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Let's learn a little
bit about springs.

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So let's say I have a spring.

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Let me draw the ground so that
we know what's going on with

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the spring.

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So let me see, this
is the floor.

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That's the floor, and
I have a spring.

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It's along the floor.

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I'll use a thicker one, just
to show it's a spring.

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Let's say the spring looks
something like this.

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Whoops, I'm still using
the line tool.

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So the spring looks like this.

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This is my spring, my amazingly
drawn spring.

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Let's say at this end it's
attached to a wall.

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That's a wall.

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And so this is a spring when I
don't have any force acting on

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it, this is just the natural
state of the spring.

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And we could call this, where it
just naturally rests, this

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tip of the spring.

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And let's say that when I were
to apply a force of 5 Newtons

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into the spring, it looks
something like this.

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Redraw everything.

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So when I apply a force of 5
Newtons-- I'll draw the wall

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in magenta now.

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When I apply a force
of 5 Newtons, the

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spring looks like this.

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It compresses, right?

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We're all familiar with this.

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We sit on a bed every
day or a sofa.

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So let's say it compresses
to here.

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If this was the normal resting--
so this is where the

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spring was when I applied no
force, but when I applied 5

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Newtons in that direction, let's
say that this distance

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right here is 10 meters.

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And so a typical question that
you'll see, and we'll explain

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how to do it, is a spring
compresses or elongates when

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you apply a certain force
by some distance.

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How much will it compress when
you apply a different force?

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So my question is how much will
it compress when I apply

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a 10-Newton force?

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So your intuition that it'll
compress more is correct, but

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is it linear to how much
I compress it?

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Is it a square of how
much I compress it?

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How does it relate?

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I think you probably
could guess.

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It's actually worth
an experiment.

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Or you could just keep
watching the video.

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So let's say I apply
a 10-Newton force.

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What will the spring
look like?

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Well, it'll be more
compressed.

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Drop my force to 10 Newtons.

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And if this was the natural
place where the spring would

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rest, what is this distance?

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Well, it turns out that
it is linear.

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What do I mean by linear?

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Well, it means that the more
the force-- it's equally

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proportional to how much the
spring will compress.

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And it actually works
the other way.

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If you applied 5 Newtons in this
direction, to the right,

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you would have gone 10 meters
in this direction.

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So it goes whether you're
elongating the spring or

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compressing the spring within
some reasonable tolerance.

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We've all had this experience.

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If you compress something too
much or you stretch it too

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much, it doesn't really go back
to where it was before.

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But within some reasonable
tolerance, it's proportional.

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So what does that mean?

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That means that the restoring
force of the spring is minus

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some number, times the
displacement of the spring.

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So what does this mean?

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So in this example right here,
what was the displacement of

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the spring?

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Well, if we take positive x to
the right and negative x to

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the left, the displacement
of the spring was what?

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The displacement, in this
example right here, x is equal

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to minus 10, right?

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Because I went 10 to the left.

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And so it says that the
restorative force is going to

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be equal to minus K times
how much it's

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distorted times minus 10.

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So the minuses cancel out,
so it equals 10K.

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What's the restorative force
in this example?

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Well, you might say, it's 5
Newtons, just because that's

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the only force I've drawn here,
and you would be to some

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degree correct.

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And actually, since we're doing
positive and negative,

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and this 5 Newton is to the
left, so to the negative

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x-direction, actually, I should
call this minus 5

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Newtons and I should call this
minus 10 Newtons, because

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obviously, these are vectors and
we're going to the left.

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I picked the convention that
to the left means negative.

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So what's the restorative
force?

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Well, in this example-- and we
assume that K is a positive

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number for our purposes.

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In this example, the restorative
force is a

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positive number.

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So what is the restorative
force?

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So that's actually the force,
the counteracting force, of

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the spring.

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That's what this formula
gives us.

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So if this spring is stationary
when I apply this

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5-Newton force, that means that
there must be another

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equal and opposite force that's

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positive 5 Newtons, right?

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If there weren't, the spring
would keep compressing.

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And if the force was more than 5
Newtons, the spring would go

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back this way.

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So the fact that I know that
when I apply a 5-Newton force

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to the left, or a negative
5-Newton force, the spring is

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no longer moving, it means that
there must be-- or no

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longer accelerating, actually,
it means that there must be an

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equal and opposite force to
the right, and that's the

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restorative force.

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Another way to think about it is
if I were to let-- well, I

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won't go in there now.

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So in this case, the restorative
force is 5

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Newtons, so we can
solve for K.

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We could say 5 is
equal to 10K.

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Divide both sides by 10.

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You get K is equal to 1/2.

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So now we can use that
information to figure out what

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is the displacement
when I apply a

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negative 10-Newton force?

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When I push the spring
in with 10 Newtons in

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the leftward direction?

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So first of all, what's the
restorative force here?

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Well, if the spring is no longer
accelerating in either

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direction, or the tip of
the spring is no longer

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accelerating in either
direction, we know that the

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restorative force must be
counterbalancing this force

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that I'm compressing
with, right?

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The force that the spring wants
to expand back with is

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10 Newtons, positive
10 Newtons, right?

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And we know the spring constant,
this K for this

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spring, for this material,
whatever it might be, is 1/2.

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So we know the restorative force
is equal to 1/2 times

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the distance, right?

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And the formula is
minus K, right?

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And then, what is
the restorative

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force in this example?

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Well I said it's 10 Newtons, so
we know that 10 Newtons is

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equal to minus 1/2x.

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And so what is x?

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Well, multiply both sides
by minus 1/2, and

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you get minus 20.

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I'm sorry, multiply both sides
by minus 2, you get minus 20

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is equal to x.

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So x goes to the
left 20 units.

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So that's all that
it's telling us.

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And this law is called Hooke's
Law, and it's named after--

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I'll read it-- a physicist in
the 17th century, a British

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physicist. And he figured out
that the amount of force

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necessary to keep a spring
compressed is proportional to

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how much you've compressed it.

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And that's all that
this formula says.

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And that negative number,
remember, this formula gives

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us the restorative force.

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So it says that the force is
always in the opposite

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direction of how much
you displace it.

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So, for example, if you were
to displace this spring in

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this direction, if you were to
apply a force and x were a

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positive and you were to go in
that direction, the force-- no

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wait, sorry.

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This is where the
spring rests.

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If you were to apply some force
and take the spring out

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to here, this negative number
tells us that the spring will

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essentially try to pull back
with the restorative force in

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the other direction.

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Let's do one more problem
and I think this

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will be clear to you.

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So let's say I have a spring,
and all of these problems kind

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of go along.

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So let's say when I apply a
force of 2 Newtons, so this is

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what I apply when I apply
a force of 2 Newtons.

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Well, let's say it this way.

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Let's say when I stretch
the spring.

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Let's say this is the spring,
and when I apply a force of 2

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Newtons to the right, the spring
gets stretched 1 meter.

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So first of all, let's
figure out what K is.

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So if the spring is stretched
by 1 meter, out here, its

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restorative force will be 2
Newtons back this way, right?

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So its restorative force, this
2 Newtons, will equal minus K

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times how much I displaced it.

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Well I, displaced it by 1 meter,
so then we multiply

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both sides by negative 1, and we
get K is equal to minus 2.

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So then we can use Hooke's Law
to note the equation for

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this-- to figure out the
restorative force for this

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particular spring, and
it would be minus 2x.

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And then I said, well, how much
force would I have to

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apply to distort the
spring by 2 meters?

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Well, it's 2 times
2, it would be 4.

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4 Newtons to displace it by 2
meters, and, of course, the

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restorative force will then be
in the opposite direction, and

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that's where we get the
negative number.

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Anyway, I've run out of time.

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I'll see you in the
next video.

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