A given spring with a given value of k, and a given value
for the mass that hangs on this spring, has one and only
one period.
If I change the amplitude of the screen, that period is not
going to change.
If I changed my definition of t equals 0, which would
introduce a value for alpha, that period is
not going to change.
Omega and p are completely independent of a and alpha.
What is the meaning of a and alpha?
Well, a and alpha are what we call the initial conditions.
I discussed one initial condition: if you define that
t equals 0, that the object is at x plus a, and you release
it with 0 speed, then out comes alpha.
You could also have released it at t equals 0, you could
release it at x equals 0, and you could release it with a
certain speed v--
just give it a kick.
Out of this information will follow both alpha and a--
that's why we call them initial conditions.
Now, in our simple harmonic oscillation, which has the
form of x in its most general form, times a cosine omega t
plus alpha--
notice that this oscillation will go on forever
and ever and ever.
Here's x equals 0-- it will reach plus a, and we'll go
back and reach minus a, and there never
comes an end to this.
This is, of course, not very realistic.
In practice, there will be damping--
that means there will be some form of air friction--
and then, the oscillation will gradually die out, but we have
not taken that into account.
It will oscillate maybe many times, and ultimately, it will
come to a halt.
In the problems that I will be doing in this section, I will
always ignore friction and air drag of any kind.