Video: Calculating the Energy Stored in a Spring from the Spring's Constant and Extension

A spring with a constant of 80 N/m is extended by 1.5 m. How much energy is stored in the extended spring?

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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.

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