Law told us that the restorative force- I'll writeĪ little r down here- is equal to negative K, where K is State, right? And we know from- well, Hooke's So when the spring was initiallyĪll the way out here, to compress it a littleīit, how much force do I have to apply? Well, this was its natural So this axis is how much I'veĬompressed it, x, and then this axis, the y-axis, is how Right, so that you can- well, we're just worrying about the So let's look at- I know I'mĬompressing to the left. Graph to maybe figure out how much work we did in compressing I've applied at different points as I compress So what I want to do is thinkĪ little bit- well, first I want to graph how much force Magnitude, so we won't worry too much about direction. Spring, it would stretch all the way out here. Here, and let's see, there's a wall here. I'm sorry if this was too long of a response, I have had so much trouble with this topic and I still do. To make it a little easier, we can revert to integration (which is a b word), because when we integrate a function (say, a function for a force), then we are able to account for the differing Y. Because F is changing, there will be different y values for different x's. When we are given an equation for the force, that means the force is changing and we cannot straight up do W=Fd because F is changing. The force needed CHANGES this is why we are given an EQUATION for the force: F = kx, yes? If the F = a constant, we would, indeed, have a rectangle. Why is it not the whole rectangle in the case of a spring? In the case of a spring, the force that one must exert to compress a spring 1m is LESS than the force needed to compress it 2m or 3m, etc. If you apply a CONSTANT force of 13N, then the graph of work (which would be a graph of force on the y axis and displacement on the x axis) would be a rectangle the vertical component (the force) would be all the way up to 13N (again, on the y axis), while the x component would just be all the way up to 5m. Basically, we would only have a rectangle graph if our force was constant! Unfortunately, the force changes with a spring. We only have a rectangle-like graph when the force is constant. We are looking for the area under the force curve.
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