8: Potential Energy and Conservation of Energy

*Shown here is part of a Ball Machine sculpture by George Rhoads. A ball in this contraption is lifted, rolls, falls, bounces, and collides with various objects, but throughout its travels, its kinetic energy changes in definite, predictable amounts, which depend on its position and the objects with which it interacts. (credit: modification of work by Roland Tanglao)*

In George Rhoads’ rolling ball sculpture, the principle of conservation of energy governs the changes in the ball’s kinetic energy and relates them to changes and transfers for other types of energy associated with the ball’s interactions. In this module, we introduce the important concept of potential energy. This will enable us to formulate the law of conservation of mechanical energy and to apply it to simple systems, making solving problems easier. In the final section on sources of energy, we will consider energy transfers and the general law of conservation of energy. Throughout this course, the law of conservation of energy will be applied in increasingly more detail, as you encounter more complex and varied systems, and other forms of energy.

The central idea of this module (and much of modern physics) is conservation of energy. Let’s begin by considering cases where there are only conservative forces acting. First, we will define what we mean by “**conservative force**“. Then we will relate the work done by a conservative force to the change in energy. This will lead us to a special kind of function called a **Potential Energy function**, represented by the letter *U*. All conservative forces have an associated potential energy function. In this course, we will only discuss two types of conservative forces: the force of gravity and the force in a spring, and therefore only two types of potential energy.

#### 8.1 Potential Energy of a System

- Relate the difference of potential energy to work done on a particle for a system without friction or air drag
- Explain the meaning of the zero of the potential energy function for a system
- Calculate and apply the gravitational potential energy for an object near Earth’s surface and the elastic potential energy of a mass-spring system

#### 8.2 Conservative and Non-Conservative Forces

- Characterize a conservative force in several different ways
- Specify mathematical conditions that must be satisfied by a conservative force and its components
- Relate the conservative force between particles of a system to the potential energy of the system
- Calculate the components of a conservative force in various cases

#### 8.3 Conservation of Energy

- Formulate the principle of conservation of mechanical energy, with or without the presence of non-conservative forces
- Use the conservation of mechanical energy to calculate various properties of simple systems