Okay, now that we have some of the mathematical prerequisites that we need,
we can go on and talk about the simple harmonic oscillator.
Now this a model that's used over and over again for oscillatory systems and
we're going to look at this in the context of just a simple mechanical,
model of the simple harmonic oscillator. Se let's say we have a mass m of certain
amount of inertia connected to a spring with its spring constant k, that is then
attached to a rigid support. Now What I'm going to do is say, the
resting position of that mass. We'll measure it right here on this side
of the block and call that x equals zero. And let's assume that we walk up to this
and give it. A impulsive force at time equals zero.
So you could walk up and give this thing a kick.
Now, the instant you do that, the mass does not move instantaneously, but you do
instantaneously change its velocity by imparting a an impulse to it.
Now what I want to do is then examine the resulting motion of this.
Now, your, your intuition,uh, tells you that the mass is going to go out to some
maximum displacement and then turn around and go back the other way, pass through
the equilibrium position compress the spring so that it's moved as far as it's
going to move to the left, and then repeat that motion, and just oscillate
back and forth. So I want to take a little bit closer
look at that. So, at the initial instant when I first
provide the impulsive force to this mass, I now have, it's still in position x
equals 0 and it has velocity v-max. Now I'm going to make a plot of.
X versus time and the velocity versus time.
And so at this initial time, the position is zero and the velocity is maximum in
what I'm calling the positive direction to the right.
Now, a quarter cycle later. The mass reaches its maximum displacement
from the equilibrium position and the string is stretched as far as it's going
to string stretch. And at that instant, the position is
ex-max, the maximum displacement, and the velocity is zero.
The simple harmonic oscillator or the mass Is just turning around, it reached
its maximum displacement, it turns around then it's going to head back the other
way. And so at that instant the velocity is
zero. So, if I plot that the displacement is
now positive maximum value and the velocity is 0.
Now, the mass is now on its way back through the equilibrium position, and
when it passes the starting point, x equals 0, it has a maximum velocity in
the negative direction. V max if I plot that it's maximum
negative velocity and displacement is 0 then the spring compresses the mass
continues to move to the left until the spring is compressed to the point that
it's able to stop the mass. And at that instant in time I've rearched
the mass has reached a maximum displacement in the negative direction
and the velocity is 0 and so at point D, it's maximum negative displacement
velocity 0. Then the mass turns around and is headed
back in the positive direction. And as it passes the zero point, the
velocity is now back to the maximum velocity.
This is how we started. And so it's, it's back to where it
started, with the same velocity that it started with.
now we've ignored damping in this, and so there hasn't been any loss of energy and
it has the same velocity. At the end of one cycle as it had at the
beginning. Now the, if we connect these points it
turns out that resulting motion of the harmonic oscillator is a sign wave.
And so we just looked a few points in the sign wave.
But if we, when we solved the problem in just a minute, we'll see that the
solution really is a sign wave. And the velocity is a cosine curve, and
so this is what the harmonic oscillator is going to do.
Now, let's go back now and use a little bit of calculus that we know to actually
solve this problem. So now it's, we just need to review a
couple of definitions from mechanics, or hopefully you've had this in high school
physics. If I take the time rate of change of
displacement, x is velocity. So the derivative, with respect to time,
of the position of that harmonic oscillator is going to be its velocity.
And the time rate of change of velocity is the acceleration.
And the, we can combine those and say if I take the time rate of change of the
time rate of change of the displacement. That's what we call the second derivative
of the displacement. And so the second derivative of the
displacement is acceleration. So that's just a little kinematics
review. Now let's go back to the harmonic
oscillator. It's our mass on this spring, and we're
going to set up a coordinate system, and say x is measured from this point on the
left of the, the mass, and increases to the right.
Now, there are two pieces of physics that we need to find the differential equation
for this system. The first one is Newtons second law that
says the force equals the mass times acceleration.
