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The graph shows the extension of a spring as the force applied to it changes.
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What is the spring constant?
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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.
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So let’s imagine that this is our spring.
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Now, as we would expect, when the force applied to the spring is zero newtons, the extension of the spring is also zero newtons.
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Because the spring is simply sitting at its natural length when there is no force applied to it.
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It’s not gonna be extended if there’s no force on it.
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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.
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So we get the spring going from its natural length to now an extended length, where this length is its extension.
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Now, we’ve been asked to find the spring constant.
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So we need an equation that relates the force applied to the spring, the extension of the spring, and the spring constant.
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The equation that we’re looking for is known as Hooke’s law.
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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.
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Now, if we’re trying to find the spring constant, then we need to rearrange the equation.
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We do this by dividing both sides by the extension 𝑥.
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This way, the extension on the right-hand side cancels.
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And we’re just left with the spring constant.
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In other words, 𝐹 divided by 𝑥 is equal to 𝑘.
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Now, by definition, the spring constant is a constant.
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So if we want to find out its value, then we can choose any one of these points on the graph.
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We can choose whatever we want.
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It doesn’t matter which one we pick.
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And let’s say we’ve chosen this second point here.
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We need to work out, first of all, the force exerted at that point.
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So, in this case, it’s 30 newtons.
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And we need to work out the extension to the spring caused by that force, in this case, 0.1 meters.
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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.
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And at this point, we can just sub in those values to our equation here.
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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.
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And before we evaluate the fraction, it’s important to notice that the units of 𝑘 are going to be newtons per meter.
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Now, when we do evaluate the fraction, we find that the value of 𝑘 is 300 newtons per meter.
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And it’s important to know that we would’ve found this value regardless of which point on the graph we had chosen.
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For example, let’s say we’d picked this point.
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Well, the force applied at this point is 120 newtons.
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That’s this value here.
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And the extension caused by this force is 0.4 meters.
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So now, we can say that when 𝐹, the force applied, is 120 newtons, the extension, 𝑥, is 0.4 meters.
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Then, we could sub it into our equation for 𝑘, as we did earlier.
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And once again, we would find 300 newtons per meter as the value for 𝑘.
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And so, we have a final answer.
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The spring constant is 300 newtons per meter.