A spring that is 0.36 meters long is pulled by a force of 18 newtons. The spring’s length increases by 0.012 meters. What is the constant of the spring?
Okay, so let’s start by drawing a diagram to represent the spring before the forces acting on it and after the forces acting on it. So here’s the spring before any forces acting on it and we’ve been told that the length of the spring is 0.36 meters. Now, at this point, we exert a force of 18 newtons on the spring. As a result, the length of the spring increases by 0.012 meters. We need to find the spring’s constant or the spring constant.
So let’s say that the spring constant of the spring which we’ll call 𝑘 is equal to question mark cause we don’t know what it is. Now, this question is one of those red herring questions, where there’s an extra information that we don’t need. To figure out what this is, let’s first recall the relationship between the force applied, the spring constant, and extension of the spring. This is the relationship we’re looking for: the force exerted on a spring is equal to the spring constant multiplied by the extension of the spring.
Now, there are a couple of important points here. First of all, the force exerted is actually not the force exerted on the spring, but rather is the force exerted against this motion by the spring itself. Whenever we extend the spring, it exerts a force against this extension. The same thing with a compression — whenever we compress the spring, the spring tries to resist this. And the way it does this is by exerting a force against the compression.
So in this case, we’re extending the spring. We’re exerting a force of 18 newtons to the right and the spring counter this by exerting a force to the left. Now, the spring only stops extending when the force to the left is also equal to 18 newtons because this way the forces on the spring are balanced. And hence, it’s not going to move in either direction. So that’s what the force 𝐹 is talking about.
Now, the other important thing is to talk about the extension 𝑥. This very specifically refers to this length here — the extension of the spring. In other words, how much does the length of the spring change by? So for this reason, the original length of the spring is not relevant, neither is the length of the spring after it’s extended. What is relevant is the extension here, which we’ll call 𝑥.
So now that we’ve discussed all of this, we can safely go about finding out the spring constant. To do this however, we need to rearrange the equation first. We need to divide both sides of the equation by 𝑥. Doing this leaves us with 𝐹 divided by 𝑥 is equal to 𝑘. Now at this point, we just need to substitute in the values. So we say that 𝑘 is equal to firstly the force exerted by the spring — that’s 18 newtons. And we divide this by the extension of the spring 𝑥, which is 0.012 meters.
Now, it’s important to note that we’re working in standard units. The standard unit of force is newtons and the standard unit of distance is meters. Therefore, the spring constant that we find is also going to be in its standard unit of newtons per meter. And so when we evaluate the right-hand side of the equation, we find that the constant of the spring is 1500 newtons per meter. And this is our final answer.