### Purpose

To determine the relationship between force, displacement, potential energy, kinetic energy, and work by using a force sensor to pull a spring and also to push a cart.

### Hypothesis

The work done on the spring will be greatest at its furthest displacement and that the greater the work done on the cart, the greater its acceleration will be.

### Labeled Diagrams

See attached sheet.

### Data

**Part One**

Time (s) | Position (m) | |

Start Pulling | 0 | 0 |

Stop Pulling | 5 | 0.23 |

Spring Constant (N/m) | 81.668 |

Stretch

7 cm | 14 cm | Maximum (23 cm) | |

Integral (during pull) (J) | 0.1779 | 0.8080 | 2.152 |

∆PE (J) | 0.20 | 0.80 | 2.16 |

**Part Two**

Weight of cart: 5.29 N

Mass of cart: 0.54 kg

Time (s) | Position (m) | |

Start Pushing | 0.46 | 0.0 |

Stop Pushing | 0.82 | 0.065 |

Mass (kg) | 0.54 |

Final Velocity (m/s) | 0.3149 |

Integral during push (J) | 0.04796 |

∆KE of cart (J) | 0.027 |

### Graphs

See attached sheets.

### Questions

**1. In Part 1 you did work to stretch the spring. The graph of force vs. distance depends on the particular spring you used, but for most springs will be a straight line. This corresponds to Hooke’s law, of 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 81.668 N/m. From the graph, it does appear that the spring follows Hooke’s law as it produced a fairly straight line. I think the spring would follow Hooke’s law until it is all the way stretched out and cannot be stretched any more, or if breaks. The slope of the graph is positive as it is showing force applied *on* the spring. Hooke’s law shows the force applied *by* the spring, so that would be in the opposite direction in which it is pulled, thus being negative.

**2. The elastic potential energy stored by a spring is given by ∆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?**

The ∆PE for my intervals was 0.20 J at 7 cm, 0.80 J at 14 cm, and 2.16 J at 23 cm. The ∆PE increased as the displacement is increased. This should be expected, as the spring becomes harder to pull the more it is stretched out.

**3. In Part 2 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, ∆KE = ½ mv ^{2} where m is the total mass of the cart and any added weights, and v is the final velocity. Record you values in the data table.**

The work done, 0.04796 J, is greater than the ∆KE, which is 0.027 J.

### Conclusion

**Lab Summarized**

The overall goal of the lab was to investigate the relationship between work, potential energy, and kinetic energy. The goal was achieved using a spring and force sensor along with a motion detector to determine the work done on the spring when pulling it. Using the motion detector, displacement was determined, which could then be used to determine the spring constant from Hooke’s law. The force and acceleration were also collected using the force sensor and motion detector. Graphs produced of force versus position could be integrated to find the work done on the spring over certain intervals. The slope of the linear fit of this graph could also be used to produce the spring constant. Finally, the elastic potential energy stored in the spring could be determined from the aforementioned data.

During the second part of the experiment, the force sensor was used to push and thus accelerate a cart on the frictionless track toward a motion detector. The measured weight of the cart and final velocity could then be used to determine the change in kinetic energy of the cart. The work applied could again be determined by taking an integral over the time period in which the cart was pushed and accelerated with the force sensor.

The data collected for part one seems fairly conclusive. The determined spring constant of 81.668 N/m is comparable to known spring constants. The values for work extrapolated by integrating the graphs of force versus position are extremely close to the calculated values for work, or potential energy. The values of 0.1779 and 0.20 J, 0.8080 and 0.80 J, and 2.152 and 2.16 J are nearly identical, which shows part one of the experiment was performed rather well (or luckily). In regards to part two of the experiment, the work done on the cart found by integrating the graph of force versus position, 0.04796 J, is almost twice as large as the calculated change in kinetic energy, 0.027 J. The only way that this could be accounted for is if the wrong interval was used for the integral on the graph. Looking at the graph, it seems like the integral taken should have been from the start to the top of the peak, and not the whole peak.

### Equations

W = F * s

W = ∆PE + ∆KE

∆PE = ½ kx^{2}

∆KE = ½ mv^{2}

F = -kx