Definition of Potential Energy

Potential energy is the capacity of an object to perform work because of the position of the object in space the forces acting on the object. In this section, the mechanical potential energies are study. The mechanic potential energy study here are of two kinds, gravitational potential energy, and elastic potential energy.

 For example, when considering the relative position of an object with respect to the floor or a table, the object can do work on any object placed below. Such an example is the figurative case of having a stone (or physics book) above a glass table hanging from a string. If the string is cut off, the stone will fall on top of the glass and, most likely, it will crack the glass. Another example, it is to drive a nail with a sledge hammer. In this case, the proper use by the operator of the sledge hammer is to make an effort only when lifting the hammer above the nail and then, to let gravity pull down on the hammer over the nail. Since the nail is displaced inside the wood by the force applied by the sledge hammer, clearly, work is being done by the hammer. This ability of the hammer to do work is associated to the position of the hammer above the nail. Thus, it is said that the hammer has gravitational potential energy.

Intuitively, how much work  can the sledge hammer do in a single strike depends on how height the hammer is lifted and how heavy is the sledge hammer (assuming that you do not miss the nail). As discussed in the section of work, when the operator lift the hammer of mass a given height , the operator does an overall work given by . This work is just the negative of the work done by gravity while the operator is lifting the sledge hammer. Therefore, a proper definition of potential energy can be obtained from the previous analysis as shown below.

Suppose that there is a box of mass placed at a height with respect to an arbitrary level, , and that this box is lifted to a different level, , by an external force, . In such a case, the external force as well as the gravitational force do work on the box. Since the box is a rest before the external force is applied and also it is at rest when the object ends at the level , the net force on the object is zero, , implying that the work done by the external force and the work done by the gravitational force are the negative of each other, . As discussed in the section about the work done by different forces, the actual external force is not constant during the processes of lifting the box; however, near the surface of the Earth, the force of gravity remains constant at all times, in particular, when lifting the object.  Thus, even when the acquire ability to do work by the box is due to its change in position, this is a consequence of lifting the object by the external force. Nevertheless, the work done by gravity is a more understandable quantity (a constant) from where to define the mechanical potential energy acquire (change) by the box when moved to a different position. In order to calculate this work, the initial location, , and final location, , of the box are important, . In fact, considering that a horizontal displacement of the box does not represent actual work done by the gravitational  force, the work done by gravity is only related to the change in the vertical displacement of the box, , Therefore, the work done by gravity is just, where the negative sign is a consequence of the force of gravity being opposite to the vertical direction of the box displacement.

In general, the definition of the change in potential energy of an object when moved from the initial position, , to the final position, , near the surface of the Earth is given by

 Change in Potential Energy with

Thus, in the present case, the corresponding change in potential energy is .

The change in gravitational potential energy can be written as where is the gravitational potential energy of the object (the box in this case) at the initial position and is the final potential energy of the object.

 In the sequence of drawings at the left, a teacher is holding an apple at about shoulder level. The drawing shows her on the second floor of the campus, . What is the gravitational potential energy of the apple?In order to respond to this question, it is necessary to know with respect to which level should the gravitational potential energy by given. In fact, in order to define the potential energy, it is necessary to select a level, typically called the ground, where the gravitational potential energy is, by definition, zero. That is, at ground level the gravitational potential energy is zero, . This level can be selected at any arbitrary horizontal plane such as, the top of the second floor desk, ; in which case, everything above the surface of the desk would have a positive gravitational energy while everything below the desk would have a negative gravitational potential energy, . Other possible ground levels are the floor of the classroom on the second floor, , or the floor on the first floor of the drawing, . These are a few of the infinite number of possible elections for the ground level of gravitational potential energy. Clearly, the gravitational potential energy of the apple is different dependent of where the zero level of gravitational potential energy is selected. In other words, just as in the case of the kinetic energy, the gravitational potential energy depends of the frame of reference. Thus, in response to the question, in order to state the gravitational potential energy of the apple it is necessary to select the ground level of the gravitational potential energy and; then, based on the mass of the apple and the position of the apple with respect to the selected ground level, the gravitational potential energy of the apple can be calculated, .

 Scale, from bottom to top of drawing, 400 m. Mass of the box, 100 Kg. The drawing on the left allows to relocate the zero level of the gravitational potential energy, , as well as the position of the box, of mass , the bar on the right of the drawing shows the amount of gravitational potential energy of the box with respect to the selected ground level of gravitational potential energy. A red bar indicates negative gravitational potential energy while a blue bar indicates a positive gravitational energy. Click at least once on the line representing the ground level of the potential energy to activate. Place the mouse pointer over the box or line representing the ground level to receive further instructions about operating this drawing. When the mouse pointer is over the background, the first coordinate represents the coordinate of the box with respect to the ground level. This coordinate is given in meters accordingly with the scale given above. The second coordinate provides the gravitational potential energy of the block with respect to the ground level, , if the box was to be placed at that location of the mouse pointer. The mass of the box is set to 100 Kg.

As represented in the previous simulation, after the ground level of potential energy has been set, the gravitational potential energy of an object is given by

 Gravitational Potential Energy where is measured with respect to the ground level of potential energy.

The concept of field in physics is associated to the following idea, suppose that

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com