### Video Transcript

A spring with a constant of 80
newtons per meter is extended by 1.5 meters. How much energy is stored in the
extended spring?

In this example, we start out with
this spring. Let’s say this is it right
here. And we’re told that our spring has
a spring constant. We’ll symbolize it with a lowercase
𝑘 of 80 newtons per meter. This means that if we want to
extend or compress the spring by a distance of one meter, we would need to apply a
force of 80 newtons. But at the moment, our spring isn’t
extended or compressed. We can say that it’s at its natural
or its equilibrium length. But we’re told that it doesn’t stay
that way. Instead, the spring is extended
from this original length. And that extension, we can call it
𝑥, is given as 1.5 meters. Based on all this, we want to know
how much energy is stored in the extended spring.

Now, the reason there’s energy at
all is because the spring wants to return to its natural length. It has a native capacity to do
that. In doing so, this currently
extended spring is capable of doing work. And that amount of work it can do
is equal to how much energy is stored in it while it’s extended. When we consider what type of
energy this is then, we know that it’s potential energy. The energy isn’t manifest now. But it will be if the spring is
released from its extended length. And we also know the energy is
elastic potential energy. That’s because it has its source in
the compression or, in our case, extension of an elastic body away from
equilibrium.

To solve for this amount of energy
then, we can recall a mathematical relationship describing elastic potential
energy. An object’s elastic potential
energy is equal to one-half its spring constant multiplied by its displacement from
equilibrium squared. As we solve for the elastic
potential energy stored in our spring. The great thing is that we’re given
𝑘, the spring constant, as well as the spring’s extension. So the elastic potential energy
stored in this extended spring is equal to one-half the spring constant, 80 newtons
per meter, multiplied by the displacement from equilibrium, 1.5 meters squared. When we calculate the right-hand
side of this expression, we find a result of 90 newton meters. But at this point, we can recall
that one newton, the base unit of force, multiplied by a meter, the base unit of
distance, is equal to a joule, the base unit of energy. So our final answer is 90
joules. That’s how much energy is stored in
this extended spring.