A block is initially at rest on a frictionless tabletop. One end of the block is attached to a spring. The other end of the spring is attached to a wall, as shown in the figure. The block is pulled away from the wall. Use the graphs shown to answer the following question.
Before we look at that question, let’s look for a moment at this sketch of the block, spring, and wall. We’re told that the surface our block is on is frictionless, which means it can move without any resistance from the tabletop. The block starts out at rest. And then, we’re told that it’s pulled away from the wall to the right. So, our block is starting to move. And now let’s get to the question about that movement.
Which graph correctly represents the elastic potential energy stored by the spring as a function of the displacement of the block from its initial position?
This question is referring to these five graphs a, b, c, d, and e, and asks which one correctly represents this relationship, spring elastic potential energy and block displacement. As we start out answering this question, let’s clear a bit of space on screen. Now, with these five graphs, we can see that none of the axes have labels on them. But our question tells us what those axes are meant to represent.
We’re to consider the elastic potential energy stored by the spring. So, that’s on the vertical axis of each of these graphs. We can write that in this way as 𝑃𝐸 sub 𝐸. And then, this elastic potential energy is being plotted as a function of the displacement of the block from its initial position. Going back to our sketch of the block, let’s say that this is its initial position. We’re told that the block is displaced from this position. And we can call that displacement 𝑋. It’s that displacement that the horizontal axis of each of our graphs is meant to indicate.
Now, one important point about displacement is that the displacement of our block when it’s in this position is zero. It’s only movement away from this initial position that counts as displacement. So, even though the block starts out this distance away from the wall, its initial displacement is still zero until it’s been moved.
Getting back to our question, we want to know which of these five graphs correctly represents the elastic potential energy stored in the spring versus the displacement of the block from its starting position. To figure this out, let’s recall the mathematical relationship for Elastic potential energy. This is also sometimes called spring potential energy because elastic potential energy is equal to one-half multiplied by this constant 𝐾 called the spring constant and that is multiplied by the displacement of the spring from equilibrium squared.
Now, this spring, the spring we’re working with in this scenario, has some spring constant. But as it turns out, we don’t need to know that to answer this question. The important thing is not what is 𝐾 the spring constant, but rather what’s the overall shape of this mathematical function. Here’s what we mean by that. Whatever 𝐾 is, whatever the spring constant is for our particular spring, this term one-half multiplied by 𝐾 is a constant value. In this equation, then, we have a constant value multiplied by our variable, the displacement 𝑋 squared. In other words, this is a quadratic function.
To make that more clear, we can write it this way. We can say that the elastic potential energy of a spring is a function of the spring’s displacement from equilibrium. And, going further, that elastic potential energy is directly proportional to that displacement squared. In other words, just like we said, we have a quadratic function where our variable is the displacement of our block from equilibrium.
So, let’s recall, what does a quadratic function in general look like? Say we have a function, we call it 𝑌. And 𝑌 is equal to 𝑋 squared. If we were to plot out 𝑋- and 𝑌-axes, then this function 𝑌 is equal to 𝑋 squared would look something like this. The rule is that we take whatever our 𝑋 value is, square it, and that’s equal to 𝑌. This is a quadratic function. All this tells us that, as we pick which of these five graphs correctly shows the relationship between elastic potential energy and displacement of the block, we’ll be looking for one that shows this quadratic relationship. After all, that’s the type of relationship there is between elastic potential energy and displacement.
Starting with answer option a, we see that this curve does not follow the 𝑌 equals 𝑋 squared overall shape. We won’t choose that one as our answer. Likewise, for options b and c, graph c looks like a sine relationship, different from a quadratic relationship. And we can see that graph b is linear rather than quadratic.
Considering then answer options d and e, if we compare the general shape of our 𝑌 equals 𝑋 squared curve to d, we see that there isn’t a match there. But for graph e, there is. We see the slope of the line in graph e is constantly increasing just like it is in our plot of 𝑌 equals 𝑋 squared. This tells us that graph e matches the functional shape that we’re looking for. It indicates that elastic potential energy is proportional to the square of the displacement of the block from its initial position. Our answer choice, then, is graph e.