Video: Finding the Spring Constant of a Spring Using a Graph

The graph shows the length of a spring as the force applied to it changes. What is the spring constant?

04:13

Video Transcript

The graph shows the length of a spring as the force applied to it changes. What is the spring constant?

Okay, so as we can see here, in the graph, we’ve been given the length of the string on the horizontal axis and the force applied to the spring on the vertical axis. Now, we can clearly see that when no force — zero force — is applied to the spring, the length of the spring is 0.5 meters. Therefore, this is what the spring looks like when no forces have been applied to it.

So here’s the spring and its length is 0.5 meters. Now, subsequently, the force applied to the spring increases. And we can see that as the force applied increases, so does the length of the spring. In other words, we exert a force on the spring — let’s say on the right end of the spring — and we call this force 𝐹.

Now because of this, the length of the spring increases. So now, the total length of the spring is from here to here. Now, if we’ve been asked to find the spring constant of the spring, then we need to recall Hooke’s law.

Hooke’s law tells us that the force applied to a spring 𝐹 is equal to the spring constant of the spring 𝑘 multiplied by the extension of the spring 𝑥. Now, note that we’re talking about the extension of the spring and not the length of the spring. In other words, when the force 𝐹 has been applied to the spring, this force is directly proportional to the extension; that’s this length here. So let’s call that length 𝑥.

The force is not directly proportional to the entire length of the spring, which we’ll call 𝐿. But because we’ve been given the length of the spring on the horizontal axis, we need to find a relationship between the length of the spring and the extension so that we can work out the spring constant.

The way we do this is to say that the total length of the spring — this whole distance here — is equal to 0.5 — that’s the natural length of the spring when no force is acting on it — plus 𝑥, the extension. And so we can say that 𝐿 is equal to 𝑥 plus 0.5 meters or if we rearrange this, 𝑥 is equal to 𝐿 minus 0.5 meters.

This means that for any point on the graph, we can now work out the force exerted on the spring, which we can simply read off by moving left to the vertical axis. And we can work out the extension of the spring. Because we know the length of the spring at that point and if we subtract 0.5 meters from it, then we’ll have the extension.

So let’s just pick any random point and rearrange Hooke’s law to work out the spring constant. We’re gonna divide both sides of the equation by 𝑥, the extension, so that the extension cancels out on the right-hand side. What we’re then left with is that the spring constant 𝑘 is equal to the force applied 𝐹 divided by the extension 𝑥.

And now, let’s assume we’re considering this point here. Well, we can see that the force applied at this point, if we go left to the vertical axis, is 200 newtons. So we can say that when the force 𝐹 is 200 newtons, the length of the spring 𝐿 is going to be 3.0 meters, which is what we write down over here. But then, we can use this equation to give us the extension of the spring.

Essentially, we subtract 0.5 meters from the length 𝐿. And so we say that 𝑥, the extension, is equal to 3.0 meters, the length, minus 0.5 meters, the natural length of the spring. And when we evaluate this, we find that the extension is 2.5 meters.

So now, we’ve got the force applied to the spring on one point of the graph and the extension of the spring at that same point. Therefore, we can substitute in the values to find the value of 𝑘. Now, note that what we’re basically doing is finding the gradient of this line of best fit because what we’re doing here is saying that 𝑘 is equal to the force applied to the spring which is 200 newtons. But that’s also equivalent to 200 minus zero newtons. And we’re dividing this by 3.0 — that’s the length of the spring — minus 0.5 — that’s the natural length of the spring.

So coincidentally, by doing this calculation, we’re actually finding the gradient of the line of best fit. But this is irrelevant. Now, when we sub in the values, we can say that the value of 𝑘 is equal to the force 200 newtons divided by the extension 2.5 meters. And once we evaluate the fraction, on the right, we find that the value of 𝑘 is 80 newtons per meter.

Hence, we have a final answer. The spring constant of the spring is 80 newtons per meter.

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