Learn about Potential energy stored in a spring Video

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Learn about Potential energy stored in a spring

Learn about Potential energy stored in a spring Welcome back, so we have this green spring here and let’s see there’s a wall here and let’s say that this is where the spring is naturally—so if I were not to push on the spring you would stretch all the way out here but in this situation I push on the spring so it has the displacement of x to the left and what is worried about magnitudes? We won’t wait 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 have applied at different points as I compress the 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 and I know I’m compressing to the left. Maybe I should compress to the right so that you can—well we’ll just worried about the magnitude of the x-axis. So let’s draw a little graph here. That’s my y-axis, my x-axis and so this axis is how much I’ve compressed it x and then this y-axis is how much force I have to apply. So when the spring was initially all the way out here. To compressed a little bit how much force I have to apply? Well this was its natural state right and we know from—well what we have done with the rest to the force is equal to negative k where k is a spring constant times the displacement. That’s the restore to the force. So that’s the force that the spring applies to whoever is pushing on it. The force to compress it is just the same thing but it’s going into 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 call it 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 of the 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’d understand that x is increasing to the left in my example right. This is where x = 0 right here and say you know this might be x = 10 because we’ve compressed it by 10 meters. So let’s see how much force we’ve applied. So when x is zero 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 bit of force, if we just give like infinitesimals 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 the spring is barely compressed were going to apply a little bit of force so almost at zero. Pretty much when to displace the spring zero we apply zero force, the sprigs the spring a little bit we need to apply a little bit more force. To displace the spring one meter, so if this is say one meter, how much force we’ll have to apply I guess to keep it there? So let’s say if this is one meter the force of compression is going to be k times 1. So it’s going to be k. And realize you didn’t apply zero and then click 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 one meter 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’ve been up to 2k etcetera and I think you see a line is forming. 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 rested. 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 rested force and so it might accelerate in the little spring bracket actually well 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 to the force. Hopefully that make sense that you understand that the force just increases proportional as a function of the distance and that just a linear equation and what’s the slope of this? Well slope is rise over run right. So if I run one what’s my rise? Its 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 the spring—let’s say this is the x not. So x is where it’s a general variable. X is not a particular value for x. It could be 10 or whatever. Let’s see how much worked we need. So what’s the definition of work? Work is equal to the force and the direction of you displacement times the displacement right. So let’s see how much we’ve displace. So when we go from zero to here we displace 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 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 I’m doing another car and then to displace the next little distance 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 this rectangles right because the height of the rectangle is a force I’m applying and the width is the distance. 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 that 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 one-half right that’s just the area of a triangle. So what’s the base? So this is just x not, what’s the height? Well we know the slope is k so this height is going to be x not times k. So this point right here is a point x not and then x not times k and so what’s the area under the curve which is the total work I did to compress the spring x not meter. Well it’s the base x not times the height x not times k and then of course multiply by 1/2 because we are dealing with the triangle so that equals 1/2k x not² and for those of you who know calculus that of course the same thing as the integral of kx dx and it should make sense. Each of these is a little dx. Spend on the code too much on the calculus now to confuse people, so that’s the total work necessary to compress the spring by distance of x not or if set the distance of x you just get rid of this not here and y is that useful? Because the work necessary to compress the spring that much is also how much potential energy there stored in the spring. So if I told you that I had a spring and its by hooks—spring constant is 10 and I compressed it—I don’t know 5 meters, so x = 5 meters at the time that is compress, how much potential energy is in that spring? Well we could just say the potential energy is equal to 1/2k times x² = ½, k = 10 x 25 = 125 and of course work in potential energy are measured in joules. So this is really is what you have to memorized or I hope you don’t memorize. I hope 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 stored once it is compressed to that point or actually stretched that magic. It could also—you know we’ve been compressing but you could also stretch the spring. And if you know that then we can start doing some problems with potential energy in springs which I will do in the next video. See you soon.

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Learn about Potential energy stored in a spring

Welcome back, so we have this green spring here and let’s see there’s a wall here and let’s say that this is where the spring is naturally—so if I were not to push on the spring you would stretch all the way out here but in this situation I push on the spring so it has the displacement of x to the left and what is worried about magnitudes? We won’t wait 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 have applied at different points as I compress the 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 and I know I’m compressing to the left. Maybe I should compress to the right so that you can—well we’ll just worried about the magnitude of the x-axis. So let’s draw a little graph here. That’s my y-axis, my x-axis and so this axis is how much I’ve compressed it x and then this y-axis is how much force I have to apply. So when the spring was initially all the way out here. To compressed a little bit how much force I have to apply? Well this was its natural state right and we know from—well what we have done with the rest to the force is equal to negative k where k is a spring constant times the displacement. That’s the restore to the force. So that’s the force that the spring applies to whoever is pushing on it. The force to compress it is just the same thing but it’s going into 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 call it 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 of the 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’d understand that x is increasing to the left in my example right. This is where x = 0 right here and say you know this might be x = 10 because we’ve compressed it by 10 meters.

So let’s see how much force we’ve applied. So when x is zero 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 bit of force, if we just give like infinitesimals 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 the spring is barely compressed were going to apply a little bit of force so almost at zero. Pretty much when to displace the spring zero we apply zero force, the sprigs the spring a little bit we need to apply a little bit more force. To displace the spring one meter, so if this is say one meter, how much force we’ll have to apply I guess to keep it there? So let’s say if this is one meter the force of compression is going to be k times 1. So it’s going to be k. And realize you didn’t apply zero and then click 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 one meter 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’ve been up to 2k etcetera and I think you see a line is forming. 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 rested. 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 rested force and so it might accelerate in the little spring bracket actually well 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 to the force. Hopefully that make sense that you understand that the force just increases proportional as a function of the distance and that just a linear equation and what’s the slope of this?

Well slope is rise over run right. So if I run one what’s my rise? Its 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 the spring—let’s say this is the x not. So x is where it’s a general variable. X is not a particular value for x. It could be 10 or whatever. Let’s see how much worked we need. So what’s the definition of work? Work is equal to the force and the direction of you displacement times the displacement right. So let’s see how much we’ve displace. So when we go from zero to here we displace 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 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 I’m doing another car and then to displace the next little distance 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 this rectangles right because the height of the rectangle is a force I’m applying and the width is the distance. 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 that 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 one-half right that’s just the area of a triangle. So what’s the base? So this is just x not, what’s the height? Well we know the slope is k so this height is going to be x not times k. So this point right here is a point x not and then x not times k and so what’s the area under the curve which is the total work I did to compress the spring x not meter.

Well it’s the base x not times the height x not times k and then of course multiply by 1/2 because we are dealing with the triangle so that equals 1/2k x not² and for those of you who know calculus that of course the same thing as the integral of kx dx and it should make sense. Each of these is a little dx. Spend on the code too much on the calculus now to confuse people, so that’s the total work necessary to compress the spring by distance of x not or if set the distance of x you just get rid of this not here and y is that useful? Because the work necessary to compress the spring that much is also how much potential energy there stored in the spring.

So if I told you that I had a spring and its by hooks—spring constant is 10 and I compressed it—I don’t know 5 meters, so x = 5 meters at the time that is compress, how much potential energy is in that spring? Well we could just say the potential energy is equal to 1/2k times x² = ½, k = 10 x 25 = 125 and of course work in potential energy are measured in joules. So this is really is what you have to memorized or I hope you don’t memorize. I hope 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 stored once it is compressed to that point or actually stretched that magic. It could also—you know we’ve been compressing but you could also stretch the spring. And if you know that then we can start doing some problems with potential energy in springs which I will do in the next video. See you soon.

Transcription by: Scribe4you Transcription Services