservative, and therefore it will derive from a potential energy in such a way that the negative of the gradient of the potential energy is the force:

The **elastic potential energy** is given by the parabola:

From the previous expression it can be seen that the elastic potential energy (or shortened to just potential energy) in a simple harmonic oscillator is proportional to the square of the position *x*. Since the total energy (E) is conserved, the sum of the kinetic energy (K) and potential energy is constant:

In the figure below is shown the mass attached to the spring as well as its energy for different values of its displacement *x*. The potential energy is plotted in red, its kinetic energy in blue and the sum of both in green.

Let’s assume that at time t = 0 the initial position of the mass is *x = A*. At t = 0 the mass is at rest, so its kinetic energy is zero. Therefore its total energy equals its initial potential energy, and has the value:

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When the mass is released, it will move to the left under the restorative force. When it arrives at any coordinate *x*_{0}, it will have both kinetic and potential energy, and the sum will be its total energy:

At the equilibrium position *x = 0* the potential energy will be zero, and its kinetic energy will equal its total energy:

At *x = -A* the mass will stop, and therefore its kinetic energy will be zero at this point. All its energy will be in the form of potential energy. Since the motion is periodic, this same sequence will repeat itself again and again cyclically.

You can also check out the page simple harmonic motion to see how to calculate the position, velocity and acceleration for a mass undergoing a simple harmonic motion.