# Potential Engery as a Function of X

1. Nov 5, 2006

### muna580

I am doing my homewrok which relates to Work/Engery/Power stuff. Well, I came along this question and I didn't understand how to do it. I went to my book, and the book had the exact example of this problem. The book said to take the derivitive of this function and set it equal to 0 and solve for x. I did that and put in the answern and I got the answr right.

As a good student, wanting to learn and understand phsyics, I wanted to undertand why I would take the derivitive of this function and set it equal to 0. Can someone please explain to me, but I don't really undertand how to do this problem.

So far, I know the following things

$$\Sigma W = - \Delta U$$
$$\Sigma W = \Delta K$$

$$\Delta K = \frac{1}{2}mv^2 - \frac{1}{2}mvo^2$$

[Broken]

Last edited by a moderator: Apr 22, 2017 at 1:50 PM
2. Nov 5, 2006

You may want to put similar questions in the same thread in the future.

Well, your book should contain the fact (and the explanation of it) that potential energy has a stationary value for the state of equilibrium.

3. Nov 5, 2006

### OlderDan

Some forces are called conservative forces because they have the property that if you move an object that is acted on by such a force, and then return it to its original position, the force that was acting does no net work. An example of such a force is gravity. The force a spring exerts on an object is another. For all such forces, you can calculate the potential energy associated with the force by calculatiing the work done against the force to change the position of the object. For gravity, the force is -mg (downward). To raise an object from y = 0 to y = h requires that you do work against gravity of F*h = mgh. This calculation is really a very simple integral starting with

dW = Fdy

Integrating gives

W = Fy = mgy because the force is constant.

A derivative is the inverse operation to an integral. In fact, the indefinite integral is often called an antiderivative. If potential energy is found by integrating a force over a distance, it follows that if you know the potential energy you can find the force related to that potential energy by taking the derivative of the potential energy with respect to distance. For gravity F = -d(mgy)/dy = -mg(dy/dy) = -mg. The - sign comes in because potential energy is the work done against the force related to the potential energy, which is the opposite of the force itself.

Any point where the derivative of the potential energy is zero is a point where the force related to that potential energy is zero. It is an equilibrium point. If the derivative at nearby points (or the second dervivative) indicate that the point is a minimum in the potential energy, then the force is toward the equilibrium point and the equilibrium is stable. If the object is given a small displacement it is attracted back toward the equilibrium point. If the derivatives nearby indicate that the point is a maximum in the potential energy, the equilibrium is unstable and a slight displacement will cause the object to be pushed away.

Last edited: Nov 5, 2006