## Potential Energy of a System

8.1 Potential Energy of a System

### Learning Objectives

By the end of this section, you will be able to:

- 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

### Gravitational Potential Energy

The work done on an object by the constant gravitational force, near the surface of Earth, over any displacement is a function only of the difference in the positions of the end-points of the displacement.

You can justify this statement by answering the following question:

Practice 8.1.1 |
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A 1-kg object is moved part-way around a square loop as shown. The square is 1 m on a side. The final position is 0.5 m lower than where it started. How much work has gravity done on the object during its journey? (Use g ≈ 10 m/s^{2}) |

(a) +5 J |

(b) -5 J |

(c) +10 J |

(d) -10 J |

(e) 0 J |

Again, consider the work done by the force of gravity in this problem. Use the work-energy theorem: to answer the question.

Practice 8.1.2 |
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Three objects of mass m begin at height h with zero velocity. One falls straight down, one slides down a frictionless inclined plane, and one swings on the end of a pendulum. What is the relationship between their velocities when they have fallen to a final height of zero? |

(a) v_{1} > v_{2} > v_{3} |

(b) v_{1} > v_{3} > v_{2} |

(c) v_{1} = v_{2} = v_{3} |

**Discuss!**

Consider how you would answer these questions. Then bring this to class for a group discussion.

A good, professional baseball pitcher throws a ball straight up in the air. * Using the Work-Energy theorem*, estimate how high the ball will go. (A good throw can reach 90 mph.)

The force of gravity is a special type of force called a conservative force. This is something we will discuss in the next section of the module. Without going into details about conservative forces just yet, I will give you a definition for potential energy:

The change in potential energy (*U*) of an object is equal to the negative of the work (*W*_{cons}) done by conservative forces.