If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Potential energy stored in a spring

## Video transcript

welcome back so we have this green spring here and let's see there's a wall here let's connect to the wall and let's say that this is that 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've 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 I want to graph how much force I have applied at different points as I compress the spring and then we I want to use that graph to to maybe figure out how much work we did in compressing the spring so let's look at let's look at and I know I'm compressing to the left maybe I should compress to the right so that you can well well we're just worried about the magnitude of the x axis so let's let's draw a little graph here let's make my y-axis x axis and 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 the spring was initially all the way out here to compress it a little bit how much force to have to apply well this was its natural state right and we know from well Hookes law told us that the the rest root of force the rest root of force all right a little are down here the rest root of force is equal to negative K where K is the spring constant times the displacement right that's the rest root of force so that's the force that the spring applies to whoever is pushing on it the force to compress it is just it's just the same thing but it's going in the same direction as the X so 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 rest root of force so what I want to hear it do plot the force of compression with respect to X and I know this I should have drawn it the other way but I think you understand that X is increasing to the left and this in my example right this is where X is equal to zero this is say X is equal to zero right here this is x equals zero and say you know this might be X is equal to ten 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 if we 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 like in an infinitesimal super small amount of force will 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 when the spring is very compressed we're going to apply a little bit of force so almost at zero pretty much it what to displace the spring zero we apply zero force just breaks the splitting a little bit we have to apply a little bit more force to displace the spring 1 meter so if this is say one meter one meter how much force how much force will we have to apply I guess to keep it there so let's say if this is one one meter the force of compression is going to be K times one so it's just going to be K okay and realize you didn't apply zero and then play apply a force you keep applying a little bit more a little bit more force every time 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 have been up to 2k 2k etc and I think you see a line is forming let me draw that line the line looks like 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 alright 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 Inc 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 a much larger force than its rest rooted force and so it might accelerate and then it'll spring Brak 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 rest root of force hopefully that makes sense that you understand that the force just increases proportional 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 one what's this is 1 what's my rise it's K so the slope of this equation the slope of this graph is K so using this graph let's figure out how much work we need to do to compress the spring you know I don't know let's say this is X naught so X is where it's a general variable X naught is a particular value for X that could be 10 or whatever let's see how much work it 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 when we go from zero to here we've deployed 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 roughly 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 moved in another color and then to displace the next little distance 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 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 rectangles 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 rectangles 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 is the work we're doing hopefully I showed you it's 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 one half right that's just the area of a triangle so what's the base so this is just X naught what's the height well we know the slope is K so this height is going to be X naught times K oops so this this this point right here is the point X naught and then X naught times K and so what's the area under the curve which is the total work I did to compress the spring X naught meters well it's the base X naught times the height X naught times K and that and then of course multiplied by one half because were you dealing with the triangle right so that equals one-half K X naught squared and for those of you know who know calculus that of course is the same thing as the integral of K X DX and it should make sense each of these are little D X's but I don't want to go too much into calculus now to confuse people so that's the total work necessary to compress this bring by a distance of X naught or if we've set a distance of X you just get rid of this knot 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 spring and it's by hooks it's it's spring constant it's spring constant is 10 and I compressed it I don't know 5 meters so X is equal to 5 meters at the time that is compressed how much potential energy is in that spring well we could just say the potential energy is equal to 1/2 K times x squared it 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 I hope you don't memorize 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 a spring to that point and how much potential energy is stored once it is compressed to that point or actually stretched that much it could also you know we've been compressing but you could also stretch a spring and and if you know that then we can we can start doing some problems with potential energy in Springs which I will do in the next video see you soon