# Spring Potential Energy

by Jared Rovny

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00:01 We have one last thing to discuss with regard to the potential energy and that is that energy can be stored in more than one sort of place.

00:08 We talked about the gravitational potential energy which is to say that if you lift something in the air, it has a certain energy associated with it that is sort of stored because I could let go of the object and it would fall and return to a kinetic energy but we could also store energy in springs.

00:24 So, if I have a spring, for example this one.

00:27 It's always got what we call an equilibrium location.

00:30 A certain length at which it's not going to push or pull.

00:33 It's the happy location for the spring.

00:35 If I pulled the spring, it would to stretch and want to pull in.

00:39 If I compress the spring, it's going to push out to try to get back to that equilibrium position.

00:44 So let's just walk through.

00:45 At the very top of this graph, have this spring compressed and so the red arrow shows that there is a force outwards to try to get back to our equilibrium position.

00:55 And if I stretch the spring as I have on the bottom, the spring is going to try to pull back and it's going to try to, again bring it back into the equilibrium.

01:03 When this is the case, when we have some equilibrium position which we usually call x-naught, the x of zero, as I've written it on the top left here.

01:10 Then the expression for the potential energy stored in the spring is one-half times k times the distance you are from your equilibrium x minus x-naught squared.

01:22 This k term tells us basically how strong our spring is.

01:24 So we can think of, first of all, x zero, as the, as what we call sort of the happy spot, the place where its equilibrium is not pushing or pulling.

01:32 And we often call that, that happy location, the equilibrium location, x equals zero, just by defining a coordinate system in which x zero equals zero and again you can always define your own coordinate system.

01:43 If we do this, if we call x-naught equal to zero, then our spring expression simplifies a little bit which it says that the potential energy stored in the spring is equal to 1/2 times k times the distance you are from zero squared.

01:56 Again, this k that I mentioned is the spring constant.

02:00 This tells us how stiff our spring is, how strong it is, whether it's very weak like a slinky or whether it's very strong like a more industrial type of spring something very thick and very intuitively, you can sort of think that a small value of k would be a very weak spring.

02:14 It's very easy to move in and out and not much energy that you would store there.

02:18 The very big k, a very high value of k for a spring, the high value of the spring constant would be for a very thick or strong spring that doesn't really want to compress or stretch and it will fight very hard if we try to pull it or push it.

The lecture Spring Potential Energy by Jared Rovny is from the course Energy of Point Object Systems.

### Included Quiz Questions

1. The spring with the greater spring constant.
2. The spring with the smaller spring constant.
3. They must be the same.
4. This will depend on the actual spring constant.
5. This will depend on the chosen equilibrium position x0.
1. By a factor of 9
2. By a factor of 6
3. By a factor of 3
4. This will depend on the spring constant
5. This will depend on the equilibrium location

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