## Conservative and Non-Conservative Forces

8.2 Conservative and Non-Conservative Forces

### Learning Objectives

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

- 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

### Conservative Forces

The work done by a conservative force is independent of the path; in other words, the work done by a conservative force is the same for any path connecting two points:

The work done by a non-conservative force depends on the path taken.

Equivalently, a force is conservative if the work it does around any closed path is zero:

**Discuss!**

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

A single conservative force acts on a 4.80-kg particle within a system due to its interaction with the rest of the system. The equation *F _{x}* = 2

*x*+ 4 describes the force, where

*F*is in newtons and

_{x}*x*is in meters. As the particle moves along the

*x*axis from

*x*= 1.00 m to

*x*= 4.51 m, calculate the following.

(a) the work done by this force on the particle

(b) the change in the potential energy of the system

(c) the kinetic energy the particle has at *x* = 4.51 m if its speed is 3.00 m/s at *x* = 1.00 m

### Gravitational Potential Energy and Elastic Potential Energy

**Discuss!**

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

A hockey puck slides without friction along a frozen lake toward an ice ramp and plateau as shown. The speed of the puck is 4 m/s and the height of the plateau is 1 m. Will the puck make it all the way up the ramp?

**Practice!**

Practice 8.2.1 |
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An athlete jumping vertically on a trampoline leaves the surface with a velocity of 8.5 m/s upward. What maximum height does she reach? |

(a) 13 m |

(b) 2.3 m |

(c) 3.7 m |

(d) 0.27 m |

(e) The answer can’t be determined because the mass of the athlete isn’t given. |

Practice 8.2.2 |
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A 2.0-kg block sliding on a horizontal frictionless surface is attached to one end of a horizontal spring (k = 200 N/m) which has its other end fixed. If the block has a speed of 4.0 m/s as it passes through the equilibrium position, what is its speed when it is 20 cm from the equilibrium position? |

(a) 2.6 m/s |

(b) 3.1 m/s |

(c) 3.5 m/s |

(d) 1.9 m/s |

(e) 2.3 m/s |

Practice 8.2.3 |
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All springs and masses shown are identical. (Gravity acts downward). Which of the systems below has the most potential energy stored in its spring(s), relative to the relaxed position? |

(a) system 1 |

(b) system 2 |

(c) same |

Practice 8.2.4 |
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A spring (k = 200 N/m) is suspended with its upper end supported from a ceiling. With the spring hanging in its equilibrium configuration, an object (mass = 2.0 kg) is attached to the lower end and released from rest. What is the speed of the object after it has fallen 4.0 cm? |

(a) 90 cm/s |

(b) 79 cm/s |

(c) 96 cm/s |

(d) 83 cm/s |

(e) 57 cm/s |

### Force = Gradient of Potential Energy

**Practice!**

Practice 8.2.5 |
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The graph shows a conservative force F _{x} as a function of x in the vicinity of x = a. As the graph shows, F_{x} = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? |

(a) U = 0 at x = a |

(b) U is a maximum at x = a. |

(c) U is a minimum at x = a. |

(d) U is neither a minimum nor a maximum at x = a, and its value at x = a need not be zero. |

(e) Not enough information is given to decide. |

Practice 8.2.6 |
---|

The graph shows a conservative force F _{x} as a function of x in the vicinity of x = a. As the graph shows, F_{x} = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? |

(a) U = 0 at x = a |

(b) U is a maximum at x = a. |

(c) U is a minimum at x = a. |

(d) U is neither a minimum nor a maximum at x = a, and its value at x = a need not be zero. |

(e) Not enough information is given to decide. |

Practice 8.2.7 |
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The graph shows a conservative force F _{x} as a function of x in the vicinity of x = a. As the graph shows, F_{x} > 0 and dF_{x}/dx < 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? |

(a) dU/dx > 0 at x = a |

(b) dU/dx < 0 at x = a |

(c) dU/dx = 0 at x = a |

(d) Any of the above could be correct. |

Practice 8.2.8 |
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The graph shows the potential energy U for a particle that moves along the x-axis. The only force that acts on the particle is the force associated with U. The particle is initially at x = d and moves in the negative x-direction. At which of the labeled x-coordinates does the particle have the greatest speed? |

(a) x = a |

(b) x = b |

(c) x = c |

(d) x = d |

(e) more than one of the above |

Practice 8.2.9 |
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The graph shows the potential energy U for a particle that moves along the x-axis. The only force that acts on the particle is the force associated with U. The particle is initially at x = d and moves in the negative x-direction. At which of the labeled x-coordinates is the particle slowing down? |

(a) x = a |

(b) x = b |

(c) x = c |

(d) x = d |

(e) more than one of the above |

Practice 8.2.10 |
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The graph shows the potential energy U for a particle that moves along the x-axis. The only force that acts on the particle is the force associated with U. At which of the labeled x-coordinates is there zero force on the particle? |

(a) at x = a and x = c |

(b) at x = b only |

(c) at x = d only |

(d) at x = b and x = d |

(e) Misleading question—there is a force at all values of x |