 In this video, I want to talk about conservative forces and potential energy. Well, first of all, what are conservative forces? Well, if we have forces, we can kind of divide them up in two types of forces. One is conservative forces and one is non-conservative. Now, let me give you an example. Conservative forces that you know are gravity, the spring force, and one that you might not know yet is the electrostatic force, which will play quite a big role in your electromagnetism classes. A typical non-conservative force is friction. Now, what does a conservative force have to fulfill in order to be called conservative? Well, the first point is that the work needed to do something against the conservative force does not depend on the path taken. This means the work that I need to move an object against the conservative force does not matter how I move the object. For example, my cup here, if I move it from here to here, I need the exact same amount of energy than if you move it first here, then here, and then here. All that matters is where we are ending up. The other part is that the energy that is used to work against the conservative force can be completely recovered. This is what we actually mean by conservative. The energy is conserved, I can get it back. In the example of gravity, if I move my cup up, once I let go, the cup will by itself accelerate and all the potential energy I have up here is converted back into kinetic energy. Nothing is lost. Now, in the example of a non-conservative force, this is not the case. First of all, if you look at friction, the work I need to move my cup from here to here is not the same as if I go around the table once. This will need much more work to do so. Also, the energy cannot be recovered. Once I moved it, some energy is converted into heat, so I cannot completely recover it. My cup will not automatically go back to its original position. What are some consequences of a force being conservative? First of all, it can help us a lot by simplifying problems. For example, if you have a problem where a ball is launched from ground and follows some projectile motion to here and you're asked to calculate the work done by gravity, you do not need to kind of integrate around the path because, remember, gravity always points down. If we would go with our work equation, work is the integral of the force.ds, the path. The angle would change, so you would have to integrate step-by-step or you would have to split it up into several small pieces. This kind of work is path-independence. Instead of calculating along the real path that the project will follow, let's do something simple. Let's go from here to here first. I can use, in this case, work is force times distance times cosine of the angle in between. For the first part, I have zero work done because I have an angle of 90 degrees between my direction of travel and the direction of the force. Then I can do the work here because gravity will always be at 180 degrees, so I will get that it is minus, the work done is minus m g h, mass times gravity being the force times height being the distance times cosine of 180, giving me the minus sign. The fact that something is path-independent, you probably have already used that when you did your Kirchhoff's loops rule in high school. Remember, when you were having some kind of electric circuits with some batteries, some resistors inside, and you were saying the sum of our voltage going around from point A back all around the point A, that's the same if I was just staying at point A here in the corner, so the sum of the energies must be zero, and we're going to see in a minute how voltage is actually connected to energies, how the Kirchhoff's loop rule is a direct consequence of the fact that it is path-independent. Apart from being able to simplify a lot of physics problems, because it is path-independent, we can also do another trick. We can assign potential energies to conservative forces. The definition for potential energy is that the change in potential energy is equal to minus the work done by the conservative force. Sometimes in a more international setting, the letter U is also used, so this is the same as U, a more international way of saying potential energy. Now, in our example from before, when I was lifting something from here to here against gravity, I had that the work done by gravity from A to B is minus MGH, and if you look at this, that means that my potential energy final is equal to minus my potential energy initial, is equal to minus MGH, minus minus gives me plus, so I have potential energy final is potential energy initial, plus MGH, something that you already have used. So instead of using the work done by conservative force, you can simply consider their potential energies for which we have quite simple equations. We have the one for gravity, we have the one for spring force, and you're going to learn the one for an electric force as well. Now, there's one thing that you have to be very careful about when you use conservation of energy in mechanics. If you do conservation of mechanical energy, we had the standard setup of energy final plus energy initial plus work done by any external forces, which then for mechanics we could split up in potential energy final plus kinetic energy final equals potential energy initial plus kinetic energy initial plus work done by any external force. Now, if you use this equation, if you use here and here the potential energy of gravity, then do not use the work done by gravity because remember the definition of potential energy was that the change in potential energy is minus the work done by the conservative force, which is attributed. So if you use the potential energy of gravity here, then do not use it in the work, otherwise you will count it twice. Now, if you are using with the spring potential energy, if you use the spring potential energy, which is the formula for the spring potential energy, by the way, P s is one half k x squared. Very simple, spring constant times the displacement squared gives you your potential energy stored in the spring. So if you use the potential energy of a spring, do not put the work done by the spring force or the elastic force in the same equation. One last note, I was saying that potential energies are related to voltages, and that is the potential energy of the electrostatic force divided by the charge of a particle Q is what we call a volt. So the voltage is defined as potential energy per unit of charge. So one volt is a true per coulomb.