Video: Spring Force

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

03:01

Video Transcript

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

Alright, so on this graph, we’ve got the force applied to a spring on the horizontal axis and the extension of a spring as a result of that force on the vertical axis. So let’s imagine that this is our spring. Now, as we would expect, when the force applied to the spring is zero newtons, the extension of the spring is also zero newtons. Because the spring is simply sitting at its natural length when there is no force applied to it. It’s not gonna be extended if there’s no force on it. However, as we increase the force applied to the spring, that is we start applying a force, let’s say, to the right-hand end and we’ll call the force 𝐹, the extension of the spring also increases. So we get the spring going from its natural length to now an extended length, where this length is its extension.

Now, we’ve been asked to find the spring constant. So we need an equation that relates the force applied to the spring, the extension of the spring, and the spring constant. The equation that we’re looking for is known as Hooke’s law. Hooke’s Law tells us that the force applied to the spring is equal to the spring constant of the spring multiplied by the extension of the spring.

Now, if we’re trying to find the spring constant, then we need to rearrange the equation. We do this by dividing both sides by the extension 𝑥. This way, the extension on the right-hand side cancels. And we’re just left with the spring constant. In other words, 𝐹 divided by 𝑥 is equal to 𝑘. Now, by definition, the spring constant is a constant. So if we want to find out its value, then we can choose any one of these points on the graph. We can choose whatever we want. It doesn’t matter which one we pick. And let’s say we’ve chosen this second point here.

We need to work out, first of all, the force exerted at that point. So, in this case, it’s 30 newtons. And we need to work out the extension to the spring caused by that force, in this case, 0.1 meters. And so, we can say that when the force applied to the spring is 30 newtons, the extension of the spring is 0.1 meters. And at this point, we can just sub in those values to our equation here. So we say that 𝑘 is equal to the force applied to the spring divided by the extension of the spring caused by that force. And before we evaluate the fraction, it’s important to notice that the units of 𝑘 are going to be newtons per meter.

Now, when we do evaluate the fraction, we find that the value of 𝑘 is 300 newtons per meter. And it’s important to know that we would’ve found this value regardless of which point on the graph we had chosen. For example, let’s say we’d picked this point. Well, the force applied at this point is 120 newtons. That’s this value here. And the extension caused by this force is 0.4 meters. So now, we can say that when 𝐹, the force applied, is 120 newtons, the extension, 𝑥, is 0.4 meters. Then, we could sub it into our equation for 𝑘, as we did earlier. And once again, we would find 300 newtons per meter as the value for 𝑘. And so, we have a final answer. The spring constant is 300 newtons per meter.

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