### 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.