Potential energy stored in a spring Work needed to compress a spring is the same thing as the potential energy stored in the compressed spring.
Potential energy stored in a spring
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- Welcome back.
- So we have this green spring here, and let's see,
- there's a wall here.
- This connected to the wall.
- And let's say that this is where the spring is naturally.
- So if I were not to push on the spring, it would stretch
- all the way out here.
- But in this situation, I pushed on the spring, so it
- has a displacement of x to the left.
- And we'll just worry about magnitude, so we won't worry
- too much about direction.
- So what I want to do is think a little bit-- well, first I
- want to graph how much force I've applied at different
- points as I compress this spring.
- And then I want to use that graph to maybe figure out how
- much work we did in compressing the spring.
- So let's look at-- I know I'm compressing to the left.
- Maybe I should compress to the right, so that you can-- well,
- we're just worrying about the magnitude of the x-axis.
- Let's draw a little graph here.
- That's my y-axis, x-axis.
- So this axis is how much I've compressed it, x, and then
- this axis, the y-axis, is how much force I have to apply.
- So when the spring was initially all the way out
- here, to compress it a little bit, how much force
- do I have to apply?
- Well, this was its natural state, right?
- And we know from-- well, Hooke's Law told us that the
- restorative force-- I'll write a little r down here-- is
- equal to negative K, where K is the spring constant, times
- the displacement, right?
- That's the restorative force, so that's the force that the
- spring applies to whoever's pushing on it.
- The force to compress it is just the same thing, but it's
- going in the same direction as the x.
- If I'm moving the spring, if I'm compressing the spring to
- the left, then the force I'm applying is also to the left.
- So I'll call that the force of compression.
- The force of compression is going to be
- equal to K times x.
- And when the spring is compressed and not
- accelerating in either direction, the force of
- compression is going to be equal to
- the restorative force.
- So what I want to do here is plot the force of compression
- with respect to x.
- And I should have drawn it the other way, but I think you
- understand that x is increasing to the left in my
- example, right?
- This is where x is equal to 0 right here.
- And say, this might be x is equal to 10 because we've
- compressed it by 10 meters.
- So let's see how much force we've applied.
- So when x is 0, which is right here, how much force do we
- need to apply to compress the spring?
- Well, if we give zero force, the spring won't move, but if
- we just give a little, little bit of force, if we just give
- infinitesimal, super-small amount of force, we'll
- compress the spring just a little bit, right?
- Because at that point, the force of compression is going
- to be pretty much zero.
- So when the spring is barely compressed, we're going to
- apply a little, little bit of force, so almost at zero.
- To displace the spring zero, we apply zero force.
- To displace the spring a little bit, we have to apply a
- little bit more force.
- To displace soon. the spring 1 meter, so if this is say, 1
- meter, how much force will we have to
- apply to keep it there?
- So let's say if this is 1 meter, the force of
- compression is going to be K times 1, so it's
- just going to be K.
- And realize, you didn't apply zero and then apply K force.
- You keep applying a little bit more force.
- Every time you compress the spring a little bit, it takes
- a little bit more force to compress it a little bit more.
- So to compress it 1 meters, you need to apply K.
- And to get it there, you have to keep increasing the amount
- of force you apply.
- At 2 meters, you would've been up to 2K, et cetera.
- I think you see a line is forming.
- Let me draw that line.
- The line looks something like that.
- And so this is how much force you need to apply as a
- function of the displacement of the spring from its natural
- rest state, right?
- And here I have positive x going to the right, but in
- this case, positive x is to the left.
- I'm just measuring its actual displacement.
- I'm not worried too much about direction right now.
- So I just want you to think a little bit about what's
- happening here.
- You just have to slowly keep on-- you could apply a very
- large force initially.
- If you apply a very large force initially, the spring
- will actually accelerate much faster, because you're
- applying a much larger force than its restorative force,
- and so it might accelerate and then it'll spring back, and
- actually, we'll do a little example of that.
- But really, just to displace the spring a certain distance,
- you have to just gradually increase the force, just so
- that you offset the restorative force.
- Hopefully, that makes sense, and you understand that the
- force just increases proportionally as a function
- of the distance, and that's just because
- this is a linear equation.
- And what's the slope of this?
- Well, slope is rise over run, right?
- So if I run 1, this is 1, what's my rise?
- It's K.
- So the slope of this graph is K.
- So using this graph, let's figure out how much work we
- need to do to compress this spring.
- I don't know, let's say this is x0.
- So x is where it's the general variable.
- X0 is a particular value for x.
- That could be 10 or whatever.
- Let's see how much work we need.
- So what's the definition of work?
- Work is equal to the force in the direction of your
- displacement times the displacement, right?
- So let's see how much we've displaced.
- So when we go from zero to here, we've
- displaced this much.
- And what was the force of the displacement?
- Well, the force was gradually increasing the entire time, so
- the force is going to be be roughly about that big.
- I'm approximating.
- And I'll show you that you actually have to approximate.
- So the force is kind of that square right there.
- And then to displace the next little distance-- that's not
- bright enough-- my force is going to increase a little
- bit, right?
- So this is the force, this is the distance.
- So if you you see, the work I'm doing is actually going to
- be the area under the curve, each of
- these rectangles, right?
- Because the height of the rectangle is the force I'm
- applying and the width is the distance, right?
- So the work is just going to be the sum of all of these
- And the rectangles I drew are just kind of approximations,
- because they don't get right under the line.
- You have to keep making the rectangle smaller, smaller,
- smaller, and smaller, and just sum up more and more and more
- rectangles, right?
- And actually I'm touching on integral calculus right now.
- But if you don't know integral calculus,
- don't worry about it.
- But the bottom line is the work we're doing-- hopefully I
- showed you-- is just going to be the area under this line.
- So the work I'm doing to displace the spring x meters
- is the area from here to here.
- And what's that area?
- Well, this is a triangle, so we just need to know the base,
- the height, and multiply it times 1/2, right?
- That's just the area of a triangle.
- So what's the base?
- So this is just x0.
- What's the height?
- Well, we know the slope is K, so this height is going to be
- x0 times K.
- So this point right here is the point x0, and
- then x0 times K.
- And so what's the area under the curve, which is the total
- work I did to compress the spring x0 meters?
- Well, it's the base, x0, times the height, x0, times K.
- And then, of course, multiply by 1/2, because we're dealing
- with a triangle, right?
- So that equals 1/2K x0 squared.
- And for those of you who know calculus, that, of course, is
- the same thing as the integral of Kx dx.
- And that should make sense.
- Each of these are little dx's.
- But I don't want to go too much into calculus now.
- It'll confuse people.
- So that's the total work necessary to compress the
- spring by distance of x0.
- Or if we set a distance of x, you can just get
- rid of this 0 here.
- And why is that useful?
- Because the work necessary to compress the spring that much
- is also how much potential energy there is
- stored in the spring.
- So if I told you that I had a spring and its spring constant
- is 10, and I compressed it 5 meters, so x is equal to 5
- meters, at the time that it's compressed, how much potential
- energy is in that spring?
- We can just say the potential energy is equal to 1/2K times
- x squared equals 1/2.
- K is 10 times 25, and that equals 125.
- And, of course, work and potential energy
- are measured in joules.
- So this is really what you just have to memorize.
- Or hopefully you don't memorize it.
- Hopefully, you understand where I got it, and that's why
- I spent 10 minutes doing it.
- But this is how much work is necessary to compress the
- spring to that point and how much potential energy is
- stored once it is compressed to that point, or actually
- stretched that much.
- We've been compressing, but you can
- also stretch the spring.
- If you know that, then we can start doing some problems with
- potential energy in springs, which I will
- do in the next video.
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