Now that we have an idea of what some basic wave properties are
and how to talk about and measure waves.
We're ready to discuss two particular mechanical examples
of periodic motion which are springs and pendulums.
Starting with springs, we can introduce a new force.
So this is harking once again back to the Newton's laws,
as we discussed them.
You might remember we had a whole lecture
where we discussed all kinds of important forces.
We talked about gravity, we talked about a number of others
including friction and the normal force.
This would be one more example of a very important force to be aware of.
This is a force from a spring.
What this equation is telling us
is that the force that the spring exerts when you move an object
attached to that spring is equal to minus k or k is what is called the spring constant
times x where x would be the displacement of your object
from the equilibrium point of your spring.
What we could do is stretch a spring away from this equilibrium point
and talk a little bit in more detail about this position coordinate.
So if I took this object on a spring and stretched it
and pull the spring away from this x equal zero point
which is called the spring sort of equilibrium point or where it's happy,
where it's not exerting a force in either direction,
what's going to happen is the spring is always going to try
to restore the object to that position
and this is the reason for the minus sign in that force equation.
If I pull this object,
the minus sign will ensure that this spring is trying to push it back
to its equilibrium point whereas if I push it,
it will try to push back and again go back to the equilibrium point.
If I grab the object and pull it away from the equilibrium point to some position,
we would call that position the amplitude.
If that's the point I pulled it back to and then I let go,
we would have this oscillation that this object would move left to right to left to right.
And just by thinking about it you could see,
it would move past the equilibrium and then come to equilibrium,
go past the equilibrium, come back to equilibrium
and the furthest distance it would go from the equilibrium point,
it's called the amplitude of the motion
and this is following exactly from our definition of the amplitude
when we were discussing how to describe waves.
So again as we pull it back away from this equilibrium point the force will be on the left,
in this case, as this spring tries to push the object back to the equilibrium
and the velocity of this object right before we let go would be zero.
So let's follow the motion.
We let go, it gets pulled back in,
it passes the equilibrium point but as it passes the equilibrium point,
it has some velocity because it was pulled and it was sped up.
So then it passes this point of equilibrium,
and then it keeps moving forward and then it goes too far,
it sort of overcorrects but then you've compressed the spring
and so the spring will push back
and you'll have force to the right again trying to restore the equilibrium point.
Right when it's at the minimum amplitude,
right when it's right to the, the most compressed it can be,
there is again no velocity in our object and then we'll have the same thing.
It will get pushed back, it will go past the equilibrium point one more time
and then it will keep doing this and this will be the oscillating behavior.
This will happen over and over and this defines our periodic motion.
What we can do with this periodic motion
is try to describe exactly some of the properties
that we already introduced like the frequency and the period of the motion.
This variable k which I briefly mentioned is called the spring constant.
It has to do with how strong the spring is.
So if the spring is a very thick spring which we discussed in a few chapters back.
For a very thick spring, it's going to be a high number and for a very thin,
easily stretched spring it will be a very low number.
So this spring won't push you with much force at all.
So if this is the case, we have some mass on our spring.
So this mass has some inertia to it.
So it won't want to move.
We'll have put this into our equations of motion and Newton's second law
and we have a spring, with a spring constant k.
It turns out and we won't go through the derivation because it,
it uses some Calculus and it's somewhat long.
So just the important things to know are that the frequency for a spring motion
that's going back and forth, the number of cycles per second
that this would take is 1/2 Pi times the square root of the spring constant
divided by the mass that is attached to the spring and the period of this motion,
the time between its repeating the exact same motion over and over again.
Well, again by our definition, just be 1 divided by the frequency and so really,
you don't need to ever memorize both the frequency and the period.
You can just memorize one of them or at least be familiar with the properties of one of them
and remember that the other one is always just the inverse
which is the case with the equation for period here.
The last thing we could do also with this spring
is apply energy conservation and this is the reason that I was emphasizing the velocity.
So we said that when it's right in its maximum amplitude
or right at its minimum amplitude,
right before we let go or right when it's the most compressed,
it has no velocity at those points.
Whereas, as the object is passing the zero point, passing the equilibrium,
we said it had a non-zero velocity and in fact,
this is the maximum velocity that it has.
So by conservation of energy,
we can say that the initial energy that the object had right before we let go,
we stretched it back and then right before we let go,
that's the initial energy and then it has some final energy
which we could maybe call the point where it's passing zero.
As we did with the gravitational, kinetic and potential energy,
we can just compare the total energy at each of these points.
So initially the energy is somewhat kinetic and somewhat potential
and then finally as it's passing the zero point
it is also somewhat kinetic and somewhat potential.
But given what we just said,
the initial kinetic energy before we let go is zero
because there's no velocity and the final potential energy
when it's right at the equilibrium point is also zero
because the spring isn't stretched or compressed
so there's no energy stored in the spring
and so we can simplify the equation above comparing the initial
and final energy by setting to zero,
the two energies I just mentioned and we have a much simpler equation
which is just that the initial potential energy stored in the spring
which is ' k times the amplitude of the spring squared
or the amplitude of the displacement from zero squared
and this just comes from the potential energy of the spring
which we discussed a few lectures back
where we used the amplitude as its position
will be equal to the kinetic energy as the object passes the zero point
and we already know the equation for the kinetic energy is 1/2 mv squared.
So you can see this analysis is exactly like the analysis that we did
when we were talking about gravity or a few other forces
when we work on the kinetic and potential energy at a few different points
and then simplified using some things that we knew about the problem
that in one case may be you have no kinetic energy
because you're not moving and then in another case,
you maybe have no potential energy
because you're not displaced from whatever it is
that store in your potential energy.
So this is the same sort of analysis by setting these energies equal to each other
by conservation of energy or of course assuming here
that no energies lost to friction or heat or anything like that.
We can solve for the velocity that the object has as it passes through the zero point
just by knowing some properties about the spring
and how mass of the object is that we're trying to move with the spring.
So this velocity, you could solve it yourself.
We'd get something that looks like this,
it's the amplitude times the square root of k is spring constant
divided by the mass of your object that you're trying to move.
So these kinds of conservation of energy ideas
with a spring equation like this one are very, very common
especially with springs because it's considered very important
to know the force from the spring which we introduced as F equals minus kx.
Be aware that that's called Hooke's Law.
We call it a restorative force as it's trying to restore an object
to where it is and you'll also be required to know the potential energy of a spring.
This is another, a very important equation in terms of the way it's considered.
So these come up very, very often.
So be familiar with this exact analysis.
How you compare the kinetic energy
to the potential energy and then solve for some variables
including maybe flipping it around.
So for example maybe you know exactly what the velocity is,
as its passing the equilibrium point and you would like to find the amplitude,
how compressed would the spring get or how far will it stretch back out
and so again be familiar with this exact sort of analysis as we have here.