where a and b are constants. Differentiating this equation, we see that

= - ab sin(bt)

and

= - ab^{2}cos(bt)

Plugging this into our original differential equation, we see that:

- ab^{2}cos(bt) + a cos(bt) = 0

It is clear that, if b^{2} = , then the equation is satisfied. Thus the
equation governing simple harmonic oscillation is:

simple

x = a cost

The Equation for Simple Harmonic Motion

From the equation for simple harmonic motion we can tell a lot about
the motion of a harmonic system. First of all, x is maximum when the cosine
function is equal to 1, or when x = a. Thus a in this equation is the amplitude
of oscillation, which we have already denoted by x_{m}. Secondly, we can find
the period of oscillation of the system. At t = 0, x = x_{m}. Also, at t = 2Π, x = x_{m}. Since both these instances have the same
position, the time between the two gives us our period of oscillation. Thus:

T = 2Π

and

ν = =

finally,

σ = 2Πν =

Note that the values of period and frequency depend only on the mass of the
block and the spring constant. No matter what initial displacement is given to
the block, it will oscillate at the same frequency. This concept is important.
A block with a small displacement will move with slower velocity, but with the
same frequency as a block with a large displacement.

Notice also that our value for σ is the same as what we called the
constant b in our original equation. So now we know that a = x_{m} and
b = σ. In addition we can take the time derivative of our equation to
generate a full set of equations for simple harmonic motion:

x

=

x_{m}cos(σt)

v

=

- σx_{m}sin(σt)

a

=

- σ^{2}x_{m}cos(σt)

Thus we have derived equations for the motion of a given simple harmonic system.

Energy of a Simple Harmonic Oscillator

Consider a simple harmonic oscillator completing one cycle. In the jargon of
conservative vs. nonconservative forces (see Conservation of Energy
the oscillator
has completed a closed loop, and returns to its initial position with the same
energy it began with. Thus the simple harmonic oscillator is a conservative
system. Since the
velocity of the oscillator does change, however, there must be an expression for
the potential energy
of the system, such that the total energy of the system is constant.

We already know the kinetic
energy of the system
at any given time:

K

=

mv^{2}

=

m(- σx_{m}sin(σt))^{2}

=

kx_{m}^{2}sin^{2}(σt)

The kinetic energy has a maximum value when the potential energy is zero, and
sin(σt) = 1. Thus K_{max} = kx_{m}. Since the potential energy
is zero at this point, this value must give the total energy of the system.
Thus, at any time, we can state that:

E

=

U + K

kx_{m}^{2}

=

U + kx_{m}^{2}sin^{2}(σt)

Solving for U:

U = kx_{m}^{2}(1 - sin^{2}(σt))

Recall that
sin^{2}a + cos^{2}a = 1
. We can thus substitute:

U = kx_{m}^{2}cos^{2}(σt)

However, we also know that x = x_{m}cos(σt) for any simple harmonic
oscillation. Using this knowledge we can further simplify our equation for
potential energy:

simplify

U = kx^{2}

With this equation we have an expression for the potential energy of a
simple harmonic oscillator given a displacement from equilibrium. When examined
practically, this equation makes sense. Consider our example of a spring. When
the spring is stretched or compressed a large amount (i.e. the block on the
spring has a large magnitude for x), there is a great deal of energy stored in
those springs. As the spring relaxes and accelerates the block this potential
energy gets converted to kinetic energy. Shown below are three positions of the
oscillating spring, and the energies associated with each position.

This SparkNote introducing oscillation and simple harmonic motion involved a
great deal of mathematics and theoretical calculations. In the next
SparkNote we explore oscillations
on a more practical level, examining real physical situations and various types
of oscillators.