# The Relationship Between Work, Potential Energy, and Kinetic Energy

**INTRODUCTION**

This experiment was designed to investigate the relationship between work, potential energy, and kinetic energy. Applying equations learned in Physics class, it was possible to compare these to values measured by computer software. Examples used in the experiment were a cart, a spring, and a mass.

**Preliminary questions:**

1. **Lift a book from the floor to the table. Did you do work? To answer this question, consider whether you applied a force parallel to the displacement of the book.**

*Yes, I did work lifting the book from the floor to the table. My arms applied a force parallel to the displacement of the book.*

2. **What was the average force acting on the book as it was lifted? Could you lift the book with a constant force? Ignore the very beginning and end of the motion in answering the question.**

*The average force on the book was greater than 9.8 N. Also, it is possible to lift the book with constant force, and could be done easily with computer manipulation.*

3. **Holding one end still, stretch a rubber band. Did you do work on the rubber band? To answer this question, consider whether you applied a force parallel to the displacement of the moving end of the rubber band.**

*Yes, I did work on the rubber band because there was an applied force parallel to the displacement of the moving end of the rubber band.*

4. **Is the force you apply constant when you stretch the rubber band? If not, at what point in the stretch is the force the least. At what point is the force the greatest?**

*The force applied was not constant, the beginning of the stretch had the least applied force while the end of the stretch had the greatest applied force.*

**Data Table:**

Part I | ||

Time (s) | Position (m) | |

Start Moving |
1.120 | 0.444 |

Stop Moving |
3.100 | 0.840 |

Average force(N) |
1.927 |

Work done (J) |
0.763 |

Integral (during lift): force |
0.761 |

D |
0.784 |

Part II | ||

Time (s) | Position (m) | |

Start Pulling |
0.350 | 0.003 |

Stop Pulling |
4.20 | 0.292 |

Spring Constant (N/m) |
21.241 |

Stretch | |||

10 cm | 20 cm | Maximum | |

Integral (during pull) (N•m) |
0.140 | 0.480 | 0.989 |

D |
0.107 | 0.425 | 0.910 |

Mass (kg) |
0.133 |

Final velocity (m/s) |
0.534 |

Integral during push (N•m) |
0.074 |

D |
0.0190 |

**Analysis/QUESTIONS:**

1. **In Part I, the work you did lifting the mass did not change its kinetic energy. The work then had to change the potential energy of the mass. Calculate the increase in gravitational potential energy using the following equation. Compare this (use a % difference) to the average work for Part I, and to the area under the force vs. position graph: where **

**D**

*h***is the distance the mass was raised. Record your values in the data table. Does the work done on the mass correspond to the change in gravitational potential energy? Should it?**

*The work done on the mass corresponds to the change in gravitational potential energy. Using the above equation, ∆U _{g} = 0.784 J, while work done is equivalent to 0.763 J, a mere 2.8% difference. The area under the force vs. position graph is valued at 0.761, nearly the same difference at 3.02%.*

2. **If you varied the speed of your hand as you lifted the mass would the force vs. position graph change? Or will it continue to correspond to mgDh?**

*Varying the speed of the lifting hand doesn’t change the correlation between area in F vs position and mgDh, displacement will be the same in either case.*

3. **In Part II you did work to stretch the spring. The graph of force vs. position depends on the particular spring you used, but for most springs will be a straight line. This corresponds to Hooke’s law, or F = – kx, where F is the force applied by the spring when it is stretched a distance x. k is the spring constant, measured in N/m. What is the spring constant of the spring? From your graph, does the spring follow Hooke’s law? Do you think that it would always follow Hooke’s law, no matter how far you stretched it? Why is the slope of your graph positive, while Hooke’s law has a minus sign?**

*The spring constant is 21.24 J. From the graph, the spring follows Hooke’s law, albeit the positive slope. The spring will not always follow Hooke’s law, there is a point of extension where the spring will be represented as F _{tension}. The graph’s slope is positive because it measures the work done by our hand and not the spring. The work done by our hand was equal in magnitude and opposite in direction of the spring.*

4. **The elastic potential energy stored by a spring is given by ****D***PE ***= ½ kx**

**2**

**, where**

*x*is the distance. Compare the work you measured to stretch the spring to 10 cm, 20 cm, and the maximum stretch to the stored potential energy predicted by this expression. Should they be similar?*Note:*Use consistent units*.*Record your values in the data table.*The difference in work to stretch the spring (values of DPE = ½ kx2 and the integrated area) 10 cm, 20 cm, and 29.2 cm was 23.6%, 11.46%, and 7.99% respectively. The potential energy equation had closer values when the spring was stretched further. They should have similar values because the integration is approx. equal to 10x ^{2} (slope of graph was 21.241), or about ½ kx^{2}.*

5. **In Part III you did work to accelerate the cart. In this case the work went to changing the kinetic energy. Since no spring was involved and the cart moved along a level surface, there is no change in potential energy. How does the work you did compare to the change in kinetic energy? Here, since the initial velocity is zero, ****D***KE*** = ½ mv**

**2**

**where**

*m*is the total mass of the cart and any added weights, and*v*is the final velocity. Record your values in the data table.*The work done has a 74.3% difference with change in kinetic energy. This high of a discrepancy might be due to the fact that the cart was massed without the force sensor, which was quite sizable.*

**CONCLUSION**

This experiment was excellent for investigation of the relationship between work, kinetic, and potential energy. Also, it was interesting to calculate the percent discrepancy between the computer calculations and our hand-calculations.