potential energy

There is a deep connection between force and potential energy. This relationship has a useful graphical representation that will help us better understand the spring-mass potential energy and, in Chapter 3, the potential energy associated with the bonding between atoms. We know intuitively that the spring will try to snap back to it's original shape -- that is, that it will tend to return to zero deformation. How can we confirm our intuitions from the graph? To answer, we'll introduce the connection between force and potential energy. The minus sign means that if the slope is positive, the force is to the left, and visa versa. This should make sense, because it says that the force will try to push the object back to lower potential. If it were the other way around, springs would stretch themselves out spontaneously and planets would fly away from each other! This is a general result that is true for the force associated with any potential energy[ [i] ]( . Disregarding the minus sign for a moment, this tells us that the steeper the slope of a PE curve plotted against its position variable, the greater the magnitude of the force . The restoring force of the spring (or anything that oscillates) will be zero when the slope is zero, which occurs at the equilibrium point, i.e., where the object comes to rest when it stops vibrating. Using this knowledge, let's see how it makes perfect sense for the spring-mass potential graphed above. Starting qualitatively, the force should be maximum when the spring is most out of shape. The parabolic function has a slope which increases in magnitude the further from zero we go. Applying the idea that the magnitude of the slope tells us the magnitude of the force, we can tell from the graph that the force increases the further the spring is from equilibrium. Next, the force should be zero when the spring is in its equilibrium position. A quick look at the graph tells us that the slope of the curve at the origin is in fact zero. So far so good! Lastly, there's the minus sign. As the spring-mass is pulled to the right, the graph's slope becomes positive. The potential-energy-force relationship tells us that the force should then be negative , which means to the left. This should make perfect sense: the spring is stretched to the right, so it pulls left in an attempt to return to equilibrium. We can quantitatively show just how right this relationships is. If we take our spring-mass potential energy function, then the force is The final result is Hooke's law, the first equation introduced in our discussion of springs! Now that you're convinced that this relationship is real, let's see if we can understand why. It all boils down to the concepts of work and energy conservation. To understand, consider the following situation (pictured below). Imagine a box is lifted through a potential field, like lifting an object against gravity. Imagine also that the field is associated with a force, which is constant and pointing downwards at all points in space. However, what we wish to know isn't the force I exerted, but the force the field exerted. By the design of the situation, the force the field exerts is equal and opposite the force I exert, so Pictured is the real potential energy vs. separation relationship for two hydrogen atoms. What does the shape of the curve tell us about the behavior of the bond? By applying the relationship between force and potential energy, you will eventually arrive upon an intuition which is akin to treating the curve like the tracks of a roller coaster. That is, you can visualize the behavior of the system by imagining the object as riding the curve like a cart subject to gravity. What that would tell us here is that hydrogen atoms easily fall into bonds with each other. We know this because the hydrogen "cart" would easily slide into the bond well. We can also tell that it's nigh impossible to bring two hydrogen atoms too close together. Imagine trying to push a ca