### Video Transcript

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

Alright so we’ve been told that the graph is showing the length of a spring, and we’ve been told that we are applying a force to the spring which changes. What we need to do is to find the spring constant. Now let’s imagine that the spring we’re considering is this spring here. Initially, we’re not exerting any force on it, so it’s just sitting there at its natural length.

Now from our graph we can actually see what the natural length of the spring is, because we can see that when a force of zero newtons, that is no force is exerted on the spring, the length of the spring is 0.5 meters. Therefore, because no force is being exerted on the spring, this has to be its natural length. Now why is this important? Well when we want to find the spring constant and relate it to the force applied on the spring, we need to use Hooke’s law to do this.

Now Hooke’s law tells us that the force applied on a spring is equal to the spring constant of the spring multiplied by the extension of the spring, so not the length of the spring, which is what we’ve been given at the graph, but the extension of the spring. In other words, let’s say we apply a force 𝐹 to the spring. And due to this force 𝐹, the spring extends by a certain amount. So the right- hand end of the spring goes from here to here. Well then, the force 𝐹 that we’ve applied is directly proportional to this extension; that is, the distance that we’ve labelled 𝑥, not the length of the spring.

Now interestingly we can see that the extension of the spring 𝑥 is given by finding the total length of the spring minus the natural length of the spring. In other words, this distance the extension is equal to the whole length of the string minus the natural length of the spring. Therefore, we can say that the extension 𝑥 of the spring is equal to the length of the spring at any point, which is what will call 𝐿, that’s this total distance here, minus the natural length of the spring, which is 0.5 meters. And then, we’ll take our Hooke’s law equation and rearrange it to find the spring constant. We can do this by dividing both sides of the equation by the extension 𝑥.

This way, the extension on the right-hand side cancels out. And on the right, we’re just left with the spring constant. In other words, 𝐹 divided by 𝑥 is equal to 𝑘. Now in order to find the spring constant of the spring, we can see what the extension of the spring is for any given value of force that we exert on it. So for example, let’s take the final point on the curve here. This final point tells us that when the force exerted on the spring is 50 newtons, the total length of the spring becomes one meter. And so we can say that when the force is 50 newtons on the spring, the length of the spring is one meter.

But from earlier, we can see that the extension of the spring is equal to the length minus 0.5 meters. And so we can say that the extension is one minus 0.5, or just 0.5 meters. Now at this point, we know the force exerted on the spring for that particular point here, and we know the extension this causes. So we can substitute those values into our equation here to work out the spring constant. Hence, we can say that 𝑘, the spring constant, is equal to 50 newtons, the force exerted on the spring, divided by 0.5 meters, the extension of the spring caused by this force.

And when we evaluate the right-hand side, we find that the spring constant ends up being 100 newtons per meter because, remember, we had our force in newtons and our extension in meters. So the unit for 𝑘 is going to be in newtons per meter. And so we have a final answer: the spring constant of the spring is 100 newtons per meter.