Suppose that an oscillating spring has one end firmly attached to a base of support and a mass attached to its free end. As the mass vibrates back and forth, we can track the behavior of three instantaneous quantities: the mass' displacement, velocity, and acceleration. Note in the diagrams shown below that when the mass' displacement is at a maximal positive position, its velocity is zero, and its acceleration, which is acting to restore the mass to its undisturbed equilibrium position, has a maximum negative value. Notice that at the endpoints, when v = 0, the mass has no kinetic energy, KE = ½mv^{2}. Therefore, all of its energy is in the form of elastic potential energy, PE_{e} = ½kx^{2}. When PE_{e} is maximum, the restoring force within the spring is also maximized resulting in the mass' acceleration also being maximized as the spring acts to return the mass to its equilibrium position. There are two formulas at our disposal to quantify the restoring force within the spring as it oscillates: Newton's 2nd Law, net F = ma, and Hooke's Law, F =  ks: This results tells us that the mass' instantaneous acceleration is directly proportional to, but in the opposite direction as, its instantaneous displacement. To help us understand the substitution which we will need to use next, we are going to return to some relationships which we learned for uniform circular motion. In the following video, note how the motion of the ball's shadow emulates the motion of a mass on the end of a vibrating spring. Additional videos and a physlet animations that will clarify further the relationships between position, velocity and acceleration are provided in the chart below.
In general, the sinusoidal equations for each property graphed at the top of this page are summarized in the following equations. where represents the frequency measured in hertz and ω, or the angular velocity, equals measured in rad/sec Returning to our previous result of please note:

that the magnitude of a_{max} is equivalent to the magnitude of the mass' centripetal acceleration, a_{c} = v^{2}/r

the fact that s_{max} equals the radius of the circle, r, or the amplitude, A, of the sine graph
Remembering the relationship from circular motion that we can substitute A for r and complete our derivation.
Summary of SHM
The following list summarizes the properties of simple harmonic oscillators.

The oscillator's motion is periodic; that is, it is repetitive at a constant frequency.

The restoring force within the oscillating system is proportional to the negative of the oscillator's displacement and acts to restore it to equilibrium.

The velocity of the oscillator is maximum as it passes through equilibrium, and zero as it passes through the extreme positions in its oscillation.

The acceleration experienced by the oscillator is proportional to the negative of its displacement from the midpoint of its motion.
