# Work, Energy, and Energy Resources

## Gravitational Potential Energy


### Learning Objectives

By the end of this section, you will be able to:
1. Explain gravitational potential energy in terms of work done against gravity.
2. Show that the gravitational potential energy of an object of mass  at height  on Earth is given by .
3. Show how knowledge of the potential energy as a function of position can be used to simplify calculations and explain physical phenomena.

### Work Done Against Gravity

Climbing stairs and lifting objects is work in both the scientific and everyday sense—it is work done against the gravitational force. When there is work, there is a transformation of energy. The work done against the gravitational force goes into an important form of stored energy that we will explore in this section.
Let us calculate the work done in lifting an object of mass  through a height , such as in , near Earth's surface. If the object is lifted straight up at constant speed, then the force needed to lift it is equal to its weight . The work done on the mass is then . We define this to be the gravitational potential energy  put into (or gained by) the object-Earth system. This energy is associated with the state of separation between two objects that attract each other by the gravitational force. For convenience, we refer to this as the  gained by the object, recognizing that this is energy stored in the gravitational field of Earth. Why do we use the word “system”? Potential energy is a property of a system rather than of a single object—due to its physical position. An object’s gravitational potential is due to its position relative to the surroundings within the Earth-object system. The force applied to the object is an external force, from outside the system. When it does positive work it increases the gravitational potential energy of the system. Because gravitational potential energy depends on relative position, we need a reference level at which to set the potential energy equal to 0. We usually choose this point to be Earth’s surface, but this point is arbitrary; what is important is the difference in gravitational potential energy, because this difference is what relates to the work done. The difference in gravitational potential energy of an object (in the Earth-object system) between two rungs of a ladder will be the same for the first two rungs as for the last two rungs. 

### Converting Between Potential Energy and Kinetic Energy

Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. If we release the mass, gravitational force will do an amount of work equal to  on it, thereby increasing its kinetic energy by that same amount (by the work-energy theorem). We will find it more useful to consider just the conversion of  to  without explicitly considering the intermediate step of work. (See .) This shortcut makes it is easier to solve problems using energy (if possible) rather than explicitly using forces.
More precisely, we define the change in gravitational potential energy  to be
where, for simplicity, we denote the change in height by  rather than the usual . Note that  is positive when the final height is greater than the initial height, and vice versa. For example, if a 0.500-kg mass hung from a cuckoo clock is raised 1.00 m, then its change in gravitational potential energy is 
Note that the units of gravitational potential energy turn out to be joules, the same as for work and other forms of energy. As the clock runs, the mass is lowered. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, without directly considering the force of gravity that does the work.

### Using Potential Energy to Simplify Calculations

The equation  applies for any path that has a change in height of , not just when the mass is lifted straight up, as long as  is small compared to the radius of Earth. Note that, as we learned in Uniform Circular Motion and Gravitation, the force of Earth’s gravity does decrease with distance from Earth. The change is negligible for small changes in distance, but if you want to use potential energy in problems involving travel to the moon or even further, Newton’s universal law of gravity must be taken into account. This more complete treatment is beyond the scope of this text and is not necessary for the problems we consider here. (See .) It is much easier to calculate  (a simple multiplication) than it is to calculate the work done along a complicated path. The idea of gravitational potential energy has the double advantage that it is very broadly applicable and it makes calculations easier. From now on, we will consider that any change in vertical position  of a mass  is accompanied by a change in gravitational potential energy , and we will avoid the equivalent but more difficult task of calculating work done by or against the gravitational force.
We have seen that work done by or against the gravitational force depends only on the starting and ending points, and not on the path between, allowing us to define the simplifying concept of gravitational potential energy. We can do the same thing for a few other forces, and we will see that this leads to a formal definition of the law of conservation of energy.

### Test Prep for AP Courses


### Section Summary

1. Work done against gravity in lifting an object becomes potential energy of the object-Earth system.
2. The change in gravitational potential energy, , is , with  being the increase in height and  the acceleration due to gravity. 
3. The gravitational potential energy of an object near Earth’s surface is due to its position in the mass-Earth system. Only differences in gravitational potential energy, , have physical significance.
4. As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increasing speed, so that .

### Conceptual Questions


### Problems & Exercises
