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.
