## The Work-Energy Theorem

7.3 The Work-Energy Theorem

### Learning Objectives

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

- Apply the work-energy theorem to find information about the motion of a particle, given the forces acting on it
- Use the work-energy theorem to find information about the forces acting on a particle, given information about its motion

### The Work-Energy Theorem

The net work done on a particle equals the change in the particle’s kinetic energy:

#### Problem-Solving Strategy: Work-Energy Theorem

- Draw a free-body diagram for each force on the object.
- Determine whether or not each force does work over the displacement in the diagram. Be sure to keep any positive or negative signs in the work done.
- Add up the total amount of work done by each force.
- Set this total work equal to the change in kinetic energy and solve for any unknown parameter.
- Check your answers. If the object is traveling at a constant speed or zero acceleration, the total work done should be zero and match the change in kinetic energy. If the total work is positive, the object must have sped up or increased kinetic energy. If the total work is negative, the object must have slowed down or decreased kinetic energy.

**Discuss!**

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

(a) A 5.33 kg block initially at rest is pulled to the right along a horizontal, frictionless surface by a constant horizontal force of 8.0 N. Find the block’s speed after it has moved 7.9 m.

(b) Suppose that the magnitude of the force is doubled. Through what displacement will the block travel before reaching the same final speed found above?

Pause & Predict 7.3.1 |
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If you push a 20 N block 1.5 m up a smooth incline with an 11 N force that is parallel to the incline, what is the net work done on the block? |

(a) 17 J |

(b) 30 J |

(c) 8.5 J |

(d) 11 J |

(e) 3.8 J |

Pause & Predict 7.3.2 |
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You apply an 11-N force to push a 20-N block up a rough incline, while a 2-N frictional force acts to impede the motion. If the block started from rest, what is its speed after moving 1.5 m up the incline? |

(a) 3.7 m/s |

(b) 0.9 m/s |

(c) 2.6 m/s |

(d) 0.5 m/s |

(e) 1.9 m/s |

**Practice!**

Practice 7.3.1 |
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On a snowy day, Max (mass = 15 kg) pulls his little sister Maya (mass = 12 kg) in a sled (mass = 8 kg) through the slippery snow. When Max pulls on the sled with 12 N of force, directed at an angle of 15° above the ground, how much work does Max do on the sled as he pulls his sister 25 m in the snow? |

(a) 300 J |

(b) 11.6 J |

(c) 290 J |

(d) 77.6 J |

Practice 7.3.2 |
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On a snowy day, Max (mass = 15 kg) pulls his little sister Maya (mass = 12 kg) in a sled (mass = 8 kg) through the slippery snow. Max pulls on the sled with 12 N of force, directed at an angle of 15° above the ground. When he comes to a recently plowed section of road, he continues to pull the sled with the same force across the road while the road exerts a frictional force of 4 N on sled. What is the net work done on the sled while Max pulls it 5 m across the road? |

(a) 38 J |

(b) 78 J |

(c) 7.6 J |

(d) 16 J |

(e) 58 J |

Practice 7.3.3 |
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On a snowy day, Max (mass = 15 kg) pulls his little sister Maya (mass = 12 kg) in a sled (mass = 8 kg) through the slippery snow. Max pulls on the sled with 12 N of force, directed at an angle of 15° above the ground. When he comes to a recently plowed section of road, he continues to pull the sled with the same force across the road while the road exerts a frictional force of 4 N on sled. If the sled’s speed is 1.5 m/s as Max pulls it onto the road, what is the sled’s speed after Max pulls it 5 m across the road? |

(a) 2.0 m/s |

(b) 2.5 m/s |

(c) 6.0 m/s |

(d) 2.8 m/s |

(e) 3.2 m/s |

**Practice!**

Practice 7.3.4 |
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Two blocks have masses m _{1} and m_{2}, where m_{1} > m_{2}. They are sliding on a frictionless floor and have the same kinetic energy when they encounter a long rough stretch (i.e. µ > 0) which slows them down to a stop. Which one will go farther before stopping? |

(a) m_{1} |

(b) m_{2} |

(c) they will go the same distance |

Practice 7.3.5 |
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How much distance do you need to come to a complete stop when you slam on the brakes while first going at 90 km/h compared to 45 km/h? |

(a) Half the distance |

(b) The same distance |

(c) Twice the distance |

(d) Four times the distance |

(e) One fourth the distance |

Practice 7.3.6 |
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An 800 kg car experiences a positive force from the road that propels the car forward, as shown in the graph. While traveling the full 900 meters, the car’s speed: |

(a) first increases and then decreases. |

(b) first decreases and then increases. |

(c) continuously increases. |

(d) drops to zero at 900 m. |

(e) None of the above. |

Practice 7.3.7 |
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An 800 kg car initially at rest at 0 meters subsequently experiences a positive static frictional force from the road that propels the car forward. At 900 meters, the car’s speed is: |

(a) 17 m/s. |

(b) 24 m/s. |

(c) 34 m/s. |

(d) 38 m/s. |

(e) None of the above. |

**Discuss!**

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

A 1-kg particle moving along the x-axis experiences the force shown in the graph. If the particle’s speed is 2 m/s at x = 0 m, what is its speed when it gets to x = 5 m?