- [Voiceover] This is Walter the penguin.
And Walter the penguin gets bored in Antarctica,
so he likes to run, jump, and then slide
across the ice to a stop.
But Walter is a clever and curious penguin,
so while he's sliding, he's thinking
about energy conservation and he's confused,
'cause he knows that he starts off over here,
with some amount of kinetic energy.
But he knows that he ends over here
with no kinetic energy since he slides to a stop.
So he wonders, how can energy be conserved,
when he seems to be losing kinetic energy?
Now if you would have asked this question
when we dealt with forces, you would have said,
"Oh, well obviously, this penguin is coming to a stop
"'cause there must be some amount of friction
"between the penguin and the ice.
"Maybe the ice is very slippery,
"but it can't be frictionless, or this penguin
"would probably keep sliding forever.
"There might be some air resistance
"causing the penguin to slow down
"but it's probably mostly friction
"between the penguin and the ice."
But we're talking about conservation of energy here,
so how can we put this idea in terms of energy?
Well, the way we do it, is we just say
that this force of friction is doing
negative work on the penguin.
And we know the work is negative
because the force of friction is directed
in the opposite direction to the penguin's motion.
The other way we could see this is that,
we could just use the formula for work done by any force.
That formula's f d cosine theta.
If we want to find the work done by the force of friction,
we would plug in the force of friction for our force,
the magnitude of it, times the distance
the penguin slid to the right, and then this theta
in cosine theta is always the angle
between the force and the direction of motion.
So this penguin's sliding to the right,
the forces directed to the left,
you might think that's zero, but that's not zero.
Think about it, the angle between leftward
and rightward is not zero,
that's actually 180 degrees.
So this angle would be 180 right here, or pi radians.
And cosine of 180 is going to give you a negative one,
so the work done by the force of friction on this penguin
is going to be negative f k d.
Negative the force of friction,
times the distance the penguin slid to the right.
But this still doesn't answer Walter's question.
Where did the kinetic energy go?
Friction may have done negative work on this penguin,
but where did that energy end up?
And you probably have a good idea,
'cause when two surfaces rub together,
some of that energy of motion is going to get
transformed into thermal energy in those two surfaces.
In other words, this sheet of ice
is going to have a little more thermal energy,
it's going to heat up just a little bit.
And Walter's feathery coat is going to heat up
just a little bit, and they're going to have
more thermal energy to end with,
than what they started with.
Just like when you rub your hands together vigorously
on a cold day to get warm, you're turning
some of that kinetic energy into thermal energy
that warms up your hands.
And you might be like, "Alright, that's all well and good,
"but how do we put this all together?
"I mean I've got this idea of work over here,
"I've got this idea of kinetic energy over here.
"What unifying framework lets me think about this
"all in one package?"
And that would be the statement of conservation of energy,
so we can say that the initial energy of any system
plus any external work that's done on that system
has to equal the final energy of that system.
In other words we could say that Walter the penguin
started with kinetic energy, so Walter starts
with 1/2 m v squared, and we don't have to worry
about gravitational potential energy
'cause Walter's not changing his height,
he's just sliding straight along the ice
at a horizontal level.
And we can add the external work that was done,
which we've just figured that out.
We know the external work would be the work
done by friction, so we'd have a minus,
'cause it was negative work, f k d,
and it's negative again because this force
is taking energy out of the system.
We could set that equal to the final energy,
but Walter ends with no kinetic energy, so we'd say
that there's no final kinetic energy to end with,
there's no energy in our system to end with,
in which case we find out,
something that might be obvious to you is that
1/2 m v squared, the initial kinetic energy
that Walter started with, has to equal,
if we add to both sides f k d,
the magnitude of the work done by friction.
But some people might object, they might say,
"Wait a minute, we just said there was thermal energy
"to end with, how come we didn't include that
"in our final energy?"
And the reason is, that in this calculation here,
we assumed that Walter, and only Walter,
was part of our energy system.
In other words, Walter, and only his motional energy,
his kinetic energy, was the only energy
we were keeping track of, that's why we said that,
initially, there was just Walter's kinetic energy.
And this sheet of ice was external to our system,
not part of our system, that's why it exerted
a negative external work, removed the energy
from the system, and Walter ended up with no kinetic energy.
But there's an alternate way to go about this calculation.
You could say, "Alright, instead of just considering
"Walter and Walter alone to be part of our system,
"let's go ahead and include the ice as part of our system."
And all those places where the thermal energy can go,
like Walter's feathery coat, if we include all the places
energy can go, then there won't be any external work.
So an alternate way to solve these problems,
is to use this same formula, but now,
Walter and the ice are both part of our system.
Our system would still start with the kinetic energy
that Walter had at the beginning, that doesn't change.
But now there would be no external work,
not because force of friction isn't acting,
there's still a force of friction,
but that's an internal force between objects in our system.
'Cause Walter and the ice are part of our system.
So there's no external work done now.
That might be a new or confusing idea to some people,
so let me just say, if there's forces between objects
within your system, then those forces cannot exert
external work and they cannot change
the total energy of your system.
Only forces exerted on objects within your system
from outside of your system, can change
the total energy of your system.
So when this ice was not part of our system,
it was exerting an outside external force on Walter,
and the energy of our system changed.
We started with kinetic energy, we ended with no energy.
But now that the ice and Walter are part of our system,
this force of friction is no longer external.
It's internal, exerted between objects within our system,
and so it does not exert any external work.
It's just going to transform energies
between different objects within our system.
So that's why we write this zero here,
there'd be no external work done if we choose the ice
and Walter as part of our system.
And this would have to equal the final energy,
and we know where this energy ends up.
It started with kinetic energy, and it ends
as this extra thermal energy in the snow,
and Walter's feathery coat.
So I could write that as e thermal.
But I know how much thermal energy was generated,
this just has to equal the amount of work done by friction.
So even though this work done is not external,
it still transfers energy between objects
within our system, so when we write that the work
was negative f k d down here, we mean that
the force of friction took f k d from something
and turned it into something else,
and that's all we need up here.
We need an expression for thermal energy.
But if friction took f k d and turned it
into something else, the thing it turned it into
was the thermal energy so that value of f k d,
that magnitude of the work done, was how much energy
ended up as thermal energy.
People might find that confusing, they might be like,
"Wait a minute, why do we have this
"with a positive here and not a negative?"
Well in this work formula, this negative is just saying
that the force of friction is taking energy from something.
If you take energy from something,
you're doing negative work on it.
If I gave energy to something, I'd be doing positive work.
So this negative sign in the work done,
just means that the force of friction took this much energy
from something, and turned it into thermal energy.
So when we want to write down how much thermal energy
did we end with, well, we ended with
the amount that we took.
So we took f k d, the thermal energy ended with f k d.
And I can still set this equal to the kinetic energy
that Walter started with, and I get the same formula
I ended up with over here, because I had to.
Because we're describing the same universe
and the same situation, so no matter what story you tell,
you should get the same physics in the end.
And we do, but some people prefer one to the other.
Some people like thinking of friction
as a negative external work, and not including
the energy within the surfaces as part of their system.
And some people like including those surfaces
as part of their energy system, and just including
that thermal energy on the e final side.
Which is fine, you can do either, you just can't do both.
Either the surface is part of your system,
and you include it in your final energy,
or the surface is not part of your system,
and you include it as external work.
But you can't say it does external work
and it gains some final energy over here
because it's got to be either part of your system,
or not part of your system.
So long story short, you can basically just think
about the thermal energy generated by friction,
as f k d, this is a formula that'll let you solve
for the amount of thermal energy generated
when two surfaces rub against each other.
And we can take this idea a little further.
The force of kinetic friction is going to be equal to
the coefficient of kinetic friction
times the normal force between the two surfaces.
So we could rewrite this.
This thermal energy term can be rewritten as
mu k times f n times d.
And you might say, "Well that's not all that remarkable,
"it just looks even worse than it did before."
But if you keep going, you realize that the normal force
for something just sitting on a surface,
is just going to be m times g.
And we still multiply by the d,
but now that we've replaced the normal force
with m g, we notice that the mass cancels.
And this should blow your mind!
This means, no matter what the mass of this penguin is,
if he starts with the same speed as some other penguin
that's more or or less massive,
he'll slide the exact same distance.
Now some people will object, they'll be like,
"Wait, a really massive penguin's going to have
"a lot of inertia, it really wants to keep moving,
"it should slide farther."
But other people would say, "No, no.
"The less massive penguin should slide farther,
"because it has less frictional force."
But that's why it doesn't matter,
these two confounding effects exactly cancel.
In other words, the more massive penguin
does have more inertia, and has more friction.
And the less massive penguin has lass inertia,
but it has less friction, so the mass ends up canceling,
and all penguins, no matter what their mass are,
slide the same amount if they start with the same speed.
And this also means that two cars,
a really tiny Smart Car and a huge SUV,
if they've got the same tires,
they'll have the same coefficient of friction,
and if they start with the same speed
and slam on their brakes, they'll both skid to a stop
in the same distance.
Again, a lot of people would think
that the really massive SUV has to slide farther,
but that massive SUV that has more inertia
also has more friction, so it stops in the same distance
as the smaller Smart Car, that has less inertia,
and less frictional force.
So even though the coefficients of friction
would be the same for both cars, the force
of friction's going to be bigger for the larger car.
Alright, so to wrap this up, let's do an example problem.
Let's get rid of all this.
Let's say Walter steps it up, he's going
to the extreme games of sliding and he's going
to slide down, starting at the top of this ramp,
that's completely icy, this ramp has no friction.
But Walter has no fear so he goes to the top,
it's four meters tall, he starts at rest.
Walter slides down, speeds up, but now
he hits a patch that does have friction
so Walter slides across this patch,
over some distance, d, and then comes to a stop.
And if we know Walter started four meters high,
and the coefficient of friction along this path is 0.2,
we could figure out what was the distance Walter slid
on this horizontal surface before coming to a stop.
So let's do this, we're going to use
conservation of energy, same formula we did before,
the initial energy of our system
plus any external work done, that is to say,
energy added to or subtracted from our system,
has to equal the energy that we end up with in our system.
Now what do we want to pick as our system?
Personally, I like choosing everything that could
get energy as part of my system.
That way, I never really have to worry about any
external work, because every force will just be internal.
So let's choose Walter, the Earth, the ice, the snow,
the incline, everything's going to be part of our system.
And let's consider our initial point to be
when Walter started at rest at the top of this incline.
And our final point over here, where Walter slid to a stop.
So what kind of energy does our system start with?
Well Walter was up here, he was at rest,
so he had no kinetic energy, but he did have
gravitational potential energy, so that's m g h.
And again, there was no external work done
'cause even though there was friction down here,
I'm going to include this surface and Walter's feathery coat
as part of our system, so there was no external work done
but, there will be thermal energy generated,
and it'll end up in our system.
So the thermal energy generated is going to be
the force of friction, times the distance that Walter slid
across the surface that had friction,
so I'm not going to include this surface right here,
'cause there was no friction there.
I'm only including this distance right here.
And we know that the force of friction
is going to be the coefficient of friction
times the normal force, and then we still multiply
by the distance that Walter slid.
And because Walter was sliding over a horizontal surface,
the normal force on Walter is just going to be equal
to the force of gravity on Walter.
So we can say m g h is going to equal
the coefficient of kinetic friction and then
f n, we're going to replace with m g,
since the normal force is just equal to m g.
And I still multiply by d.
Again, something miraculous happens,
the m's end up canceling, Walter could have been
100 kilograms or 2 kilograms.
Would have slid the same amount.
Finally I can solve this for d.
I'm going to say that d is going to equal
g h, oh, actually.
Turns out the g's cancel too!
Could have done this on the moon, wouldn't have mattered.
Even if the g, the gravitational acceleration,
was different, Walter would still slide the same amount.
So I'm just going to get that d equals h,
and then I divide both sides by this
coefficient of kinetic friction, and there's my result.
It's really simple, in other words,
h was four meters, and then I divide by
the coefficient of friction, which was 0.2,
and I get that Walter's going to slide 20 meters
before coming to a stop.
So recapping, when there's a force of friction
acting on an object, you can use conservation of energy
by treating that frictional surface
as part of your system, in which case,
you would include it as a thermal energy on the final side.
Or, not including that surface as part of your system.
In which case, you would include the same term
with a negative sign as the external work done
on your system, regardless of what you do,
the thermal energy generated in such a situation
is going to be the force of friction
multiplied by the distance through which the object slides.