## Purpose

To see how the acceleration of a cart depends on the resultant force acting on the cart and the mass of the cart, and how this relation can be expressed in a single equation. We hypothesized that with a constant mass of the system (a decrease in cart mass) and an increased net force, acceleration would increase. Also, with an increased mass of the cart and a constant net force, acceleration would stay the same.

## Apparatus and Materials

- Dynamics cart
- Three 200.0 g masses
- Two 1.0 kg masses
- String
- Pulley/Photogate
- 2 clamps
- LabPro
^{TM}Interface - LoggerPro
^{TM}Analysis - Digital Balance

## Method

“Investigation 2.3.1” Physics 11. Toronto: Nelson Thomson Learning, 2002. 68-70.

## Observations and Calculations

Run Number | Mass of System (M_{sys}) | 1/ Mass of System | Net Force (of Hanger) | Net Force/Mass of System | Acceleration |

A-1 | 1.54 kg | n/a | 1.96 N | 1.27 N/kg | 1.08 m/s^{2} |

A-2 | 1.54 kg | n/a | 3.92 N | 2.55 N/kg | 2.24 m/s^{2} |

A-3 | 1.54 kg | n/a | 5.88 N | 3.82 N/kg | 3.42 m/s^{2} |

B-1 | 1.54 kg | 0.649 1/kg | 5.88 N | 3.82 N/kg | 3.42 m/s^{2} |

B-2 | 2.54 kg | 0.394 1/kg | 5.88 N | 2.31 N/kg | 2.12 m/s^{2} |

B-3 | 3.54 kg | 0.282 1/kg | 5.88 N | 1.66 N/kg | 1.45 m/s^{2} |

## Questions

**Analysis (pg. 69-70):**

c) See “Observations and Calculations”

d) See graph “Acceleration vs. F_{net}”. The graph of acceleration as a function of the net force indicates that there is a direct relationship between the net force and the acceleration such that when the net force is doubled the acceleration is also doubled.

e) The acceleration as a function of the mass of the system indicates that acceleration decreases as the mass of the system increases. The graph suggests that a point will be reached where there is the possibility of putting so much mass on the cart that the cart will approach the point of zero acceleration.

f) See graph “Acceleration vs. Reciprocal of M_{sys}”. The slope of the acceleration vs. reciprocal of the mass of the system (1/Mass of System)is in the unit m/s^{2}/1/kg. One can change this to say kg m/s^{2} which is also known as N. Therefore, the slope of the graph represents the net force acting on the cart.

g) See “Observations and Calculations”. When comparing the calculated acceleration with the ratio F_{net}/m one discovers that they are essentially the same. An equation to describe this relation is: a = F_{net}/m

h) See graph “Acceleration vs. F_{net}/m”. The acceleration points are essentially the same as the F_{net}/m points, so: a = F_{net}/m

## Discussions

k) In this lab, we were only looking at the relative difference between force and acceleration and mass and acceleration. In this case, the frictional forces were practically constant throughout the lab, so they had no role in determining the outcome of the lab.

l) Some cases of systematic errors in this lab are that we don’t know if the balance or LoggerPro^{TM} were completely accurate due to possible calibration errors if the wheels and the pulley are not always turning consistently, so more or less force would be needed to pull it.

A case of random error is that estimates were made to obtain the last significant digit when weighing the cart on the balance because the balance only goes to a certain amount of digits and rounds off the last one.

Some cases of human error could be that maybe the cart was released before the LoggerPro^{TM} started recording the run, the “stop” button on the LoggerPro^{TM} might have been pressed too early before the computer had finished recording the run, maybe the cart was not perfectly aligned each time, so more or less force would be needed to pull it.

m) Yes the equation a = F_{net}/m applies to Newton’s First Law because if there is zero net force there is a constant velocity or zero acceleration.

To improve our results, we could have made absolutely sure that the person releasing the cart let go after the other person pressed the “collect” button on the LoggerPro^{TM}. We also could have made sure that there was enough time for the LoggerPro^{TM} to record all of the results before we pressed the “stop” button. We probably could have done each of the runs a few times in order for the data to be completely accurate.

## Conclusions

i) The acceleration of the cart will increase/decrease when the net force increases/decreases (the mass of the cart remains constant) or when the mass of the cart decreases/increases (the net force remains constant). From the graphs we see that: a µ F_{net} , and, a µ 1/m. Therefore, a = F_{net}/m.

j) The first part of our hypothesis stated, “…with a constant mass of the system (a decrease in cart mass) and an increased net force, acceleration would increase.” This prediction was shown to be true on the graph “Acceleration vs. F_{net}” which is from Part A of our experiment.

This graph shows the linear relationship between net force and acceleration such that when the net force increases acceleration increases as well. The second part of our hypothesis stated, “…with an increased mass of the cart and a constant net force, acceleration would stay the same.”

This prediction was shown to be incorrect on the graphs “Acceleration vs. Mass of System” and “Acceleration vs. Reciprocal of M_{sys}” which are both from Part B of our experiment. These graphs show the inverse relationship between acceleration and the mass of the system such that when the mass of the system increases the acceleration decreases. Therefore, our hypothesis of the experiment was partially correct.

*SchoolWorkHelper*, 2019, https://schoolworkhelper.net/investigation-2-3-1-the-relationship-involving-acceleration-net-force-and-mass-answers/.

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