Video: Determining the Spring Constant of a Spring

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

02:35

Video Transcript

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

Okay, so as we’ve been told in the graph, we can see that, on the horizontal axis, we’ve been given the extension of a spring and, on the vertical axis, we’ve got the force applied to that spring. So let’s imagine that this is our spring. Now at the moment, the spring has no force applied to it. So its extension is zero. And we can see this from the graph because we can see that when the force applied is zero newtons the extension is also zero meters.

However, as we start to apply a force to the spring, which we’ll call 𝐹, the spring starts to extend. And the extension of the spring is this distance here, which we’ll call 𝑥. In other words, the extension of the spring is how much longer the spring is compared to its natural length, where of course the natural length of the spring is this distance here. It’s the length of the spring when no force is applied to it.

But, we don’t need to worry about that. All we need to worry about is the force applied and the extension of the spring. And of course, we’ve been asked to work out the spring constant. So to do this, we need to recall something known as Hooke’s law. Hooke’s law tells us that the force applied to a spring 𝐹 is equal to the spring constant 𝑘 multiplied by the extension of the spring caused by the force, where we’ll call the extension 𝑥.

Now if we want to find the value of the spring constant, then we need to rearrange this equation. We need to divide both sides by the extension 𝑥 so that 𝑥 on the right-hand side cancels out. Then we’re just left with 𝐹 over 𝑥 on the left-hand side and 𝑘 the spring constant on the right-hand side. Now since the spring constant is a constant — in other words, it doesn’t change with force or with the extension 𝑥 — all we need to do is to pick any random point on this graph.

So let’s say we’ve picked the last one here. And then we need to find the value of the force exerted on the spring at that point, which happens to be 100 newtons. And we need to know the extension of the spring caused by that force. Now in this case, the extension happens to be 2.0 meters. And so we can say that when the force applied 𝐹 is equal to 100 newtons, the extension is 2.0 meters, which means that we can plug these values into our equation here to find the value of the spring constant.

So we’ve substituted the value of the force, which is 100 newtons, and the extension, 2.0 meters. Now before we evaluate the fraction, we can see that the units of the spring constant are going to be newtons per meter. And once we’ve noticed that about the unit, we can evaluate the fraction. And when we do, we find that the value of 𝑘 is 50 newtons per meter. So we found our final answer: the spring constant is 50 newtons per meter.

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