We're now ready to talk about energy,
but we've covered a lot of topics up to this point.
So before we get into the energy of point object systems.
Let's do a brief overview of where we've been.
We started with the equations of motion and introduced these three equations,
which describe how positions, velocities,
and accelerations are related to each other as an object moves
through time from one position to another.
We then asked how these accelerations arose, where they came from.
And went through Newton's laws
and very especially Newton's second law,
as we see it here F equals ma.
Then we just now recently discussed the Torque,
which is what happens when we take these forces
and apply them to cause an object to rotate,
as well as what happens in an equilibrium situation.
And now, we're going to discuss energy.
With our discussion of energy, and a few topics after,
we will have finished our mechanics section.
This is all sort of the beginning part of this course,
so up to this point, we should really be comfortable with,
is what the equations of motion mean, and how to use them in a problem.
What a Newton's second law means,
and how to put different quantities on the left
and right hand side of the equation, F equals ma,
and then how to use Newton's second law with the equations of motion.
As well as how to solve basic Torque problems when objects are rotating,
especially understanding with this r and F
and sine of theta are in a given problem.
Now we're going to discuss energy,
a few different types, kinetic and potential,
and then the total energy for an object which will be the sum of these two.
After we've done this, we'll discuss work,
which is a way of measuring, how energy changes.
In this equation for work, we'll go over a little bit later.
And finally, to finish our entire mechanics part of this course,
we'll discuss momentum and talk about how momentum
and energy are related and conserved.
So first, before we get into the broader discussion of energy and momentum,
what I'd like to do is quickly motivate, why energy and momentum are related
and both important and both conserved quantities.
We have here, for example, an object,
an apple, mass, m, moving at velocity, v.
And it turns out historically, there were a lot of questions
as to which of these quantities, mass times velocity squared
or mass time velocity should be more important.
And these were all simple measurable things.
You can measure the mass of your object.
You could see how fast it was moving and some people would take that mass,
and how fast it was moving and square it,
and ask about that quantity and how it changed or didn't changed.
And some people would take the mass times the velocity
and do calculations about that quantity.
And so historically, a long, long time ago,
we had questions about and discussions about
which of these two quantities that you see here are more important
or more physically significant.
This picture of a woman you see here, this is Emmy Noether.
And I can't pronounce the name properly
but you can see the spelling of it, N-O-E-T-H-E-R.
And she came up with a very important theory relating these two quantities
that ended up being derived related to these two that were debated in history.
One we call energy that has the velocity squared term in it,
which we'll discuss shortly.
And then the other was momentum, which is just the mass times the velocity.
And what Emmy Noether was able to do, was to understand that the reason,
that there are conservation laws anywhere in nature,
like conservation of energy or conservation of momentum,
is that there are symmetries in nature.
And she was able to show that for any symmetry in nature.
Meaning all the laws of Physics are the same, at this position in space,
as well as that position in space.
For any such symmetry, whether it's a symmetry in space or symmetry in time,
there's a corresponding conservation law.
And it turns out that energy is a conservation law
related to symmetries in time.
And momentum is a conservation law related to symmetries in space.
So really energy and momentum,
these two debated quantities throughout history
that we have a modern forms for,
are not different at all, and neither are they in anyway contradictory.
They're actually talking about space and time,
and the conservation coming from the symmetries
and those that we understand now much better.
Thanks to the work of people like Emmy Noether.
All of the stuff that I put on the slide here.
The symmetry and the conservations and the ideas of these,
none of this is something that you have to memorize.
The important thing to understand is that energy and momentum
are both conserved quantities
and ways that we're about to describe.
But that they're both important and related quantities,
not just arbitrary things that we're sort of adding to the list of things
that you have to memorize.
There's actually a relationship between the two.
And as I discuss the energy was a symmetry in time
and the momentum was a symmetry in space.
Again, thanks to the work of Emmy Noether that we understand that.
So now first, let's talked about the kinetic energy.
What we're going to do
is basically talk about the different kinds of energy of any object.
And one kind of energy will come from the fact
that it's moving and has sort of a kinetic energy.
And the other kind of energy will come from its position in a given scenario,
it's called the potential energy, which we'll discuss in a minute.
But first the kinetic energy, the kinetic energy of an object,
so for example, of a mass, m and a velocity, v, is equal to 1/2,
the mass of that object times its velocity squared.
This is a very important equation to know
and to be able to use which we're going to do shortly.
This quantity of energy, a few things to understand about it.
This kinetic energy we're going to call K.
The units of K, the kinetic energy as you can see from the equation
are the units of mass times the units of velocity squared.
So we have kilograms, meter squared per second squared
and we call these units joules.
Joules, the name Joule comes from the person
and we abbreviated with a letter J for joules.
The very, very important thing to understand
about this quantity and its units,
is that unlike some of the quantities we've already discussed like velocity,
it's just a scale or it's just a number.
So for example, I couldn't ask you in which direction is kinetic energy.
I could ask in which direction is an object moving,
but its kinetic energy is just the simple quantity,
a single number that tells me something
about how much energy the object has.
So I could say it has 10 joules
or 5 joules of energy by virtue of its motion,
but I can't again tell you anything about the kinetic energy
in a certain direction.
So this is a scale of quantity, just a number and no vectors.
Let's do a quick example,
just so that we really have a grasp on this idea of kinetic energy.