Introduction to mechanical advantage

WEBVTT
Kind: captions
Language: en

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Welcome back.

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We'll now use a little bit of
what we've learned about work

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and energy and the conservation
of energy and

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apply it to simple machines.

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And we'll learn a little bit
about mechanical advantage.

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So I've drawn a simple
lever here.

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And you've probably
been exposed to

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simple levers before.

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They're really just kind
of like a seesaw.

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This place where the
lever pivots.

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This is called a fulcrum.

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Just really the pivot point.

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And you can kind of view this
as either a seesaw or a big

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plank of wood on top of a
triangle, which essentially is

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what I've drawn.

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So in this example, I have
the big plank of wood.

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At one end I have this 10
newton weight, and I've

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written 10 in there.

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And what we're going to figure
out is one, how much force--

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well, we could figure out
a couple of things.

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How much force do I have
to apply here to

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just keep this level?

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Because this weight's going
to be pushing downwards.

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So it would naturally
want this whole

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lever to rotate clockwise.

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So what I want to figure out is,
how much force do I have

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to apply to either keep the
lever level or to actually

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rotate this lever
counterclockwise?

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And when I rotate the lever

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counterclockwise, what's happening?

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I'm pushing down on this
left-hand side, and I'm

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lifting this 10 newton block.

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So let's do a little thought
experiment and see what

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happens after I rotate this
lever a little bit.

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So let's say, what I've drawn
here in mauve, that's our

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starting position.

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And in yellow, I'm going to draw
the finishing position.

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So the finishing position
is going to look

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something like this.

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I'll try my best to draw it.

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The finishing position is
something like this.

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And also, one thing I want to
figure out, that I wanted to

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write, is let's say that the
distance, that this distance

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right here, from where I'm
applying the force to the

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fulcrum, let's say that
that distance is 2.

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And from the fulcrum to the
weight that I'm lifting, that

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distance is 1.

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Let's just say that, just for
the sake of argument.

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Let's say it's 2 meters and 1
meter, although it could be 2

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kilometers and 1 kilometer,
we'll soon see.

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And what I did is I pressed down
with some force, and I

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rotated it through
an angle theta.

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So that's theta and this
is also theta.

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So my question to you, and
we'll have to take out a

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little bit of our trigonometry
skills, is how much did this

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object move up?

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So essentially, what
was this distance?

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What's its distance in the
vertical direction?

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How much did it go up?

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And also, for what distance did
I have to apply the force

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downwards here-- so that's this
distance-- in order for

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this weight to move up this
distance over here?

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So let's figure out
either one.

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So this distance is what?

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Well, we have theta.

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This is the opposite.

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This is a 90 degree
angle, because we

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started off at level.

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So this is opposite.

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And this is what?

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This is the adjacent angle.

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So what do we have there?

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Opposite over adjacent.

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Soh Cah Toa.

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Opposite over adjacent.

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Opposite over adjacent.

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That's Toa, or tangent.

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So in this situation, we know
that the tangent of theta is

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equal to-- let's call
this the distance

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that we move the weight.

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soon.

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So that equals opposite over
adjacent, the distance that we

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moved the weight over 1.

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And then if we go on
to this side, we

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can do the same thing.

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Tangent is opposite
over adjacent.

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So let's call this the distance
of the force.

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So here the opposite of the
distance of the force and the

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adjacent is this 2 meters.

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Because this is the hypotenuse
right here.

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So we also have the tangent of
theta-- now you're using this

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triangle-- is equal to
the opposite side.

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The distance of the force
over 2 meters.

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So this is interesting.

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They're both equal to
tangent of theta.

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We don't even have to
figure out what the

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tangent of theta is.

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We know that this quantity is
equal to this quantity.

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And we can write it here.

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We could write the distance of
the force, that's the distance

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that we had to push down on
the side of the lever

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downwards, over 2, is equal to
the distance of the weight.

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The distance the weight traveled
upwards is equal to

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the distance, the weight,
divided by 1.

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Or we could say-- this
1 we can ignore.

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Something divided
by 1 is just 1.

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Or we could say that the
distance of the force is equal

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to 2 times the distance
of the weight.

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And this is interesting, because
now we can apply what

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we just learned here to figure
out what the force was.

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And how do I do that?

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Well, when I'm applying a
force here, over some

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distance, I'm putting energy
into the system.

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I'm doing work.

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Work is just a transfer of
energy into this machine.

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And when I do that, that
machine is actually

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transferring that energy
to this block.

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It's actually doing work on the
block by lifting it up.

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So we know the law of
conservation of energy, and

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we're assuming that this is a
frictionless system, and that

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nothing is being lost to
heat or whatever else.

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So the work in has to be
equal to the work out.

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And so what's the work in?

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Well, it's the force that I'm
applying downward times the

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distance of the force.

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So this is the work in.

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Force times the distance
of the force.

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I'm going to switch colors
just to keep things

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interesting.

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And that has to be the same
thing as the work out.

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Well, what's the work out?

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It's the force of the weight
pulling downwards.

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So we have to-- it's essentially
the lifting force

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of the lever.

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It has to counteract the force
of the weight pulling

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downwards actually.

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Sorry I mis-said it
a little bit.

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But this lever is essentially
going to be

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pushing up on this weight.

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The weight ends up here.

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So it pushes up with the
force equal to the

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weight of the object.

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So that's the weight of the
object, which is -- I said

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it's a 10 newton object -- So
it's equal to 10 newtons.

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That's the force.

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The upward force here.

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And it does that for
a distance of what?

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We figured out this object, this
weight, moves up with a

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distance d sub w.

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And we know what the distance
of the force is in terms of

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the distance of w.

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So we could rewrite this as
force times, substitute here,

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2 d w is equal to 10 d w.

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Divide both sides by 2 you d w
and you get force is equal to

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10 d w 2 two d w, which is
equaled to, d w's cancel out,

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and you're just left with 5.

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So this is interesting.

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And I think you'll see where
this is going, and we did it

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little complicated this time.

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But hopefully you'll realize
a general theme.

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This was a 10 newton weight.

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And I only had to press down
with 5 newtons in order to

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lift it up.

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But at the same time, I pressed
down with 5 newtons,

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but I had to push down
for twice as long.

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So my force was half as much,
but my distance that I had to

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push was twice as much.

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And here the force is twice as
much but the distance it

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traveled is half as much.

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So what essentially just
happened here is, I

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multiplied my force.

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And because I multiplied
my force, I

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essentially lost some distance.

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But I multiplied my
force, because I

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inputted a 5 newton force.

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And I got a 10 newton force out,
although the 10 newton

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force traveled for
less distance.

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Because the work was constant.

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And this is called mechanical
advantage.

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If I have an input force of 5,
and I get an output force of

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10, the mechanical
advantage is 2.

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So mechanical advantage is equal
to output force over

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input force, and that should
hopefully make a little

00:07:59.260 --> 00:08:00.910
intuitive sense to you.

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And another thing that maybe
you're starting to realize

00:08:03.220 --> 00:08:07.110
now, is that proportion of the
mechanical advantage was

00:08:07.110 --> 00:08:13.480
actually the ratio of this
length to this length.

00:08:13.480 --> 00:08:15.660
And we figured that out by
taking the tangent and doing

00:08:15.660 --> 00:08:16.670
these ratios.

00:08:16.670 --> 00:08:20.200
But in general, it makes sense,
because this force

00:08:20.200 --> 00:08:23.340
times this distance has
to be equal to this

00:08:23.340 --> 00:08:25.820
force times this distance.

00:08:25.820 --> 00:08:29.410
And we know that the distance
this goes up is proportional

00:08:29.410 --> 00:08:34.169
to the length of from the
fulcrum to the weight.

00:08:34.169 --> 00:08:36.340
And we know on this side the
distance that you're pushing

00:08:36.340 --> 00:08:39.260
down, is proportional to the
length from where you're

00:08:39.260 --> 00:08:42.020
applying the weight
to the fulcrum.

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And now I'll introduce you
to a concept of moments.

00:08:45.640 --> 00:08:46.890
In just a moment.

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So in general, if I have, and
this is really all you have to

00:08:52.240 --> 00:08:54.350
learn, that last thought
exercise was just

00:08:54.350 --> 00:08:55.370
to show it to you.

00:08:55.370 --> 00:09:03.740
If I have a fulcrum here, and
if we call this distance d 1

00:09:03.740 --> 00:09:06.590
and we called this
distance d 2.

00:09:06.590 --> 00:09:12.335
And if I want to apply
an upward force here,

00:09:12.335 --> 00:09:15.770
let's call this f 1.

00:09:15.770 --> 00:09:20.830
And I have a downward force,
f 2, in this machine.

00:09:20.830 --> 00:09:27.970
f 2 times d 2 is equal
to d 1 times f 1.

00:09:27.970 --> 00:09:30.220
And this is really all
you need to know.

00:09:30.220 --> 00:09:33.630
And this just all falls
out of the work in is

00:09:33.630 --> 00:09:34.880
equal to the work out.

00:09:34.880 --> 00:09:37.540
Now, this quantity isn't
exactly the work in.

00:09:37.540 --> 00:09:41.520
The work in was this force--
sorry, F2-- is this force

00:09:41.520 --> 00:09:43.130
times this distance.

00:09:43.130 --> 00:09:47.370
But this distance is
proportional to this distance,

00:09:47.370 --> 00:09:48.610
and that's what you
need to realize.

00:09:48.610 --> 00:09:51.610
And this quantity right here is
actually called the moment.

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In the next video, which I'll
start very soon because this

00:09:54.560 --> 00:09:55.730
video is about to end.

00:09:55.730 --> 00:09:56.660
I'm running out of time.

00:09:56.660 --> 00:09:59.330
I will use these quantities to
solve a bunch of mechanical

00:09:59.330 --> 00:10:01.560
advantage problems. See


Work/energy problem with friction

WEBVTT
Kind: captions
Language: en

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Welcome back.

00:00:00.000 --> 00:00:01.580
Welcome back.
Welcome back.

00:00:01.580 --> 00:00:03.540
I'll now do another conservation
of energy

00:00:03.540 --> 00:00:06.110
problem, and this time I'll
add another twist. So far,

00:00:06.110 --> 00:00:08.530
everything we've been doing,
energy was conserved by the

00:00:08.530 --> 00:00:09.970
law of conservation.

00:00:09.970 --> 00:00:11.740
But that's because all of the
forces that were acting in

00:00:11.740 --> 00:00:13.810
these systems were conservative
forces.

00:00:13.810 --> 00:00:15.750
And now I'll introduce you to
a problem that has a little

00:00:15.750 --> 00:00:17.810
bit of friction, and we'll see
that some of that energy gets

00:00:17.810 --> 00:00:18.720
lost to friction.

00:00:18.720 --> 00:00:20.340
And we can think about
it a little bit.

00:00:20.340 --> 00:00:22.470
Well where does that
energy go?

00:00:22.470 --> 00:00:23.980
And I'm getting this problem
from the University of

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Oregon's zebu.uoregon.edu.

00:00:27.990 --> 00:00:30.590
And they seem to have some nice
physics problems, so I'll

00:00:30.590 --> 00:00:31.120
use theirs.

00:00:31.120 --> 00:00:32.530
And I just want to make
sure they get credit.

00:00:32.530 --> 00:00:32.860
So let's see.

00:00:32.860 --> 00:00:35.910
They say a 90 kilogram
bike and rider.

00:00:35.910 --> 00:00:38.725
So the bike and rider combined
are 90 kilograms. So let's

00:00:38.725 --> 00:00:45.920
just say the mass is 90
kilograms. Start at rest from

00:00:45.920 --> 00:00:48.840
the top of a 500 meter
long hill.

00:00:48.840 --> 00:00:50.860
OK, so I think they mean
that the hill is

00:00:50.860 --> 00:00:51.710
something like this.

00:00:51.710 --> 00:00:57.550
So if this is the hill, that
the hypotenuse here is 500

00:00:57.550 --> 00:00:58.270
hundred meters long.

00:00:58.270 --> 00:01:04.160
So the length of that,
this is 500 meters.

00:01:04.160 --> 00:01:07.420
A 500 meter long hill with
a 5 degree incline.

00:01:07.420 --> 00:01:08.670
So this is 5 degrees.

00:01:11.850 --> 00:01:14.370
And we can kind of just view
it like a wedge, like we've

00:01:14.370 --> 00:01:18.070
done in other problems.
There you go.

00:01:18.070 --> 00:01:20.890
That's pretty straight.

00:01:20.890 --> 00:01:22.550
OK.

00:01:22.550 --> 00:01:26.050
Assuming an average friction
force of 60 newtons.

00:01:26.050 --> 00:01:28.490
OK, so they're not telling us
the coefficient of friction

00:01:28.490 --> 00:01:29.580
and then we have to figure
out the normal

00:01:29.580 --> 00:01:30.620
force and all of that.

00:01:30.620 --> 00:01:33.630
They're just telling us, what
is the drag of friction?

00:01:33.630 --> 00:01:36.430
Or how much is actually friction
acting against this

00:01:36.430 --> 00:01:38.750
rider's motion?

00:01:38.750 --> 00:01:40.360
We could think a little bit
about where that friction is

00:01:40.360 --> 00:01:40.880
coming from.

00:01:40.880 --> 00:01:46.660
So the force of friction is
equal to 60 newtons And of

00:01:46.660 --> 00:01:49.030
course, this is going to be
going against his motion or

00:01:49.030 --> 00:01:50.040
her motion.

00:01:50.040 --> 00:01:53.150
And the question asks us, find
the speed of the biker at the

00:01:53.150 --> 00:01:54.110
bottom of the hill.

00:01:54.110 --> 00:01:58.510
So the biker starts up
here, stationary.

00:01:58.510 --> 00:01:59.240
That's the biker.

00:01:59.240 --> 00:02:01.560
My very artful rendition
of the biker.

00:02:01.560 --> 00:02:05.230
And we need to figure out the
velocity at the bottom.

00:02:05.230 --> 00:02:07.920
This to some degree is a
potential energy problem.

00:02:11.380 --> 00:02:14.790
It's definitely a conservation
of mechanical energy problem.

00:02:14.790 --> 00:02:17.610
So let's figure out what the
energy of the system is when

00:02:17.610 --> 00:02:19.550
the rider starts off.

00:02:19.550 --> 00:02:21.360
So the rider starts off at
the top of this hill.

00:02:21.360 --> 00:02:23.930
So definitely some
potential energy.

00:02:23.930 --> 00:02:26.850
And is stationary, so there's
no kinetic energy.

00:02:26.850 --> 00:02:28.760
So all of the energy is
potential, and what is the

00:02:28.760 --> 00:02:29.600
potential energy?

00:02:29.600 --> 00:02:35.030
Well potential energy is equal
to mass times the acceleration

00:02:35.030 --> 00:02:37.950
of gravity times
height, right?

00:02:37.950 --> 00:02:41.300
Well that's equal to, if the
mass is 90, the acceleration

00:02:41.300 --> 00:02:43.830
of gravity is 9.8 meters
per second squared.

00:02:43.830 --> 00:02:44.540
And then what's the height?

00:02:44.540 --> 00:02:45.960
Well here we're going to have
to break out a little

00:02:45.960 --> 00:02:46.810
trigonometry.

00:02:46.810 --> 00:02:51.010
We need to figure out this side
of this triangle, if you

00:02:51.010 --> 00:02:53.270
consider this whole
thing a triangle.

00:02:53.270 --> 00:02:53.630
Let's see.

00:02:53.630 --> 00:02:55.080
We want to figure out
the opposite.

00:02:55.080 --> 00:02:58.630
We know the hypotenuse and
we know this angle here.

00:02:58.630 --> 00:03:01.920
So the sine of this angle is
equal to opposite over

00:03:01.920 --> 00:03:03.440
hypotenuse.

00:03:03.440 --> 00:03:04.680
So, SOH.

00:03:04.680 --> 00:03:06.910
Sine is opposite over
hypotenuse.

00:03:06.910 --> 00:03:10.490
So we know that the height--
so let me do a little work

00:03:10.490 --> 00:03:15.010
here-- we know that sine of
5 degrees is equal to

00:03:15.010 --> 00:03:17.510
the height over 500.

00:03:17.510 --> 00:03:25.420
Or that the height is equal
to 500 sine of 5 degrees.

00:03:25.420 --> 00:03:28.470
And I calculated the sine of
5 degrees ahead of time.

00:03:28.470 --> 00:03:30.380
Let me make sure I
still have it.

00:03:30.380 --> 00:03:32.820
That's cause I didn't have my
calculator with me today.

00:03:32.820 --> 00:03:34.290
But you could do this
on your own.

00:03:34.290 --> 00:03:38.400
So this is equal to 500,
and the sine of

00:03:38.400 --> 00:03:42.806
5 degrees is 0.087.

00:03:42.806 --> 00:03:46.870
So when you multiply these
out, what do I get?

00:03:46.870 --> 00:03:49.220
I'm using the calculator
on Google actually.

00:03:49.220 --> 00:03:52.400
500 times sine.

00:03:52.400 --> 00:03:54.670
You get 43.6.

00:03:54.670 --> 00:03:58.900
So this is equal to 43.6.

00:03:58.900 --> 00:04:03.970
So the height of the hill
is 43.6 meters.

00:04:03.970 --> 00:04:06.390
So going back to the potential
energy, we have the mass times

00:04:06.390 --> 00:04:07.850
the acceleration of gravity
times the height.

00:04:07.850 --> 00:04:09.740
Times 43.6.

00:04:09.740 --> 00:04:12.290
And this is equal to, and then
I can use just my regular

00:04:12.290 --> 00:04:13.640
calculator since I don't
have to figure out

00:04:13.640 --> 00:04:15.090
trig functions anymore.

00:04:15.090 --> 00:04:25.380
So 90-- so you can see the whole
thing-- times 9.8 times

00:04:25.380 --> 00:04:34.860
43.6 is equal to, let's
see, roughly 38,455.

00:04:34.860 --> 00:04:42.180
So this is equal to 38,455
joules or newton meters.

00:04:42.180 --> 00:04:44.270
And that's a lot of
potential energy.

00:04:44.270 --> 00:04:44.980
So what happens?

00:04:44.980 --> 00:04:47.690
At the bottom of the hill--
sorry, I have to readjust my

00:04:47.690 --> 00:04:52.070
chair-- at the bottom of the
hill, all of this gets

00:04:52.070 --> 00:04:53.590
converted to, or maybe I should

00:04:53.590 --> 00:04:54.260
pose that as a question.

00:04:54.260 --> 00:04:57.450
Does all of it get converted
to kinetic energy?

00:04:57.450 --> 00:05:00.560
Almost. We have a force
of friction here.

00:05:00.560 --> 00:05:03.880
And friction, you can kind of
view friction as something

00:05:03.880 --> 00:05:08.120
that eats up mechanical
energy.

00:05:08.120 --> 00:05:10.250
These are also called
nonconservative forces because

00:05:10.250 --> 00:05:12.590
when you have these forces
at play, all of the

00:05:12.590 --> 00:05:14.580
force is not conserved.

00:05:14.580 --> 00:05:19.160
So a way to think about it is,
is that the energy, let's just

00:05:19.160 --> 00:05:20.430
call it total energy.

00:05:20.430 --> 00:05:30.670
So let's say total energy
initial, well let me just

00:05:30.670 --> 00:05:40.740
write initial energy is equal
to the energy wasted in

00:05:40.740 --> 00:05:51.870
friction-- I should have written
just letters-- from

00:05:51.870 --> 00:06:00.170
friction plus final energy.

00:06:00.170 --> 00:06:03.670
So we know what the initial
energy is in this system.

00:06:03.670 --> 00:06:07.120
That's the potential energy
of this bicyclist and this

00:06:07.120 --> 00:06:12.730
roughly 38 and 1/2 kilojoules
or 38,500 joules, roughly.

00:06:12.730 --> 00:06:17.530
And now let's figure out the
energy wasted from friction,

00:06:17.530 --> 00:06:20.110
and the energy wasted from
friction is the negative work

00:06:20.110 --> 00:06:20.850
that friction does.

00:06:20.850 --> 00:06:22.470
And what does negative
work mean?

00:06:22.470 --> 00:06:29.000
Well the bicyclist is moving 500
meters in this direction.

00:06:29.000 --> 00:06:30.740
So distance is 500 meters.

00:06:30.740 --> 00:06:33.050
But friction isn't acting
along the same

00:06:33.050 --> 00:06:34.290
direction as distance.

00:06:34.290 --> 00:06:37.470
The whole time, friction is
acting against the distance.

00:06:37.470 --> 00:06:40.690
So when the force is going in
the opposite direction as the

00:06:40.690 --> 00:06:42.430
distance, your work
is negative.

00:06:45.440 --> 00:06:49.500
So another way of thinking of
this problem is energy initial

00:06:49.500 --> 00:06:56.420
is equal to, or you could say
the energy initial plus the

00:06:56.420 --> 00:06:58.370
negative work of friction,
right?

00:06:58.370 --> 00:07:02.090
If we say that this is a
negative quantity, then this

00:07:02.090 --> 00:07:05.730
is equal to the final energy.

00:07:05.730 --> 00:07:07.920
And here, I took the friction
and put it on the other side

00:07:07.920 --> 00:07:10.410
because I said this is going to
be a negative quantity in

00:07:10.410 --> 00:07:11.290
the system.

00:07:11.290 --> 00:07:13.680
And so you should always just
make sure that if you have

00:07:13.680 --> 00:07:15.770
friction in the system, just as
a reality check, that your

00:07:15.770 --> 00:07:18.100
final energy is less than
your initial energy.

00:07:18.100 --> 00:07:25.500
Our initial energy is, let's
just say 38.5 kilojoules.

00:07:25.500 --> 00:07:28.290
What is the negative work
that friction is doing?

00:07:28.290 --> 00:07:29.530
Well it's 500 meters.

00:07:29.530 --> 00:07:33.600
And the entire 500 meters, it's
always pushing back on

00:07:33.600 --> 00:07:36.420
the rider with a force
of 60 newtons.

00:07:36.420 --> 00:07:37.960
So force times distance.

00:07:37.960 --> 00:07:40.690
So it's minus 60 newtons,
cause it's going in the

00:07:40.690 --> 00:07:44.530
opposite direction of the
motion, times 500.

00:07:44.530 --> 00:07:49.000
And this is going to equal
the ending, oh, no.

00:07:49.000 --> 00:07:53.460
This is going to equal the
final energy, right?

00:07:53.460 --> 00:07:54.030
And what is this?

00:07:54.030 --> 00:08:00.100
60 times 500, that's 3,000.

00:08:00.100 --> 00:08:01.190
No, 30,000, right.

00:08:01.190 --> 00:08:04.975
So let's subtract 30,000
from 38,500.

00:08:04.975 --> 00:08:06.490
So let's see.

00:08:06.490 --> 00:08:09.580
Minus 30.

00:08:09.580 --> 00:08:10.460
I didn't have to do that.

00:08:10.460 --> 00:08:11.780
I could have done
that in my head.

00:08:11.780 --> 00:08:19.990
So that gives us 8,455 joules is
equal to the final energy.

00:08:19.990 --> 00:08:21.820
And what is all the
final energy?

00:08:21.820 --> 00:08:24.530
Well by this time, the rider's
gotten back to, I guess we

00:08:24.530 --> 00:08:25.410
could call the sea level.

00:08:25.410 --> 00:08:26.890
So all of the energy
is now going to be

00:08:26.890 --> 00:08:28.930
kinetic energy, right?

00:08:28.930 --> 00:08:30.510
What's the formula for
kinetic energy?

00:08:30.510 --> 00:08:35.059
It's 1/2 mv squared.

00:08:35.059 --> 00:08:36.580
And we know what m is.

00:08:36.580 --> 00:08:38.126
The mass of the rider is 90.

00:08:38.126 --> 00:08:42.480
So we have this is 90.

00:08:42.480 --> 00:08:44.610
So if we divide both sides.

00:08:44.610 --> 00:08:45.850
So the 1/2 times 90.

00:08:45.850 --> 00:08:48.160
That's 45.

00:08:48.160 --> 00:08:50.140
So 8,455 divided by 45.

00:08:50.140 --> 00:08:59.540
So we get v squared
is equal to 187.9.

00:08:59.540 --> 00:09:01.500
And let's take the square root
of that and we get the

00:09:01.500 --> 00:09:06.690
velocity, 13.7.

00:09:06.690 --> 00:09:09.340
So if we take the square root of
both sides of this, so the

00:09:09.340 --> 00:09:12.900
final velocity is 13.7.

00:09:12.900 --> 00:09:14.270
I know you can't read that.

00:09:14.270 --> 00:09:17.510
13.7 meters per second.

00:09:17.510 --> 00:09:20.960
And this was a slightly more
interesting problem because

00:09:20.960 --> 00:09:24.480
here we had the energy wasn't
completely conserved.

00:09:24.480 --> 00:09:27.460
Some of the energy, you could
say, was eaten by friction.

00:09:27.460 --> 00:09:28.700
And actually that energy
just didn't

00:09:28.700 --> 00:09:30.080
disappear into a vacuum.

00:09:30.080 --> 00:09:32.860
It was actually generated
into heat.

00:09:32.860 --> 00:09:33.490
And it makes sense.

00:09:33.490 --> 00:09:37.930
If you slid down a slide of
sandpaper, your pants would

00:09:37.930 --> 00:09:40.750
feel very warm by the time you
got to the bottom of that.

00:09:40.750 --> 00:09:43.400
But the friction of this, they
weren't specific on where the

00:09:43.400 --> 00:09:45.420
friction came from, but it
could have come from the

00:09:45.420 --> 00:09:46.420
gearing within the bike.

00:09:46.420 --> 00:09:48.550
It could have come
from the wind.

00:09:48.550 --> 00:09:50.470
Maybe the bike actually
skidded a little

00:09:50.470 --> 00:09:51.280
bit on the way down.

00:09:51.280 --> 00:09:52.190
I don't know.

00:09:52.190 --> 00:09:54.290
But hopefully you found that
a little bit interesting.

00:09:54.290 --> 00:09:57.230
And now you can not only work
with conservation of

00:09:57.230 --> 00:09:59.950
mechanical energy, but you can
work problems where there's a

00:09:59.950 --> 00:10:01.820
little bit of friction
involved as well.

00:10:01.820 --> 00:10:03.380
Anyway, I'll see you
in the next video.


Conservation of energy

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.550
Welcome back.

00:00:01.550 --> 00:00:02.935
At the end of the last
video, I left you

00:00:02.935 --> 00:00:03.990
with a bit of a question.

00:00:03.990 --> 00:00:08.370
We had a situation where we
had a 1 kilogram object.

00:00:08.370 --> 00:00:11.890
This is the 1 kilogram object,
which I've drawn neater in

00:00:11.890 --> 00:00:12.955
this video.

00:00:12.955 --> 00:00:15.910
That is 1 kilogram.

00:00:15.910 --> 00:00:20.110
And we're on earth, and I need
to mention that because

00:00:20.110 --> 00:00:22.380
gravity is different from
planet to planet.

00:00:22.380 --> 00:00:24.670
But as I mentioned,
I'm holding it.

00:00:24.670 --> 00:00:27.200
Let's say I'm holding it 10
meters above the ground.

00:00:27.200 --> 00:00:34.590
So this distance or this
height is 10 meters.

00:00:34.590 --> 00:00:39.600
And we're assuming the
acceleration of gravity, which

00:00:39.600 --> 00:00:42.480
we also write as just g, let's
assume it's just 10 meters per

00:00:42.480 --> 00:00:44.680
second squared just for the
simplicity of the math instead

00:00:44.680 --> 00:00:45.880
of the 9.8.

00:00:45.880 --> 00:00:48.560
So what we learned in the last
video is that the potential

00:00:48.560 --> 00:00:53.690
energy in this situation, the
potential energy, which equals

00:00:53.690 --> 00:00:59.740
m times g times h is equal to
the mass is 1 kilogram times

00:00:59.740 --> 00:01:02.300
the acceleration of gravity,
which is 10

00:01:02.300 --> 00:01:03.490
meters per second squared.

00:01:03.490 --> 00:01:05.349
I'm not going to write the units
down just to save space,

00:01:05.349 --> 00:01:08.890
although you should do this when
you do it on your test.

00:01:08.890 --> 00:01:11.920
And then the height
is 10 meters.

00:01:11.920 --> 00:01:13.910
And the units, if you work them
all out, it's in newton

00:01:13.910 --> 00:01:17.960
meters or joules and so it's
equal to 100 joules.

00:01:17.960 --> 00:01:20.400
That's the potential energy when
I'm holding it up there.

00:01:20.400 --> 00:01:22.290
And I asked you, well when
I let go, what happens?

00:01:22.290 --> 00:01:24.490
Well the block obviously
will start falling.

00:01:24.490 --> 00:01:26.370
And not only falling, it will
start accelerating to the

00:01:26.370 --> 00:01:29.870
ground at 10 meters per second
squared roughly.

00:01:29.870 --> 00:01:33.940
And right before it hits the
ground-- let me draw that in

00:01:33.940 --> 00:01:38.640
brown for ground-- right before
the object hits the

00:01:38.640 --> 00:01:43.130
ground or actually right when it
hits the ground, what will

00:01:43.130 --> 00:01:46.490
be the potential energy
of the object?

00:01:46.490 --> 00:01:48.070
Well it has no height, right?

00:01:48.070 --> 00:01:49.790
Potential energy is mgh.

00:01:49.790 --> 00:01:51.810
The mass and the acceleration of
gravity stay the same, but

00:01:51.810 --> 00:01:52.470
the height is 0.

00:01:52.470 --> 00:01:54.460
So they're all multiplied
by each other.

00:01:54.460 --> 00:01:56.540
So down here, the potential
energy is going

00:01:56.540 --> 00:01:58.360
to be equal to 0.

00:01:58.360 --> 00:02:00.120
And I told you in the last video
that we have the law of

00:02:00.120 --> 00:02:01.160
conservation of energy.

00:02:01.160 --> 00:02:02.950
That energy is conserved.

00:02:02.950 --> 00:02:04.510
It cannot be created
or destroyed.

00:02:04.510 --> 00:02:06.920
It can just be converted from
one form to another.

00:02:06.920 --> 00:02:10.919
But I'm just showing you, this
object had 100 joules of

00:02:10.919 --> 00:02:12.010
energy or, in this case,

00:02:12.010 --> 00:02:13.920
gravitational potential energy.

00:02:13.920 --> 00:02:16.820
And down here, it
has no energy.

00:02:16.820 --> 00:02:18.780
Or at least it has no
gravitational potential

00:02:18.780 --> 00:02:20.130
energy, and that's the key.

00:02:20.130 --> 00:02:22.800
That gravitational potential
energy was converted into

00:02:22.800 --> 00:02:23.920
something else.

00:02:23.920 --> 00:02:25.480
And that something else
it was converted

00:02:25.480 --> 00:02:28.070
into is kinetic energy.

00:02:28.070 --> 00:02:32.800
And in this case, since it has
no potential energy, all of

00:02:32.800 --> 00:02:35.580
that previous potential energy,
all of this 100 joules

00:02:35.580 --> 00:02:39.880
that it has up here is now going
to be converted into

00:02:39.880 --> 00:02:40.800
kinetic energy.

00:02:40.800 --> 00:02:43.620
And we can use that information
to figure out its

00:02:43.620 --> 00:02:46.980
velocity right before
it hits the ground.

00:02:46.980 --> 00:02:47.760
So how do we do that?

00:02:47.760 --> 00:02:49.770
Well what's the formula
for kinetic energy?

00:02:49.770 --> 00:02:52.615
And we solved it two videos
ago, and hopefully it

00:02:52.615 --> 00:02:53.677
shouldn't be too much
of a mystery to you.

00:02:53.677 --> 00:02:55.840
It's something good to memorize,
but it's also good

00:02:55.840 --> 00:02:58.870
to know how we got it and go
back two videos if you forgot.

00:03:01.430 --> 00:03:05.440
So first we know that all the
potential energy was converted

00:03:05.440 --> 00:03:07.450
into kinetic energy.

00:03:07.450 --> 00:03:10.330
We had 100 joules of potential
energy, so we're still going

00:03:10.330 --> 00:03:12.210
to have 100 joules, but now
all of it's going to be

00:03:12.210 --> 00:03:13.350
kinetic energy.

00:03:13.350 --> 00:03:17.810
And kinetic energy is
1/2 mv squared.

00:03:17.810 --> 00:03:21.080
So we know that 1/2 mv squared,
or the kinetic

00:03:21.080 --> 00:03:24.990
energy, is now going to
equal 100 joules.

00:03:24.990 --> 00:03:25.710
What's the mass?

00:03:25.710 --> 00:03:27.960
The mass is 1.

00:03:27.960 --> 00:03:29.930
And we can solve for v now.

00:03:29.930 --> 00:03:31.910
1/2 v squared equals
100 joules, and v

00:03:31.910 --> 00:03:36.050
squared is equal to 200.

00:03:36.050 --> 00:03:39.310
And then we get v is equal to
square root of 200, which is

00:03:39.310 --> 00:03:40.560
something over 14.

00:03:40.560 --> 00:03:41.860
We can get the exact number.

00:03:41.860 --> 00:03:46.725
Let's see, 200 square
root, 14.1 roughly.

00:03:46.725 --> 00:03:51.360
The velocity is going to
be 14.1 meters per

00:03:51.360 --> 00:03:52.870
second squared downwards.

00:03:52.870 --> 00:03:55.500
Right before the object
touches the ground.

00:03:55.500 --> 00:03:56.930
Right before it touches
the ground.

00:03:56.930 --> 00:03:59.180
And you might say, well Sal
that's nice and everything.

00:03:59.180 --> 00:04:01.170
We learned a little
bit about energy.

00:04:01.170 --> 00:04:02.890
I could have solved that or
hopefully you could have

00:04:02.890 --> 00:04:04.990
solved that problem just using
your kinematics formula.

00:04:04.990 --> 00:04:07.550
So what's the whole point
of introducing

00:04:07.550 --> 00:04:09.590
these concepts of energy?

00:04:09.590 --> 00:04:11.280
And I will now show you.

00:04:11.280 --> 00:04:14.640
So let's say they have the same
1 kilogram object up here

00:04:14.640 --> 00:04:17.529
and it's 10 meters in the air,
but I'm going to change things

00:04:17.529 --> 00:04:18.779
a little bit.

00:04:20.660 --> 00:04:24.970
Let me see if I can competently
erase all of this.

00:04:24.970 --> 00:04:26.940
Nope, that's not what
I wanted to do.

00:04:29.920 --> 00:04:31.900
OK, there you go.

00:04:31.900 --> 00:04:37.150
I'm trying my best to erase
this, all of this stuff.

00:04:37.150 --> 00:04:38.570
OK.

00:04:38.570 --> 00:04:40.770
So I have the same object.

00:04:40.770 --> 00:04:43.025
It's still 10 meters
in the air and I'll

00:04:43.025 --> 00:04:45.390
write that in a second.

00:04:45.390 --> 00:04:47.160
And I'm just holding it there
and I'm still going to drop

00:04:47.160 --> 00:04:50.130
it, but something interesting
is going to happen.

00:04:50.130 --> 00:04:51.880
Instead of it going straight
down, it's actually going to

00:04:51.880 --> 00:04:54.610
drop on this ramp of ice.

00:04:57.180 --> 00:04:58.710
The ice has lumps on it.

00:05:01.600 --> 00:05:02.660
And then this is the bottom.

00:05:02.660 --> 00:05:04.510
This is the ground down here.

00:05:04.510 --> 00:05:07.310
This is the ground.

00:05:07.310 --> 00:05:09.070
So what's going to
happen this time?

00:05:09.070 --> 00:05:11.130
I'm still 10 meters in the
air, so let me draw that.

00:05:11.130 --> 00:05:12.130
That's still 10 meters.

00:05:12.130 --> 00:05:15.360
I should switch colors just
so not everything is ice.

00:05:15.360 --> 00:05:18.190
So that's still 10 meters, but
instead of the object going

00:05:18.190 --> 00:05:21.090
straight down now, it's going to
go down here and then start

00:05:21.090 --> 00:05:21.840
sliding, right?

00:05:21.840 --> 00:05:24.080
It's going to go sliding
along this hill.

00:05:24.080 --> 00:05:27.870
And then at this point it's
going to be going really fast

00:05:27.870 --> 00:05:29.010
in the horizontal direction.

00:05:29.010 --> 00:05:31.760
And right now we don't
know how fast.

00:05:31.760 --> 00:05:34.630
And just using our kinematics
formula, this would have been

00:05:34.630 --> 00:05:35.880
a really tough formula.

00:05:35.880 --> 00:05:38.630
This would have been
difficult.

00:05:38.630 --> 00:05:40.400
I mean you could have attempted
it and it actually

00:05:40.400 --> 00:05:42.990
would have taken calculus
because the angle of the slope

00:05:42.990 --> 00:05:44.150
changes continuously.

00:05:44.150 --> 00:05:46.990
We don't even know the formula
for the angle of the slope.

00:05:46.990 --> 00:05:48.410
You would have had to break
it out into vectors.

00:05:48.410 --> 00:05:49.690
You would have to do all sorts
of complicated things.

00:05:49.690 --> 00:05:52.050
This would have been a nearly
impossible problem.

00:05:52.050 --> 00:05:55.050
But using energy, we can
actually figure out what the

00:05:55.050 --> 00:05:58.680
velocity of this object
is at this point.

00:05:58.680 --> 00:06:00.880
And we use the same idea.

00:06:00.880 --> 00:06:03.330
Here we have 100 joules
of potential energy.

00:06:03.330 --> 00:06:05.070
We just figured that out.

00:06:05.070 --> 00:06:07.190
Down here, what's the height
above the ground?

00:06:07.190 --> 00:06:08.570
Well the height is 0.

00:06:08.570 --> 00:06:10.580
So all the potential energy
has disappeared.

00:06:10.580 --> 00:06:13.270
And just like in the previous
situation, all of the

00:06:13.270 --> 00:06:16.530
potential energy is now
converted into kinetic energy.

00:06:16.530 --> 00:06:18.520
And so what is that kinetic
energy going to equal?

00:06:18.520 --> 00:06:21.440
It's going to be equal to the
initial potential energy.

00:06:21.440 --> 00:06:27.150
So here the kinetic energy
is equal to 100 joules.

00:06:27.150 --> 00:06:30.090
And that equals 1/2
mv squared, just

00:06:30.090 --> 00:06:31.300
like we just solved.

00:06:31.300 --> 00:06:34.380
And if you solve for v, the
mass is 1 kilogram.

00:06:34.380 --> 00:06:39.690
So the velocity in the
horizontal direction will be,

00:06:39.690 --> 00:06:42.400
if you solve for it, 14.1
meters per second.

00:06:42.400 --> 00:06:44.070
Instead of going straight down,
now it's going to be

00:06:44.070 --> 00:06:47.300
going in the horizontal
to the right.

00:06:47.300 --> 00:06:49.270
And the reason why I said it
was ice is because I wanted

00:06:49.270 --> 00:06:52.360
this to be frictionless and I
didn't want any energy lost to

00:06:52.360 --> 00:06:53.760
heat or anything like that.

00:06:53.760 --> 00:06:56.320
And you might say OK Sal, that's
kind of interesting.

00:06:56.320 --> 00:06:59.990
And you kind of got the same
number for the velocity than

00:06:59.990 --> 00:07:01.870
if I just dropped the object
straight down.

00:07:01.870 --> 00:07:02.660
And that's interesting.

00:07:02.660 --> 00:07:07.910
But what else can
this do for me?

00:07:07.910 --> 00:07:10.100
And this is where it's
really cool.

00:07:10.100 --> 00:07:15.080
Not only can I figure out the
velocity when all of the

00:07:15.080 --> 00:07:17.100
potential energy has
disappeared, but I can figure

00:07:17.100 --> 00:07:19.180
out the velocity of any
point-- and this is

00:07:19.180 --> 00:07:21.460
fascinating-- along
this slide.

00:07:21.460 --> 00:07:25.200
So let's say when the box is
sliding down here, so let's

00:07:25.200 --> 00:07:29.600
say the box is at this point.

00:07:29.600 --> 00:07:31.630
It changes colors
too as it falls.

00:07:31.630 --> 00:07:34.510
So this is the 1 kilogram
box, right?

00:07:34.510 --> 00:07:35.840
It falls and it slides
down here.

00:07:35.840 --> 00:07:40.480
And let's say at this point it's
height above the ground

00:07:40.480 --> 00:07:42.750
is 5 meters.

00:07:42.750 --> 00:07:44.860
So what's its potential
energy here?

00:07:44.860 --> 00:07:45.770
So let's just write something.

00:07:45.770 --> 00:07:47.810
All of the energy is
conserved, right?

00:07:47.810 --> 00:07:51.080
So the initial potential
energy plus the initial

00:07:51.080 --> 00:07:57.360
kinetic energy is equal to the
final potential energy plus

00:07:57.360 --> 00:07:59.960
the final kinetic energy.

00:07:59.960 --> 00:08:02.100
I'm just saying energy
is conserved here.

00:08:02.100 --> 00:08:05.580
Up here, what's the initial
total energy in the system?

00:08:05.580 --> 00:08:08.400
Well the potential energy is 100
and the kinetic energy is

00:08:08.400 --> 00:08:10.420
0 because it's stationary.

00:08:10.420 --> 00:08:11.880
I haven't dropped it.

00:08:11.880 --> 00:08:13.090
I haven't let go of it yet.

00:08:13.090 --> 00:08:14.360
It's just stationary.

00:08:14.360 --> 00:08:18.530
So the initial energy is going
to be equal to 100 joules.

00:08:18.530 --> 00:08:20.830
That's cause this is
0 and this is 100.

00:08:20.830 --> 00:08:22.840
So the initial energy
is 100 joules.

00:08:22.840 --> 00:08:27.500
At this point right here, what's
the potential energy?

00:08:27.500 --> 00:08:32.270
Well we're 5 meters
up, so mass times

00:08:32.270 --> 00:08:32.990
gravity times height.

00:08:32.990 --> 00:08:36.900
Mass is 1, times gravity, 10
meters per second squared.

00:08:36.900 --> 00:08:39.049
Times height, times 5.

00:08:39.049 --> 00:08:40.630
So it's 50 joules.

00:08:40.630 --> 00:08:43.409
That's our potential energy
at this point.

00:08:43.409 --> 00:08:47.130
And then we must have some
kinetic energy with the

00:08:47.130 --> 00:08:48.250
velocity going roughly
in that direction.

00:08:48.250 --> 00:08:51.950
Plus our kinetic energy
at this point.

00:08:51.950 --> 00:08:54.560
And we know that no energy
was destroyed.

00:08:54.560 --> 00:08:55.330
It's just converted.

00:08:55.330 --> 00:08:59.080
So we know the total energy
still has to be 100 joules.

00:08:59.080 --> 00:09:01.410
So essentially what happened,
and if we solve for this--

00:09:01.410 --> 00:09:03.430
it's very easy, subtract 50 from
both sides-- we know that

00:09:03.430 --> 00:09:06.120
the kinetic energy is
now also going to

00:09:06.120 --> 00:09:07.100
be equal to 50 joules.

00:09:07.100 --> 00:09:07.890
So what happened?

00:09:07.890 --> 00:09:12.370
Halfway down, essentially half
of the potential energy got

00:09:12.370 --> 00:09:14.120
converted to kinetic energy.

00:09:14.120 --> 00:09:16.120
And we can use this information
that the kinetic

00:09:16.120 --> 00:09:18.230
energy is 50 joules
to figure out the

00:09:18.230 --> 00:09:20.170
velocity at this point.

00:09:20.170 --> 00:09:24.840
1/2 mv squared is equal to 50.

00:09:24.840 --> 00:09:26.080
The mass is 1.

00:09:26.080 --> 00:09:27.530
Multiply both sides by 2.

00:09:27.530 --> 00:09:30.370
You get v squared
is equal to 100.

00:09:30.370 --> 00:09:34.570
The velocity is 10 meters
per second along

00:09:34.570 --> 00:09:37.390
this crazy, icy slide.

00:09:37.390 --> 00:09:39.970
And that is something that I
would have challenged you to

00:09:39.970 --> 00:09:42.670
solve using traditional
kinematics formulas,

00:09:42.670 --> 00:09:46.500
especially considering that we
don't know really much about

00:09:46.500 --> 00:09:49.490
the surface of this slide.

00:09:49.490 --> 00:09:51.930
And even if we did, that would
have been a million times

00:09:51.930 --> 00:09:55.440
harder than just using the law
of conservation of energy and

00:09:55.440 --> 00:09:58.380
realizing that at this point,
half the potential energy is

00:09:58.380 --> 00:10:00.620
now kinetic energy and
it's going along the

00:10:00.620 --> 00:10:02.620
direction of the slide.

00:10:02.620 --> 00:10:03.870
I will see you in
the next video.


Introduction to work and energy

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:01.760
Welcome back.

00:00:01.760 --> 00:00:02.980
I'm now going to introduce
you to the

00:00:02.980 --> 00:00:05.310
concepts of work and energy.

00:00:05.310 --> 00:00:08.490
And these are two words that
are-- I'm sure you use in your

00:00:08.490 --> 00:00:10.220
everyday life already and
you have some notion

00:00:10.220 --> 00:00:11.760
of what they mean.

00:00:11.760 --> 00:00:13.690
But maybe just not in the
physics context, although

00:00:13.690 --> 00:00:15.340
they're not completely
unrelated.

00:00:15.340 --> 00:00:18.400
So work, you know
what work is.

00:00:18.400 --> 00:00:20.280
Work is when you do something.

00:00:20.280 --> 00:00:21.870
You go to work, you
make a living.

00:00:21.870 --> 00:00:25.480
In physics, work is-- and I'm
going to use a lot of words

00:00:25.480 --> 00:00:28.500
and they actually end up being
kind of circular in their

00:00:28.500 --> 00:00:29.350
definitions.

00:00:29.350 --> 00:00:31.220
But I think when we start doing
the math, you'll start

00:00:31.220 --> 00:00:34.450
to get at least a slightly more
intuitive notion of what

00:00:34.450 --> 00:00:35.250
they all are.

00:00:35.250 --> 00:00:38.580
So work is energy transferred
by a force.

00:00:38.580 --> 00:00:43.910
So I'll write that down, energy
transferred-- and I got

00:00:43.910 --> 00:00:47.480
this from Wikipedia because
I wanted a good, I guess,

00:00:47.480 --> 00:00:49.780
relatively intuitive
definition.

00:00:49.780 --> 00:00:51.880
Energy transferred by a force.

00:00:51.880 --> 00:00:54.360
And that makes reasonable
sense to me.

00:00:54.360 --> 00:00:56.590
But then you're wondering, well,
I know what a force is,

00:00:56.590 --> 00:00:58.400
you know, force is mass
times acceleration.

00:00:58.400 --> 00:00:59.670
But what is energy?

00:00:59.670 --> 00:01:04.069
And then I looked up energy on
Wikipedia and I found this,

00:01:04.069 --> 00:01:05.190
well, entertaining.

00:01:05.190 --> 00:01:08.560
But it also I think tells you
something that these are just

00:01:08.560 --> 00:01:12.770
concepts that we use to, I
guess, work with what we

00:01:12.770 --> 00:01:16.900
perceive as motion and force
and work and all of these

00:01:16.900 --> 00:01:17.630
types of things.

00:01:17.630 --> 00:01:22.160
But they really aren't
independent notions.

00:01:22.160 --> 00:01:23.380
They're related.

00:01:23.380 --> 00:01:27.150
So Wikipedia defines energy
as the ability to do work.

00:01:27.150 --> 00:01:29.430
So they kind of use each other
to define each other.

00:01:29.430 --> 00:01:33.090
Ability to do work.

00:01:33.090 --> 00:01:37.360
Which is frankly, as good of a
definition as I could find.

00:01:37.360 --> 00:01:40.520
And so, with just the words,
these kind of don't give you

00:01:40.520 --> 00:01:41.390
much information.

00:01:41.390 --> 00:01:43.690
So what I'm going to do is move
onto the equations, and

00:01:43.690 --> 00:01:46.000
this'll give you a more
quantitative feel of what

00:01:46.000 --> 00:01:47.540
these words mean.

00:01:47.540 --> 00:01:55.470
So the definition of work in
mechanics, work is equal to

00:01:55.470 --> 00:01:58.665
force times distance.

00:02:01.890 --> 00:02:04.030
So let's say that I have a block
and-- let me do it in a

00:02:04.030 --> 00:02:05.720
different color just
because this yellow

00:02:05.720 --> 00:02:07.540
might be getting tedious.

00:02:07.540 --> 00:02:11.840
And I apply a force of--
let's say I apply

00:02:11.840 --> 00:02:17.740
a force of 10 Newtons.

00:02:17.740 --> 00:02:22.390
And I move that block
by applying

00:02:22.390 --> 00:02:24.210
a force of 10 Newtons.

00:02:24.210 --> 00:02:27.770
I move that block, let's
say I move it-- I

00:02:27.770 --> 00:02:31.960
don't know-- 7 meters.

00:02:31.960 --> 00:02:35.930
So the work that I applied to
that block, or the energy that

00:02:35.930 --> 00:02:42.650
I've transferred to that block,
the work is equal to

00:02:42.650 --> 00:02:46.880
the force, which is 10 Newtons,
times the distance,

00:02:46.880 --> 00:02:49.300
times 7 meters.

00:02:49.300 --> 00:02:54.150
And that would equal 70-- 10
times 7-- Newton meters.

00:02:54.150 --> 00:03:00.450
So Newton meters is one, I
guess, way of describing work.

00:03:00.450 --> 00:03:03.720
And this is also defined
as one joule.

00:03:03.720 --> 00:03:05.340
And I'll do another presentation
on all of the

00:03:05.340 --> 00:03:06.230
things that soon.

00:03:06.230 --> 00:03:06.920
Joule did.

00:03:06.920 --> 00:03:09.170
But joule is the unit
of work and it's

00:03:09.170 --> 00:03:10.540
also the unit of energy.

00:03:10.540 --> 00:03:12.650
And they're kind of
transferrable.

00:03:12.650 --> 00:03:14.380
Because if you look at the
definitions that Wikipedia

00:03:14.380 --> 00:03:17.620
gave us, work is energy
transferred by a force and

00:03:17.620 --> 00:03:19.880
energy is the ability to work.

00:03:19.880 --> 00:03:23.900
So I'll leave this relatively
circular definition alone now.

00:03:23.900 --> 00:03:26.680
But we'll use this definition,
which I think helps us a

00:03:26.680 --> 00:03:30.710
little bit more to understand
the types of work we can do.

00:03:30.710 --> 00:03:34.680
And then, what kind of energy
we actually are transferring

00:03:34.680 --> 00:03:37.380
to an object when we do
that type of work.

00:03:37.380 --> 00:03:40.025
So let me do some examples.

00:03:42.960 --> 00:03:44.225
Let's say I have a block.

00:03:49.120 --> 00:03:53.290
I have a block of mass m.

00:03:53.290 --> 00:03:57.610
I have a block of mass m and it
starts at rest. And then I

00:03:57.610 --> 00:04:00.200
apply force.

00:04:00.200 --> 00:04:09.070
Let's say I apply a force, F,
for a distance of, I think,

00:04:09.070 --> 00:04:10.850
you can guess what the distance
I'm going to apply it

00:04:10.850 --> 00:04:13.380
is, for a distance of d.

00:04:13.380 --> 00:04:17.500
So I'm pushing on this block
with a force of F for a

00:04:17.500 --> 00:04:18.540
distance of d.

00:04:18.540 --> 00:04:21.290
And what I want to figure
out is-- well, we know

00:04:21.290 --> 00:04:22.610
what the work is.

00:04:22.610 --> 00:04:27.630
I mean, by definition, work is
equal to this force times this

00:04:27.630 --> 00:04:30.480
distance that I'm applying
the block-- that

00:04:30.480 --> 00:04:31.930
I'm pushing the block.

00:04:31.930 --> 00:04:36.030
But what is the velocity going
to be of this block over here?

00:04:36.030 --> 00:04:36.620
Right?

00:04:36.620 --> 00:04:39.320
It's going to be something
somewhat faster.

00:04:39.320 --> 00:04:42.270
Because force isn't-- and I'm
assuming that this is

00:04:42.270 --> 00:04:43.570
frictionless on here.

00:04:43.570 --> 00:04:48.040
So force isn't just moving
the block with a constant

00:04:48.040 --> 00:04:50.420
velocity, force is equal to
mass times acceleration.

00:04:50.420 --> 00:04:52.430
So I'm actually going to be
accelerating the block.

00:04:52.430 --> 00:04:55.390
So even though it's stationary
here, by the time we get to

00:04:55.390 --> 00:04:57.960
this point over here,
that block is going

00:04:57.960 --> 00:05:00.290
to have some velocity.

00:05:00.290 --> 00:05:02.200
We don't know what it is
because we're using all

00:05:02.200 --> 00:05:03.550
variables, we're not
using numbers.

00:05:03.550 --> 00:05:06.650
But let's figure out what
it is in terms of v.

00:05:06.650 --> 00:05:10.980
So if you remember your
kinematics equations, and if

00:05:10.980 --> 00:05:12.120
you don't, you might
want to go back.

00:05:12.120 --> 00:05:14.060
Or if you've never seen the
videos, there's a whole set of

00:05:14.060 --> 00:05:17.380
videos on projectile motion
and kinematics.

00:05:17.380 --> 00:05:20.170
But we figured out that when
we're accelerating an object

00:05:20.170 --> 00:05:23.400
over a distance, that the final
velocity-- let me change

00:05:23.400 --> 00:05:27.910
colors just for variety-- the
final velocity squared is

00:05:27.910 --> 00:05:31.850
equal to the initial velocity
squared plus 2 times the

00:05:31.850 --> 00:05:33.530
acceleration times
the distance.

00:05:33.530 --> 00:05:36.210
And we proved this back then,
so I won't redo it now.

00:05:36.210 --> 00:05:38.640
But in this situation, what's
the initial velocity?

00:05:38.640 --> 00:05:40.525
Well the initial
velocity was 0.

00:05:43.250 --> 00:05:44.160
Right?

00:05:44.160 --> 00:05:50.026
So the equation becomes vf
squared is equal to 2 times

00:05:50.026 --> 00:05:54.990
the acceleration times
the distance.

00:05:54.990 --> 00:05:57.830
And then, we could rewrite
the acceleration

00:05:57.830 --> 00:05:59.180
in terms of, what?

00:05:59.180 --> 00:06:01.050
The force and the mass, right?

00:06:01.050 --> 00:06:03.010
So what is the acceleration?

00:06:03.010 --> 00:06:04.260
Well F equals ma.

00:06:07.310 --> 00:06:12.450
Or, acceleration is equal to
force divided by you mass.

00:06:12.450 --> 00:06:18.910
So we get vf squared is equal
to 2 times the force divided

00:06:18.910 --> 00:06:22.360
by the mass times
the distance.

00:06:22.360 --> 00:06:23.930
And then we could take the
square root of both sides if

00:06:23.930 --> 00:06:26.930
we want, and we get the final
velocity of this block, at

00:06:26.930 --> 00:06:37.140
this point, is going to be equal
to the square root of 2

00:06:37.140 --> 00:06:41.780
times force times distance
divided by mass.

00:06:41.780 --> 00:06:43.760
And so that's how we could
figure it out.

00:06:43.760 --> 00:06:46.000
And there's something
interesting going on here.

00:06:46.000 --> 00:06:49.060
There's something interesting
in what we did just now.

00:06:49.060 --> 00:06:52.160
Do you see something that looks
a little bit like work?

00:06:52.160 --> 00:06:52.910
Well sure.

00:06:52.910 --> 00:06:54.570
You have this force
times distance

00:06:54.570 --> 00:06:56.090
expression right here.

00:06:56.090 --> 00:06:58.660
Force times distance
right here.

00:06:58.660 --> 00:07:01.290
So let's write another
equation.

00:07:01.290 --> 00:07:07.380
If we know the given amount of
velocity something has, if we

00:07:07.380 --> 00:07:09.590
can figure out how much work
needed to be put into the

00:07:09.590 --> 00:07:12.580
system to get to
that velocity.

00:07:12.580 --> 00:07:15.135
Well we can just replace force
times distance with work.

00:07:15.135 --> 00:07:15.690
Right?

00:07:15.690 --> 00:07:17.370
Because work is equal to
force times distance.

00:07:17.370 --> 00:07:20.905
So let's go straight from this
equation because we don't have

00:07:20.905 --> 00:07:22.200
to re-square it.

00:07:22.200 --> 00:07:27.670
So we get vf squared
is equal to 2

00:07:27.670 --> 00:07:29.370
times force times distance.

00:07:29.370 --> 00:07:31.150
That's work.

00:07:31.150 --> 00:07:33.230
Took that definition
right here.

00:07:33.230 --> 00:07:37.630
2 times work divided
by the mass.

00:07:37.630 --> 00:07:40.870
Let's multiply both sides of
this equation times the mass.

00:07:40.870 --> 00:07:44.150
So you get mass times
the velocity.

00:07:44.150 --> 00:07:46.290
And we don't have to write-- I'm
going to get rid of this f

00:07:46.290 --> 00:07:48.530
because we know that we started
at rest and that the

00:07:48.530 --> 00:07:51.180
velocity is going to be--
let's just call it v.

00:07:51.180 --> 00:07:56.390
So m times V squared is equal
to 2 times the work.

00:07:56.390 --> 00:07:58.090
Divide both sides by 2.

00:07:58.090 --> 00:08:03.380
Or that the work is equal
to mv squared over 2.

00:08:03.380 --> 00:08:05.860
Just divided both sides by 2.

00:08:05.860 --> 00:08:07.880
And of course, the unit
here is joules.

00:08:07.880 --> 00:08:09.570
So this is interesting.

00:08:09.570 --> 00:08:16.840
Now if I know the velocity of
an object, I can figure out,

00:08:16.840 --> 00:08:19.590
using this formula, which
hopefully wasn't too

00:08:19.590 --> 00:08:21.200
complicated to derive.

00:08:21.200 --> 00:08:25.110
I can figure out how much work
was imputed into that object

00:08:25.110 --> 00:08:26.790
to get it to that velocity.

00:08:26.790 --> 00:08:31.270
And this, by definition, is
called kinetic energy.

00:08:31.270 --> 00:08:32.500
This is kinetic energy.

00:08:32.500 --> 00:08:35.760
And once again, the definition
that Wikipedia gives us is the

00:08:35.760 --> 00:08:41.190
energy due to motion, or the
work needed to accelerate from

00:08:41.190 --> 00:08:43.179
an object from being stationary

00:08:43.179 --> 00:08:44.980
to its current velocity.

00:08:44.980 --> 00:08:48.490
And I'm actually almost out of
time, but what I will do is I

00:08:48.490 --> 00:08:51.550
will leave you with this
formula, that kinetic energy

00:08:51.550 --> 00:08:53.600
is mass times velocity
squared divided by

00:08:53.600 --> 00:08:55.780
2, or 1/2 mv squared.

00:08:55.780 --> 00:08:57.150
It's a very common formula.

00:08:57.150 --> 00:08:59.030
And I'll leave you with
that and that

00:08:59.030 --> 00:09:00.660
is one form of energy.

00:09:00.660 --> 00:09:02.680
And I'll leave you
with that idea.

00:09:02.680 --> 00:09:04.390
And in the next video,
I will show you

00:09:04.390 --> 00:09:05.690
another form of energy.

00:09:05.690 --> 00:09:07.560
And then, I will introduce
you to the law of

00:09:07.560 --> 00:09:08.590
conservation of energy.

00:09:08.590 --> 00:09:11.220
And that's where things become
useful, because you can see

00:09:11.220 --> 00:09:13.730
how one form of energy can be
converted to another to figure

00:09:13.730 --> 00:09:15.190
out what happens to an object.

00:09:15.190 --> 00:09:16.640
I'll see


Work and energy (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:01.650
Welcome back.

00:00:01.650 --> 00:00:05.850
In the last video, I showed you
or hopefully, I did show

00:00:05.850 --> 00:00:10.230
you that if I apply a force
of F to a stationary, an

00:00:10.230 --> 00:00:14.990
initially stationary object with
mass m, and I apply that

00:00:14.990 --> 00:00:20.920
force for distance d, that that
force times distance, the

00:00:20.920 --> 00:00:24.460
force times the distance that
I'm pushing the object is

00:00:24.460 --> 00:00:30.980
equal to 1/2 mv squared, where
m is the mass of the object,

00:00:30.980 --> 00:00:34.430
and v is the velocity of the
object after pushing it for a

00:00:34.430 --> 00:00:35.690
distance of d.

00:00:35.690 --> 00:00:37.505
And we defined in that
last video, we just

00:00:37.505 --> 00:00:38.810
said this is work.

00:00:38.810 --> 00:00:42.830
Force times distance by
definition, is work.

00:00:42.830 --> 00:00:47.340
And 1/2 mv squared, I said this
is called kinetic energy.

00:00:51.680 --> 00:00:56.110
And so, by definition, kinetic
energy is the amount of work--

00:00:56.110 --> 00:00:57.870
and I mean this is the
definition right here.

00:00:57.870 --> 00:01:00.725
It's the amount of work you need
to put into an object or

00:01:00.725 --> 00:01:03.820
apply to an object to
get it from rest

00:01:03.820 --> 00:01:05.550
to its current velocity.

00:01:05.550 --> 00:01:07.100
So its velocity over here.

00:01:07.100 --> 00:01:10.740
So let's just say I looked at an
object here with mass m and

00:01:10.740 --> 00:01:14.250
it was moving with
the velocity v.

00:01:14.250 --> 00:01:16.960
I would say well, this
has a kinetic

00:01:16.960 --> 00:01:19.660
energy of 1/2 mv squared.

00:01:19.660 --> 00:01:21.365
And if the numbers are confusing
you, let's say the

00:01:21.365 --> 00:01:22.760
mass was-- I don't know.

00:01:22.760 --> 00:01:25.640
Let's say this was a 5 kilogram
object and it's

00:01:25.640 --> 00:01:29.330
moving at 7 meters per second.

00:01:29.330 --> 00:01:33.325
So I would say the kinetic
energy of this object is going

00:01:33.325 --> 00:01:40.190
to be 5-- 1/2 times the mass
times 5 times 7 squared, times

00:01:40.190 --> 00:01:40.850
velocity squared.

00:01:40.850 --> 00:01:42.890
It's times 49.

00:01:42.890 --> 00:01:43.220
So let's see.

00:01:43.220 --> 00:01:46.270
1/2 times 49, that's
a little under 25.

00:01:46.270 --> 00:01:53.170
So it'll be approximately 125
Newton meters, which is

00:01:53.170 --> 00:01:55.500
approximately-- and Newton
meter is just

00:01:55.500 --> 00:01:58.220
a joule-- 125 joules.

00:01:58.220 --> 00:02:00.420
So this is if we actually
put numbers in.

00:02:00.420 --> 00:02:02.770
And so when we immediately know
this, even if we didn't

00:02:02.770 --> 00:02:05.770
know what happened, how did this
object get to this speed?

00:02:05.770 --> 00:02:08.560
Let's say we didn't know that
someone else had applied a

00:02:08.560 --> 00:02:13.090
force of m for a distance of
d to this object, just by

00:02:13.090 --> 00:02:16.710
calculating its kinetic energy
as 125 joules, we immediately

00:02:16.710 --> 00:02:20.120
know that that's the amount of
work that was necessary.

00:02:20.120 --> 00:02:22.500
And we don't know if this is
exactly how this object got to

00:02:22.500 --> 00:02:24.920
this velocity, but we know that
that is the amount of

00:02:24.920 --> 00:02:28.530
work that was necessary to
accelerate the object to this

00:02:28.530 --> 00:02:32.620
velocity of 7 meters
per second.

00:02:32.620 --> 00:02:35.330
So let's give another example.

00:02:35.330 --> 00:02:39.420
And instead of this time just
pushing something in a

00:02:39.420 --> 00:02:41.640
horizontal direction and
accelerating it, I'm going to

00:02:41.640 --> 00:02:43.940
show you an example we're going
to push something up,

00:02:43.940 --> 00:02:46.550
but its velocity really
isn't going to change.

00:02:49.430 --> 00:02:51.280
Invert.

00:02:51.280 --> 00:02:54.140
Let's say I have a different
situation, and we're on this

00:02:54.140 --> 00:02:55.990
planet, we're not
in deep space.

00:02:55.990 --> 00:03:01.090
And I have a mass of m and
I were to apply a force.

00:03:01.090 --> 00:03:06.470
So let's say the force that I
apply is equal to mass times

00:03:06.470 --> 00:03:08.930
the acceleration of gravity.

00:03:08.930 --> 00:03:11.460
Mass times-- let's just call
that gravity, right?

00:03:11.460 --> 00:03:13.140
9.8 meters per second squared.

00:03:13.140 --> 00:03:18.740
And I were to apply this force
for a distance of d upwards.

00:03:18.740 --> 00:03:19.500
Right?

00:03:19.500 --> 00:03:20.810
Or instead of d, let's say h.

00:03:20.810 --> 00:03:23.050
H for height.

00:03:23.050 --> 00:03:27.030
So in this case, the force times
the distance is equal

00:03:27.030 --> 00:03:31.490
to-- well the force is mass
times the acceleration of

00:03:31.490 --> 00:03:33.240
gravity, right?

00:03:33.240 --> 00:03:35.880
And remember, I'm pushing with
the acceleration of gravity

00:03:35.880 --> 00:03:38.410
upwards, while the acceleration
of gravity is

00:03:38.410 --> 00:03:41.100
pulling downwards.

00:03:41.100 --> 00:03:45.045
So the force is mass times
gravity, and I'm applying that

00:03:45.045 --> 00:03:47.390
for a distance of h, right?

00:03:47.390 --> 00:03:48.050
d is h.

00:03:48.050 --> 00:03:50.810
So the force is this.

00:03:50.810 --> 00:03:52.580
This is the force.

00:03:52.580 --> 00:03:56.730
And then the distance I'm
applying is going to be h.

00:03:56.730 --> 00:04:00.900
And what's interesting is-- I
mean if you want to think of

00:04:00.900 --> 00:04:05.910
an exact situation, imagine an
elevator that is already

00:04:05.910 --> 00:04:08.140
moving because you would
actually have to apply a force

00:04:08.140 --> 00:04:10.330
slightly larger than the
acceleration of gravity just

00:04:10.330 --> 00:04:11.270
to get the object moving.

00:04:11.270 --> 00:04:12.280
But let's say that
the object is

00:04:12.280 --> 00:04:14.860
already at constant velocity.

00:04:14.860 --> 00:04:17.500
Let's say it's an elevator.

00:04:17.500 --> 00:04:20.950
And it is just going up with
a constant velocity.

00:04:20.950 --> 00:04:23.520
And let's say the mass of the
elevator is-- I don't know--

00:04:23.520 --> 00:04:29.230
10 kilograms. And it moves up
with a constant velocity.

00:04:32.270 --> 00:04:35.380
It moves up 100 meters.

00:04:35.380 --> 00:04:38.280
So we know that the work done
by whatever was pulling on

00:04:38.280 --> 00:04:41.370
this elevator, it probably was
the tension in this wire that

00:04:41.370 --> 00:04:43.750
was pulling up on the elevator,
but we know that the

00:04:43.750 --> 00:04:47.210
work done is the force necessary
to pull up on it.

00:04:47.210 --> 00:04:49.330
Well that's just going to
be the force of gravity.

00:04:49.330 --> 00:04:50.780
So we're assuming that
the elevator's not

00:04:50.780 --> 00:04:52.040
accelerating, right?

00:04:52.040 --> 00:04:55.690
Because if the elevator was
accelerating upwards, then the

00:04:55.690 --> 00:04:57.690
force applied to it
would be more than

00:04:57.690 --> 00:04:59.080
the force of gravity.

00:04:59.080 --> 00:05:01.840
And if the elevator was
accelerating downwards, or if

00:05:01.840 --> 00:05:05.250
it was slowing down upwards,
then the force being applied

00:05:05.250 --> 00:05:06.850
would be less than the
acceleration of gravity.

00:05:06.850 --> 00:05:10.360
But since the elevator is at a
constant velocity moving up,

00:05:10.360 --> 00:05:14.760
we know that the force pulling
upwards is completely equal to

00:05:14.760 --> 00:05:16.330
the force pulling downwards,
right?

00:05:16.330 --> 00:05:17.250
No net force.

00:05:17.250 --> 00:05:20.520
Because gravity and this force
are at the same level, so

00:05:20.520 --> 00:05:22.050
there's no change in velocity.

00:05:22.050 --> 00:05:24.020
I think I said that two times.

00:05:24.020 --> 00:05:26.840
So we know that this upward
force is equal to

00:05:26.840 --> 00:05:28.720
the force of gravity.

00:05:28.720 --> 00:05:31.045
At least in magnitude in
the opposite direction.

00:05:31.045 --> 00:05:35.310
So this is mg.

00:05:35.310 --> 00:05:38.500
So what's m? m is 10 kilograms.
Times the

00:05:38.500 --> 00:05:39.320
acceleration of gravity.

00:05:39.320 --> 00:05:41.900
Let's say that's 9.8 meters
per second squared.

00:05:41.900 --> 00:05:43.750
I'm not writing the units here,
but we're all assuming

00:05:43.750 --> 00:05:45.490
kilograms and meters
per second squared.

00:05:45.490 --> 00:05:50.760
And we're moving that for a
distance of 100 meters.

00:05:50.760 --> 00:05:55.490
So how much work was put into
this elevator, or into this

00:05:55.490 --> 00:05:57.980
object-- it doesn't have to be
an elevator-- by whatever

00:05:57.980 --> 00:06:00.210
force that was essentially
pushing up on it or

00:06:00.210 --> 00:06:01.770
pulling up on it?

00:06:01.770 --> 00:06:02.410
And so, let's see.

00:06:02.410 --> 00:06:05.160
This would be 98 times 100.

00:06:05.160 --> 00:06:13.820
So it's 9,800 Newton meters
or 9,800 joules.

00:06:13.820 --> 00:06:17.270
After we've moved up 100 meters,
notice there's no

00:06:17.270 --> 00:06:18.760
change in velocity.

00:06:18.760 --> 00:06:22.220
So the question is, where
did all that work get

00:06:22.220 --> 00:06:24.140
put into the object?

00:06:24.140 --> 00:06:26.780
And the answer here is, is that
the work got transferred

00:06:26.780 --> 00:06:29.940
to something called
potential energy.

00:06:29.940 --> 00:06:33.150
And potential energy is
defined as-- well,

00:06:33.150 --> 00:06:34.780
gravitational potential
energy.

00:06:34.780 --> 00:06:37.150
We'll work with other types of
potential energy later with

00:06:37.150 --> 00:06:38.810
springs and things.

00:06:38.810 --> 00:06:42.580
Potential energy is defined
as mass times the force of

00:06:42.580 --> 00:06:45.760
gravity times the height
that the object is at.

00:06:45.760 --> 00:06:47.840
And why is this called
potential energy?

00:06:47.840 --> 00:06:51.430
Because at this point, the
energy-- work had to be put

00:06:51.430 --> 00:06:53.920
into the object to get it
to this-- in the case of

00:06:53.920 --> 00:06:57.670
gravitational potential energy,
work had to be put

00:06:57.670 --> 00:07:00.090
into the object to get
it to this height.

00:07:00.090 --> 00:07:02.720
But the object now, it's not
moving or anything, so it

00:07:02.720 --> 00:07:04.110
doesn't have any
kinetic energy.

00:07:04.110 --> 00:07:06.510
But it now has a lot of
potential to do work.

00:07:06.510 --> 00:07:08.860
And what do I mean by potential
to do work?

00:07:08.860 --> 00:07:12.430
Well after I move an object up
100 meters into the air,

00:07:12.430 --> 00:07:14.420
what's its potential
to do work?

00:07:14.420 --> 00:07:19.520
Well, I could just let go of it
and have no outside force

00:07:19.520 --> 00:07:20.360
other than gravity.

00:07:20.360 --> 00:07:22.080
The gravity will
still be there.

00:07:22.080 --> 00:07:25.190
And because of gravity, the
object will come down and be

00:07:25.190 --> 00:07:27.490
at a very, very fast velocity
when it lands.

00:07:27.490 --> 00:07:31.230
And maybe I could apply this to
some machine or something,

00:07:31.230 --> 00:07:33.150
and this thing could do work.

00:07:33.150 --> 00:07:34.400
And if that's a little
confusing, let

00:07:34.400 --> 00:07:38.000
me give you an example.

00:07:38.000 --> 00:07:41.300
It all works together
with our--

00:07:41.300 --> 00:07:47.882
So let's say I have an object
that is-- oh, I don't know-- a

00:07:47.882 --> 00:07:53.060
1 kilogram object and
we're on earth.

00:07:53.060 --> 00:07:55.230
And let's say that is 10 meters
above the ground.

00:07:58.700 --> 00:08:06.070
So we know that its potential
energy is equal to mass times

00:08:06.070 --> 00:08:09.510
gravitational acceleration
times height.

00:08:09.510 --> 00:08:11.240
So mass is 1.

00:08:11.240 --> 00:08:13.300
Let's just say gravitational
acceleration is 10 meters per

00:08:13.300 --> 00:08:15.660
second squared.

00:08:15.660 --> 00:08:17.435
Times 10 meters per
second squared.

00:08:17.435 --> 00:08:19.820
Times 10 meters, which
is the height.

00:08:19.820 --> 00:08:25.720
So it's approximately equal to
100 Newton meters, which is

00:08:25.720 --> 00:08:28.020
the same thing is 100 joules.

00:08:28.020 --> 00:08:28.580
Fair enough.

00:08:28.580 --> 00:08:29.700
And what do we know
about this?

00:08:29.700 --> 00:08:33.500
We know that it would take about
100-- or exactly-- 100

00:08:33.500 --> 00:08:38.799
joules of work to get this
object from the ground to this

00:08:38.799 --> 00:08:40.809
point up here.

00:08:40.809 --> 00:08:44.730
Now what we can do now is use
our traditional kinematics

00:08:44.730 --> 00:08:47.480
formulas to figure out, well, if
I just let this object go,

00:08:47.480 --> 00:08:50.750
how fast will it be when
it hits the ground?

00:08:50.750 --> 00:08:52.645
And we could do that, but
what I'll show you is

00:08:52.645 --> 00:08:53.380
even a faster way.

00:08:53.380 --> 00:08:56.070
And this is where all of
the work and energy

00:08:56.070 --> 00:08:57.440
really becomes useful.

00:08:57.440 --> 00:08:59.840
We have something called the law
of conservation of energy.

00:08:59.840 --> 00:09:02.550
It's that energy cannot be
created or destroyed, it just

00:09:02.550 --> 00:09:04.600
gets transferred from
one form to another.

00:09:04.600 --> 00:09:06.770
And there's some minor
caveats to that.

00:09:06.770 --> 00:09:09.090
But for our purposes, we'll
just stick with that.

00:09:09.090 --> 00:09:12.110
So in the situation where I just
take the object and I let

00:09:12.110 --> 00:09:16.120
go up here, up here it has a
ton of potential energy.

00:09:16.120 --> 00:09:18.560
And by the time it's down here,
it has no potential

00:09:18.560 --> 00:09:21.530
energy because the height
becomes 0, right?

00:09:21.530 --> 00:09:27.310
So here, potential energy is
equal to 100 and here,

00:09:27.310 --> 00:09:29.870
potential energy
is equal to 0.

00:09:29.870 --> 00:09:31.796
And so the natural question is--
I just told you the law

00:09:31.796 --> 00:09:34.930
of conservation of energy, but
if you look at this example,

00:09:34.930 --> 00:09:37.330
all the potential energy
just disappeared.

00:09:37.330 --> 00:09:39.160
And it looks like I'm running
out of time, but what I'll

00:09:39.160 --> 00:09:41.130
show you in the next video is
that that potential energy

00:09:41.130 --> 00:09:42.780
gets converted into another
type of energy.

00:09:42.780 --> 00:09:44.980
And I think you might be able
to guess what type that is

00:09:44.980 --> 00:09:48.030
because this object is going to
be moving really fast right

00:09:48.030 --> 00:09:49.150
before it hits the ground.

00:09:49.150 --> 00:09:51.030
I'll see you in the
next video.


2-dimensional momentum problem (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:01.440
Welcome back.

00:00:01.440 --> 00:00:03.440
When I left off I was rushing
at the end of this problem

00:00:03.440 --> 00:00:06.510
because I tend to rush at the
end of problems when I am

00:00:06.510 --> 00:00:08.160
getting close to the YouTube
10 minute limit.

00:00:08.160 --> 00:00:10.680
But I just wanted to review the
end of it because I feel

00:00:10.680 --> 00:00:11.520
like I rushed it.

00:00:11.520 --> 00:00:14.660
And then, actually continue with
it and actually solve for

00:00:14.660 --> 00:00:16.970
the angle and then, introduce
a little bit of-- a little

00:00:16.970 --> 00:00:18.280
more trigonometry.

00:00:18.280 --> 00:00:21.240
So just to review what we did,
we said momentum is conserved

00:00:21.240 --> 00:00:24.270
and in two dimensions that means
momentum is conserved in

00:00:24.270 --> 00:00:25.650
each of the dimensions.

00:00:25.650 --> 00:00:28.040
So we figured out what the
initial momentum of the entire

00:00:28.040 --> 00:00:30.920
system was and we said, well,
in the x direction, the

00:00:30.920 --> 00:00:33.070
initial momentum-- and all the
momentum was coming from the

00:00:33.070 --> 00:00:34.440
ball A right.

00:00:34.440 --> 00:00:37.110
Because ball B wasn't moving,
so its velocity was 0.

00:00:37.110 --> 00:00:38.570
So its momentum was 0.

00:00:38.570 --> 00:00:41.300
So ball A in the x direction and
it was only moving in the

00:00:41.300 --> 00:00:42.190
x direction.

00:00:42.190 --> 00:00:44.390
So it's momentum in the x
direction was 3 meters per

00:00:44.390 --> 00:00:46.610
second times 10 kilogram
meters per second.

00:00:46.610 --> 00:00:49.050
And we got 30 kilogram
meters per second.

00:00:49.050 --> 00:00:51.960
And then there was no momentum
in the y direction.

00:00:51.960 --> 00:00:54.680
And then we knew that well after
they hit each other,

00:00:54.680 --> 00:00:56.910
ball A kind of ricochets off
at a 30 degree angle at 2

00:00:56.910 --> 00:00:58.110
meters per second.

00:00:58.110 --> 00:01:00.430
We used that information to
figure out the x and y

00:01:00.430 --> 00:01:02.700
components of A's velocity.

00:01:02.700 --> 00:01:05.840
So A's velocity in the y
direction was 1 meter per

00:01:05.840 --> 00:01:10.420
second and A's velocity
in the x direction was

00:01:10.420 --> 00:01:11.700
square root of 3.

00:01:11.700 --> 00:01:14.090
And we used that information to
figure out A's momentum in

00:01:14.090 --> 00:01:15.060
each direction.

00:01:15.060 --> 00:01:18.010
We said well, the momentum in
the y direction must be 1

00:01:18.010 --> 00:01:21.600
meter per second times A's mass,
which is 10 kilogram

00:01:21.600 --> 00:01:22.370
meters per second.

00:01:22.370 --> 00:01:27.170
Which I wrote-- what
I wrote here.

00:01:27.170 --> 00:01:31.800
And then we figured out A's
momentum in the B direction

00:01:31.800 --> 00:01:33.550
and we said well, that's
just going to be square

00:01:33.550 --> 00:01:35.700
root of 3 times 10.

00:01:35.700 --> 00:01:37.070
And that's 10 square
root of 3.

00:01:37.070 --> 00:01:39.040
And then we used that
information to

00:01:39.040 --> 00:01:40.680
solve for B's momentum.

00:01:40.680 --> 00:01:43.070
Because we said well, B's
momentum plus A's momentum in

00:01:43.070 --> 00:01:45.350
the x direction has
to add up to 30.

00:01:45.350 --> 00:01:47.680
This was the x direction
before.

00:01:47.680 --> 00:01:52.000
And we knew that B's momentum
plus A's momentum in the y

00:01:52.000 --> 00:01:54.220
direction had to add
up to 0, right?

00:01:54.220 --> 00:02:00.430
And so, since y's momentum going
upwards was 10 kilogram

00:02:00.430 --> 00:02:04.580
meters per second, we knew
that B's momentum going

00:02:04.580 --> 00:02:06.550
downwards would also
have to be 10

00:02:06.550 --> 00:02:07.350
kilogram meters per second.

00:02:07.350 --> 00:02:09.190
Or you could even say
it's negative 10.

00:02:09.190 --> 00:02:11.870
And we figure that out based
on the fact that B

00:02:11.870 --> 00:02:13.450
had half the mass.

00:02:13.450 --> 00:02:15.910
That its velocity going down
was 2 meters per second.

00:02:15.910 --> 00:02:21.140
And similarly, we knew that
A's momentum in the x

00:02:21.140 --> 00:02:24.770
direction, which was 10 square
root of 3 kilogram meters per

00:02:24.770 --> 00:02:26.830
second, plus B's momentum
in the x

00:02:26.830 --> 00:02:29.730
direction is equal to 30.

00:02:29.730 --> 00:02:32.800
And then we just subtracted out
and we got B's momentum in

00:02:32.800 --> 00:02:33.630
the x direction.

00:02:33.630 --> 00:02:36.270
And then we divided by B's
mass to get its velocity.

00:02:36.270 --> 00:02:39.150
Which we got as 2.54.

00:02:39.150 --> 00:02:41.340
So that's where I left off
and we were rushing.

00:02:41.340 --> 00:02:46.140
And already, this gives you a
sense of what B is doing.

00:02:46.140 --> 00:02:48.770
Although it's broken up into
the x and y direction.

00:02:48.770 --> 00:02:51.800
Now if we wanted to simplify
this, if we wanted to kind of

00:02:51.800 --> 00:02:54.470
write B's new velocity the same
way that the problem gave

00:02:54.470 --> 00:02:55.490
us A's velocity, right?

00:02:55.490 --> 00:02:58.090
They told us A's velocity was
2 meters per second at an

00:02:58.090 --> 00:02:59.390
angle of 30 degrees.

00:02:59.390 --> 00:03:04.010
We now have to use this
information to figure out B's

00:03:04.010 --> 00:03:06.450
velocity and the angle of it.

00:03:06.450 --> 00:03:07.150
And how do we do that?

00:03:07.150 --> 00:03:09.890
Well this is just straight up
trigonometry at this point, or

00:03:09.890 --> 00:03:13.360
really just straight
up geometry.

00:03:13.360 --> 00:03:14.400
Let me clear all of this.

00:03:14.400 --> 00:03:21.680
And let's remember these two
numbers, 2.54 and minus 2.

00:03:21.680 --> 00:03:30.590
So B, we learned that in the x
direction its velocity-- this

00:03:30.590 --> 00:03:37.260
is all for B-- is equal to 2.54
meters per second and

00:03:37.260 --> 00:03:39.900
then y direction, it
was moving down.

00:03:39.900 --> 00:03:41.910
We could write this
as minus 2.

00:03:45.970 --> 00:03:51.830
But I'll just write this as 2
meters per second downwards.

00:03:51.830 --> 00:03:52.480
Right?

00:03:52.480 --> 00:03:52.970
Same thing.

00:03:52.970 --> 00:03:55.820
Minus 2 up is the same thing as
2 meters per second down.

00:03:55.820 --> 00:03:57.410
So the resulting vector's
going to look

00:03:57.410 --> 00:03:59.240
something like this.

00:03:59.240 --> 00:04:02.725
When you add two vectors you
just put them-- put the one's

00:04:02.725 --> 00:04:07.350
end at the beginning of the
other-- put them front to end,

00:04:07.350 --> 00:04:08.320
like we did here.

00:04:08.320 --> 00:04:09.760
And then you add them
together and this is

00:04:09.760 --> 00:04:10.730
the resulting vector.

00:04:10.730 --> 00:04:14.230
And I think you're used
to that at this point.

00:04:14.230 --> 00:04:18.510
And now we have to figure out
this angle and this side.

00:04:18.510 --> 00:04:21.100
Well this side is easy because
this is a right angle, so we

00:04:21.100 --> 00:04:21.810
use Pythagorean theorem.

00:04:21.810 --> 00:04:27.530
So this is going to be the
square root of 2.54 squared

00:04:27.530 --> 00:04:29.460
plus 2 squared.

00:04:29.460 --> 00:04:31.860
And what's 2.54 squared?

00:04:31.860 --> 00:04:40.880
2.54 times-- whoops.

00:04:40.880 --> 00:04:48.840
2.54 times 2.54 is
equal to 6.45.

00:04:48.840 --> 00:04:55.530
So that's the square root of
6.45 plus 4, which equals the

00:04:55.530 --> 00:05:02.070
square root of 10.45.

00:05:02.070 --> 00:05:03.550
And take the square
root of that.

00:05:03.550 --> 00:05:07.800
So that's 3.2, roughly.

00:05:07.800 --> 00:05:11.200
So the resulting velocity in
this direction, whatever angle

00:05:11.200 --> 00:05:16.550
this is, is 3.2 meters
per second.

00:05:16.550 --> 00:05:18.250
And I just used Pythagorean
theorem.

00:05:18.250 --> 00:05:22.440
So now all we have to do is
figure out the angle.

00:05:22.440 --> 00:05:25.540
We could use really any of the
trig ratios because we know

00:05:25.540 --> 00:05:26.730
all of the sides.

00:05:26.730 --> 00:05:29.270
So I don't know, let's
use one that you

00:05:29.270 --> 00:05:30.220
feel comfortable with.

00:05:30.220 --> 00:05:32.920
Well let's use sine.

00:05:32.920 --> 00:05:39.500
So sine of theta is
equal to what?

00:05:39.500 --> 00:05:40.680
SOH CAH TOA.

00:05:40.680 --> 00:05:42.820
Sine is opposite over
hypotenuse.

00:05:42.820 --> 00:05:47.020
So the opposite side is the y
direction, so that's 2, over

00:05:47.020 --> 00:05:50.530
the hypotenuse, 3.2.

00:05:50.530 --> 00:05:58.790
So 2 divided by 2 divided by 3.2
is equal to 0.625, which

00:05:58.790 --> 00:06:01.265
equals 0.625.

00:06:01.265 --> 00:06:03.350
So sine of theta equals 0.625.

00:06:03.350 --> 00:06:05.510
And maybe you're not familiar
with arcsine yet because I

00:06:05.510 --> 00:06:07.200
don't think I actually have
covered yet in the trig

00:06:07.200 --> 00:06:09.780
modules, although I
will eventually.

00:06:09.780 --> 00:06:13.090
So we know it's just the inverse
function of sine.

00:06:13.090 --> 00:06:18.170
So sine of theta is
equal to 0.625.

00:06:18.170 --> 00:06:25.920
Then we know that theta is equal
to the arcsine of 0.625.

00:06:25.920 --> 00:06:29.235
This is essentially saying,
when you say arcsine, this

00:06:29.235 --> 00:06:32.090
says, tell me the angle whose
sine is this number?

00:06:32.090 --> 00:06:33.540
That's what arcsine is.

00:06:33.540 --> 00:06:38.460
And we can take out Google
because it actually happens

00:06:38.460 --> 00:06:44.310
that Google has a-- let's see.

00:06:44.310 --> 00:06:46.680
Google actually-- it's an
automatic calculator.

00:06:46.680 --> 00:06:52.590
So you could type in arcsine
on Google of 0.625.

00:06:52.590 --> 00:06:55.460
Although I think the
answer they give

00:06:55.460 --> 00:06:57.720
you will be in radians.

00:06:57.720 --> 00:07:00.230
So I'll take that answer that
will be in radians and I want

00:07:00.230 --> 00:07:03.300
to convert to degrees, so I
multiply it times 180 over pi.

00:07:03.300 --> 00:07:05.850
That's just how I convert
from radians to degrees.

00:07:05.850 --> 00:07:07.350
And let's see what I get.

00:07:07.350 --> 00:07:12.670
So Google, you see, Google
says 38.68 degrees.

00:07:12.670 --> 00:07:14.260
They multiplied the whole
thing times 180 and then

00:07:14.260 --> 00:07:16.630
divided by pi, but that should
be the same thing.

00:07:16.630 --> 00:07:20.225
So roughly 38.7 degrees
is theta.

00:07:20.225 --> 00:07:22.090
Hope you understand that.

00:07:22.090 --> 00:07:24.430
You could pause it here if
you don't, but let me

00:07:24.430 --> 00:07:25.300
just write that down.

00:07:25.300 --> 00:07:30.070
So it's 38 degrees.

00:07:30.070 --> 00:07:35.080
So theta is equal
to 38.7 degrees.

00:07:35.080 --> 00:07:36.360
So then we're done.

00:07:36.360 --> 00:07:39.250
We figured out that
ball B gets hit.

00:07:39.250 --> 00:07:40.950
This is ball B and it
got hit by ball A.

00:07:40.950 --> 00:07:43.900
Ball A went off in that
direction at a 30 degree

00:07:43.900 --> 00:07:47.880
angle, at a 30 degree angle
at 2 meters per second.

00:07:47.880 --> 00:07:52.040
And now ball B goes at 38.--
or we could say roughly 39

00:07:52.040 --> 00:07:56.320
degrees below the horizontal
at a velocity of 3.2 meters

00:07:56.320 --> 00:07:57.700
per second.

00:07:57.700 --> 00:08:01.270
And does this intuitively
make sense to you?

00:08:01.270 --> 00:08:03.140
Well if you remember the problem
from before-- and I

00:08:03.140 --> 00:08:03.860
know I erased everything.

00:08:03.860 --> 00:08:07.540
Ball A had a mass of 10
kilograms while ball B had a

00:08:07.540 --> 00:08:10.780
mass of 5 kilograms.
So it makes sense.

00:08:10.780 --> 00:08:12.210
So let's think about just
the y direction.

00:08:12.210 --> 00:08:16.680
Ball A, we figured out, the y
component of its velocity was

00:08:16.680 --> 00:08:19.260
1 meter per second.

00:08:19.260 --> 00:08:23.100
And ball B's y component is 2
meters per second downwards.

00:08:23.100 --> 00:08:24.060
And does that makes sense?

00:08:24.060 --> 00:08:25.000
Well sure.

00:08:25.000 --> 00:08:27.030
Because their momentums
have to add up to 0.

00:08:27.030 --> 00:08:31.470
There was no y component of the
momentum before they hit

00:08:31.470 --> 00:08:32.440
each other.

00:08:32.440 --> 00:08:37.539
And in order for B to have the
same momentum going downwards

00:08:37.539 --> 00:08:41.049
in the y direction as A going
upwards, its velocity has to

00:08:41.049 --> 00:08:44.690
be essentially double, because
its mass is half.

00:08:44.690 --> 00:08:48.680
And a similar logic, although
the cosine-- it doesn't work

00:08:48.680 --> 00:08:50.550
out exactly like that.

00:08:50.550 --> 00:08:53.450
But a similar logic would mean
that its overall velocity is

00:08:53.450 --> 00:08:59.670
going to be faster than the-
than A's velocity.

00:08:59.670 --> 00:09:03.680
And so what was I
just-- oh yeah.

00:09:03.680 --> 00:09:07.550
My phone was ringing and
I got caught up.

00:09:07.550 --> 00:09:09.730
My brain starts to
malfunction.

00:09:09.730 --> 00:09:11.100
But anyway, as I was
saying, so just

00:09:11.100 --> 00:09:12.220
intuitively it makes sense.

00:09:12.220 --> 00:09:16.330
B has a smaller mass than A, so
it makes sense that-- one,

00:09:16.330 --> 00:09:18.460
B will be going faster and
that it gets deflected a

00:09:18.460 --> 00:09:20.350
little bit more as well.

00:09:20.350 --> 00:09:21.800
The reason why it seems like
it gets deflected more is

00:09:21.800 --> 00:09:23.095
because its y component
is more.

00:09:23.095 --> 00:09:26.200
But anyway, that last piece is
just to kind of hopefully give

00:09:26.200 --> 00:09:29.510
you a sense of what's happening
and I will see you

00:09:29.510 --> 00:09:31.120
in the next video.