## Conservation of Energy

8.3 Conservation of Energy

### Learning Objectives

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

- 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

### Generalized Work-Energy Theorem

**Discuss!**

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

A 12.5 kg block is dragged over a rough, horizontal surface by a 65.9 N force acting at 21° above the horizontal. The block is displaced 5.7 m, and the coefficient of kinetic friction is 0.180.

(a) Find the work done on the block by the 65.9 N force.

(b) Find the work done on the block by the normal force.

(c) Find the work done on the block by the gravitational force.

(d) What is the increase in internal energy of the block-surface system due to friction?

(e) Find the total change in the block’s kinetic energy.

Pause & Predict 8.3.1 |
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How high would this hill need to be for the car to reach the top and stop? |

(a) 2.6 m |

(b) 6.2 m |

(c) 20.2 m |

(d) 14.6 m |

(e) 12.3 m |

Pause & Predict 8.3.2 |
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How long should the rough section of track be to bring the car to a stop? |

(a) 46 m |

(b) 43 m |

(c) 9.4 m |

(d) 3.8 m |

(e) 13 m |

**Practice!**

Practice 8.3.1 |
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You are riding on a roller coaster that starts from rest at a height of 25.0 m and moves down a frictionless track to a height of 3.00 m. How fast are you moving when you arrive at the 3.00-m height? |

(a) 23.4 m/s |

(b) 22.1 m/s |

(c) 20.8 m/s |

(d) 14.7 m/s |

The work-energy theorem states that the work done by non-conservative forces is equal to the change in kinetic energy plus the change in potential energy: In this problem (above), the only forces acting on the car are the force of gravity and the normal force of the track on the car. The force of gravity is a conservative force and the normal force is a non-conservative force. However, the normal force does no work since the force is perpendicular to the car’s displacement everywhere on the track. This means and mechanical energy is conserved.

Practice 8.3.2 |
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You are riding on a roller coaster that starts from rest at a height of 25.0 m and moves along a frictionless track. How fast are you moving when you reach the top of the loop, at point B? |

(a) 20.3 m/s |

(b) 26.9 m/s |

(c) 13.3 m/s |

(d) 16.0 m/s |

Practice 8.3.3 |
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You are riding on a roller coaster that starts from rest at a height of 25.0 m and moves along a frictionless track. However, after a bad storm some leaves settled on part of the track causing a 9.4-m length of the track to exert a frictional force of 625 N on the car. To safely make it around the loop, the 50-kg car must have a minimum speed of 7.7 m/s at the top of the loop (point B). How fast should the car be moving initially at point A to ensure that it reaches the top of the loop with the minimum required speed? |

(a) 107 m/s |

(b) 6.28 m/s |

(c) 28.0 m/s |

(d) 17.7 m/s |

(e) 8.90 m/s |

The work-energy theorem states that the work done by non-conservative forces is equal to the change in kinetic energy plus the change in potential energy: . In this problem there are three forces acting on the car: the force of gravity, which is a conservative force, the normal force of the track on the car, and the frictional force of the rough section of the track on the car. The normal force does no work since the force is perpendicular to the car’s displacement everywhere on the track. But the frictional force **does** do work on the car while the car is moving along the 9.4-m length of track that is covered in leaves. The work done by the frictional force while the car moves along the rough length (*d*) of the track is . The work is negative because the frictional force is in the opposite direction to the car’s displacement.

**Discuss!**

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

A block slides along a track from one level to a higher level, by moving through an intermediate valley. The track is frictionless until the block reaches the higher level. There a frictional force stops the block in a distance *d*. The block’s initial speed *v*_{0} is 6.0 m/s, the height difference *h* is 1.3 m, and the coefficient of kinetic friction µ_{k} is 0.60. Find *d*.

A block of mass 1.1 kg is attached to a horizontal spring that has a force constant of 2.5 × 10^{3} N/m as shown in the figure. The spring is compressed 3.7 cm and is then released from rest. Calculate the speed of the block as it passes through the equilibrium position if a constant friction force of 45 N retards its motion from the moment it is released.