The next object that we?ll discuss
that also undergoes periodic motion is a pendulum.
So if we have an object like this one,
suspended from maybe a string or maybe a bar
or something more rigid
so we don?t have to worry about it bending or anything like this.
This pendulum would swing back and forth
and what is again periodic motion.
It will just keep doing this as long as
again we assume nothing is lost to,
to friction or heat or anything like this.
What we?re going to try to do here
is write the force that the pendulum is feeling
in the same way that we wrote the spring force,
Hook?s Restorative law
that?s trying to restore the object to where it is.
In the Spring Law, what we had was that the amount of the force
dependent on how far you were from equilibrium,
how far you'll pulled yourself away.
So we saw that F was equal to minus k
times x and x there was how far have you pulled from equilibrium.
If you pull yourself twice as far, the force will double and so on.
So we?re going to try to write the pendulum of force
in the exact same way and that looks something like this.
We could write the gravitational energy or the gravitational force
rather that?s pulling that mass downwards,
just using a very typical Newton?s Law analysis.
So we have mg pulling the object downwards
and so we could try to solve for the amount of mg
that?s in the direction of motion.
So in the spring situation we had a force
if we displaced our object that was exactly in the direction of motion.
It would try to push you back towards equilibrium
and so the force is in the same direction as the motion.
So we have to try to do the same thing here.
Since the pendulum is swinging,
we want to find the amount of this gravitational force
that?s in the swinging direction just using some basic trigonometry,
we can see exactly what this is.
It would be m times g times the sine of the angle,
where the angle would be, the angle with a vertical here
and so don?t be too worried about trying to derive exactly what this angle is.
We should understand as just conceptually the next assumption I?m going to make.
The assumption that we?ll make here
is that this angle that we?re taking about is a small angle
and this is conceptually again the most important point
about this periodic motion of the pendulum.
For small angles,
we can?t have a pendulum pulled up very, very far
to like 90 degrees or something like this.
It has to be a small, little displacement.
We let go and it will oscillate.
If that angle is small,
it turns out mathematically that the sine of a small angle
is approximately equal to the angle itself
and so this is the assumption that?s made often.
You don?t need to be familiar with how the assumption is made
or being aware of how to derive a small angle.
A sine of that angle would be approximately equal to the angle itself.
But you should certainly be aware of how this approximation is made
or the fact that this approximation is made
and the fact that this periodic motion equation for the pendulum
is only true for small angles.
So when we do this though,
we end up with a force in the direction of motion
that is equal to mass times g, so this is some force,
times the angle of theta and so you can compare this to the spring force.
It?s a significant equation
but it might not be immediately apparent why it?s significant.
So let?s think about that.
This equation is something mg times theta, the displacement,
the distance you are away from your equilibrium.
But this was exactly the case with the spring force,
F was equal to minus k times x the distance you were.
And so this force that we have written here,
this mg theta is also what we could call a restorative force.
Gravity is always trying to get the pendulum to fall exactly back to its equilibrium
and so these forces, the pendulum force and the spring force are very comparable.
The equation is in fact, exactly identical in terms of its form.
Something times your displacement distance and so force that reason,
we have an equivalent formula for the frequency and the period for the pendulum
which are written here in terms of somewhat different parameters
than we had for the spring.
In this case the frequency is 1 over 2Pi
times the square root of g is the gravitational acceleration
divided by L where L is the length of the pendulum
and then the period will be 1 over this frequency
which were also in here because again,
you don?t need to always memorize both of these variables,
the frequency and the period.
They?re always gonna be one divided by the other.
There?s one last important thing about the pendulum
which is you notice that this equation
does not depend on the mass of the object on the pendulum.
This was actually discovered a long time ago for that,
that for a small oscillation.
You could have any object on the end of the pendulum
no matter how big or no matter how small
and it would have the same frequency of motion.
It would just keep doing the same sort of thing
which is somewhat a surprising result.
We can also apply the same conservation of energy idea
that we did for the spring here.
We could again equate energy at the top of this motion
as it swings to the energy at the bottom of the motion
and do the exact same sort of analysis
by looking at the kinetic and potential energy.
When we do this,
we see that the energy at the top of the motion before it?s swung
is just the gravitational potential energy
which is the energy at some height, h.
When it?s right at the bottom of that swinging path,
it no longer has any of that potential energy,
it's only kinetic energy and we can again write down
the equation for the kinetic energy.
So this completes our lecture introducing periodic motion.
We?ve talked about some of the main variables you need to be familiar with,
with how waves work and how we measure and compare those variables
with each other and then we talked about something
that wasn?t exactly a wave phenomenon in terms of energy
being transferred but was a periodic motion phenomenon
that hark back to many of the same principles
that we introduced when we?re talking about mechanics and Newton?s laws.
So with these in place we?re ready to talk about
some more specific examples of waves including sound and light.
Thanks for listening.