The work done by a conservative force is independent of the path; in other words, the work done by a conservative force is the same for any path connecting two points:. Equivalently, a force is conservative if the work it does around any closed path is zero:.
The work done going along a path from B to A is the negative of the work done going along the same path from A to B , where A and B are any two points on the closed path:. You might ask how we go about proving whether or not a force is conservative, since the definitions involve any and all paths from A to B , or any and all closed paths, but to do the integral for the work, you have to choose a particular path.
There are mathematical conditions that you can use to test whether the infinitesimal work done by a force is an exact differential, and the force is conservative. These conditions only involve differentiation and are thus relatively easy to apply.
You may recall that the work done by the force in Figure depended on the path. For that force,. Can you see what you could change to make it a conservative force?
Figure 8. Which of the following two-dimensional forces are conservative and which are not? Assume a and b are constants with appropriate units:. Apply the condition stated in Figure , namely, using the derivatives of the components of each force indicated.
If the derivative of the y -component of the force with respect to x is equal to the derivative of the x -component of the force with respect to y , the force is a conservative force, which means the path taken for potential energy or work calculations always yields the same results. The conditions in Figure are derivatives as functions of a single variable; in three dimensions, similar conditions exist that involve more derivatives.
So there is always a conservative force associated with every potential energy. We have seen that potential energy is defined in relation to the work done by conservative forces. That relation, Figure , involved an integral for the work; starting with the force and displacement, you integrated to get the work and the change in potential energy.
However, integration is the inverse operation of differentiation; you could equally well have started with the potential energy and taken its derivative, with respect to displacement, to get the force. The infinitesimal increment of potential energy is the dot product of the force and the infinitesimal displacement,. We also expressed the dot product in terms of the magnitude of the infinitesimal displacement and the component of the force in its direction. Both these quantities are scalars, so you can divide by dl to get.
It is important to note that non-conservative forces do not destroy energy they just convert it into a less useful less ordered form. It explores how friction turns macroscopic motion into microscopic. To learn more about conservative and non-conservative forces, please see hyperphysics. Fossil Fuels. Nuclear Fuels. Acid Rain. Intuitively, you might see how this definition makes sense.
Think about a situation where you are in some position where you have a certain amount of potential energy. Your position is obviously going to change, and so does your potential energy too as you get pulled closer towards Earth. This means that the force acting on you is connected to the change of your position and potential energy , which makes perfect sense. Equivalently, potential energy can be defined by simply adding up all of the conservative forces acting on an object at each point during a certain path.
Mathematically this means that the total potential energy is the integral i. This is also easy to figure out from the definition of a conservative force by simply moving around the terms and integrating both sides:. These definitions can easily be used to find conservative forces and potential energy functions that match each other. If we know either the force or the potential energy, we can derive the corresponding force or potential for that case. Now, these equations work because gravitational forces are conservative forces , meaning they can be associated with a potential energy.
On the other hand, consider something like the viscous drag force , which is a force that acts on an object moving through a liquid of some sorts. The drag force is actually a velocity dependent force , which means that there is a greater force on an object moving at higher speed.
This, of course makes sense if you think about an object that moves through a fluid. The mathematical definition for the drag force is as follows:. All frictional forces are non-conservative forces, because they are not derived from a potential. Ultimately though, this definition comes from how these forces conserve energy. This idea has particular significance in Lagrangian mechanics, which relies on the notion of conservative forces.
I go into more detail about this concept in this article. Another key characteristic for conservative forces is that they conserve the mechanical energy of a system or an object. Mechanical energy simply means the total of kinetic and potential energy. Non-conservative forces, on the other hand, do not. In fact, this property of energy conservation is where the names of conservative and non-conservative forces come from.
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