Intro to springs and Hooke's Law Introduction to Hooke's Law
Intro to springs and Hooke's Law
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- Let's learn a little bit about springs.
- So let's say I have a spring.
- Let me draw the ground so that we know what's going on with
- the spring.
- So let me see, this is the floor.
- That's the floor, and I have a spring.
- It's along the floor.
- I'll use a thicker one, just to show it's a spring.
- Let's say the spring looks something like this.
- Whoops, I'm still using the line tool.
- So the spring looks like this.
- This is my spring, my amazingly drawn spring.
- Let's say at this end it's attached to a wall.
- That's a wall.
- And so this is a spring when I don't have any force acting on
- it, this is just the natural state of the spring.
- And we could call this, where it just naturally rests, this
- tip of the spring.
- And let's say that when I were to apply a force of 5 Newtons
- into the spring, it looks something like this.
- Redraw everything.
- So when I apply a force of 5 Newtons-- I'll draw the wall
- in magenta now.
- When I apply a force of 5 Newtons, the
- spring looks like this.
- It compresses, right?
- We're all familiar with this.
- We sit on a bed every day or a sofa.
- So let's say it compresses to here.
- If this was the normal resting-- so this is where the
- spring was when I applied no force, but when I applied 5
- Newtons in that direction, let's say that this distance
- right here is 10 meters.
- And so a typical question that you'll see, and we'll explain
- how to do it, is a spring compresses or elongates when
- you apply a certain force by some distance.
- How much will it compress when you apply a different force?
- So my question is how much will it compress when I apply
- a 10-Newton force?
- So your intuition that it'll compress more is correct, but
- is it linear to how much I compress it?
- Is it a square of how much I compress it?
- How does it relate?
- I think you probably could guess.
- It's actually worth an experiment.
- Or you could just keep watching the video.
- So let's say I apply a 10-Newton force.
- What will the spring look like?
- Well, it'll be more compressed.
- Drop my force to 10 Newtons.
- And if this was the natural place where the spring would
- rest, what is this distance?
- Well, it turns out that it is linear.
- What do I mean by linear?
- Well, it means that the more the force-- it's equally
- proportional to how much the spring will compress.
- And it actually works the other way.
- If you applied 5 Newtons in this direction, to the right,
- you would have gone 10 meters in this direction.
- So it goes whether you're elongating the spring or
- compressing the spring within some reasonable tolerance.
- We've all had this experience.
- If you compress something too much or you stretch it too
- much, it doesn't really go back to where it was before.
- But within some reasonable tolerance, it's proportional.
- So what does that mean?
- That means that the restoring force of the spring is minus
- some number, times the displacement of the spring.
- So what does this mean?
- So in this example right here, what was the displacement of
- the spring?
- Well, if we take positive x to the right and negative x to
- the left, the displacement of the spring was what?
- The displacement, in this example right here, x is equal
- to minus 10, right?
- Because I went 10 to the left.
- And so it says that the restorative force is going to
- be equal to minus K times how much it's
- distorted times minus 10.
- So the minuses cancel out, so it equals 10K.
- What's the restorative force in this example?
- Well, you might say, it's 5 Newtons, just because that's
- the only force I've drawn here, and you would be to some
- degree correct.
- And actually, since we're doing positive and negative,
- and this 5 Newton is to the left, so to the negative
- x-direction, actually, I should call this minus 5
- Newtons and I should call this minus 10 Newtons, because
- obviously, these are vectors and we're going to the left.
- I picked the convention that to the left means negative.
- So what's the restorative force?
- Well, in this example-- and we assume that K is a positive
- number for our purposes.
- In this example, the restorative force is a
- positive number.
- So what is the restorative force?
- So that's actually the force, the counteracting force, of
- the spring.
- That's what this formula gives us.
- So if this spring is stationary when I apply this
- 5-Newton force, that means that there must be another
- equal and opposite force that's
- positive 5 Newtons, right?
- If there weren't, the spring would keep compressing.
- And if the force was more than 5 Newtons, the spring would go
- back this way.
- So the fact that I know that when I apply a 5-Newton force
- to the left, or a negative 5-Newton force, the spring is
- no longer moving, it means that there must be-- or no
- longer accelerating, actually, it means that there must be an
- equal and opposite force to the right, and that's the
- restorative force.
- Another way to think about it is if I were to let-- well, I
- won't go in there now.
- So in this case, the restorative force is 5
- Newtons, so we can solve for K.
- We could say 5 is equal to 10K.
- Divide both sides by 10.
- You get K is equal to 1/2.
- So now we can use that information to figure out what
- is the displacement when I apply a
- negative 10-Newton force?
- When I push the spring in with 10 Newtons in
- the leftward direction?
- So first of all, what's the restorative force here?
- Well, if the spring is no longer accelerating in either
- direction, or the tip of the spring is no longer
- accelerating in either direction, we know that the
- restorative force must be counterbalancing this force
- that I'm compressing with, right?
- The force that the spring wants to expand back with is
- 10 Newtons, positive 10 Newtons, right?
- And we know the spring constant, this K for this
- spring, for this material, whatever it might be, is 1/2.
- So we know the restorative force is equal to 1/2 times
- the distance, right?
- And the formula is minus K, right?
- And then, what is the restorative
- force in this example?
- Well I said it's 10 Newtons, so we know that 10 Newtons is
- equal to minus 1/2x.
- And so what is x?
- Well, multiply both sides by minus 1/2, and
- you get minus 20.
- I'm sorry, multiply both sides by minus 2, you get minus 20
- is equal to x.
- So x goes to the left 20 units.
- So that's all that it's telling us.
- And this law is called Hooke's Law, and it's named after--
- I'll read it-- a physicist in the 17th century, a British
- physicist. And he figured out that the amount of force
- necessary to keep a spring compressed is proportional to
- how much you've compressed it.
- And that's all that this formula says.
- And that negative number, remember, this formula gives
- us the restorative force.
- So it says that the force is always in the opposite
- direction of how much you displace it.
- So, for example, if you were to displace this spring in
- this direction, if you were to apply a force and x were a
- positive and you were to go in that direction, the force-- no
- wait, sorry.
- This is where the spring rests.
- If you were to apply some force and take the spring out
- to here, this negative number tells us that the spring will
- essentially try to pull back with the restorative force in
- the other direction.
- Let's do one more problem and I think this
- will be clear to you.
- So let's say I have a spring, and all of these problems kind
- of go along.
- So let's say when I apply a force of 2 Newtons, so this is
- what I apply when I apply a force of 2 Newtons.
- Well, let's say it this way.
- Let's say when I stretch the spring.
- Let's say this is the spring, and when I apply a force of 2
- Newtons to the right, the spring gets stretched 1 meter.
- So first of all, let's figure out what K is.
- So if the spring is stretched by 1 meter, out here, its
- restorative force will be 2 Newtons back this way, right?
- So its restorative force, this 2 Newtons, will equal minus K
- times how much I displaced it.
- Well I, displaced it by 1 meter, so then we multiply
- both sides by negative 1, and we get K is equal to minus 2.
- So then we can use Hooke's Law to note the equation for
- this-- to figure out the restorative force for this
- particular spring, and it would be minus 2x.
- And then I said, well, how much force would I have to
- apply to distort the spring by 2 meters?
- Well, it's 2 times 2, it would be 4.
- 4 Newtons to displace it by 2 meters, and, of course, the
- restorative force will then be in the opposite direction, and
- that's where we get the negative number.
- Anyway, I've run out of time.
- I'll see you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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