1-March-2017: Free Fall Lab -- Determination of g and some statistics for analyzing data

Eric Chong

Free Fall Lab -- determination of g and some statistics for analyzing data

Lab Partners: Lynel Ornedo and Harvey Thai

Purpose: to examine the validity of the statement: In the absence of all other external forces except gravity, a falling body will accelerate at 9.8 m/s^2

Introduction: For this lab, we are trying to measure the acceleration due to gravity of a falling object. To do so, we used an apparatus that can be used to drop objects in a straight line downwards while marking a piece of paper at certain intervals of time. Since gravity is expected to accelerate the object downwards, the distance between each markings will increase. We can measure the increase in distance over time with these markings, and use the data collected to find the acceleration due to gravity.

Part 1

Procedure: First off, the apparatus is overall 1.86 m tall with an electromagnet at the top, a falling distance of 1.5 m, and a tripod bottom to help the apparatus stand in place. A long strip of spark paper is made straight by a weighted clip at the bottom. When the object held by the electromagnet is released, its fall is recorded and marked on the spark paper at precise intervals of time. Here is a picture of the apparatus:

After we got introduced to the apparatus, we can now use the device in order to get our data. We set up the paper on the apparatus and made it "tight" with a paper clip at the bottom, put the object with a metal ring around it on the electromagnet, activated the spark generator and made the object on the electromagnet spark at 60 Hz, turned off the electromagnet so the object would drop and make marks on the paper, and turned off the spark generator. This is how we got the raw data for the position vs. time of the object.

Next, we made an excel file and created columns labeled TIME, DISTANCE, delta x, Mid-interval time, and Mid-interval speed. Each time interval is 1/60 s under the TIME column. For the Mid-interval time, we set each time interval to be 1/120 s, and for the Mid-interval speed, we divided each delta x by 1/60 to give us the average speed of each mid-interval. Essentially, the mid-interval speed should be the same as the average speed of the entire time interval for constant acceleration, and I will discuss more of this in the conclusions. Now we have the set up for our data table. We can now manually measure the position of each marking and analyzing it with respect to time in our data table. To start, we made the first dot marked on the paper our t = 0 s dot. By doing so, we can compare the first dot's position with those of the sequential dots afterwards with a meter stick with respect to time. Here is a picture of what we did:

From there, we simply took the positions of each dot and analyzed the change in position, how much time has passed at each dot, and put it into our excel file. After we obtained and analyzed the data, the excel file should automatically give us the Mid-interval time and the Mid-interval speed, since we set up the columns that way. Here is a picture of our data:

Now, we want to graph our mid-interval speed vs. the mid-interval time because we established that mid-interval speed at the mid-interval time is the same as average velocity of the entire interval. So, we can get a nice approximation of the velocity vs. time graph of the entire data. We curve fitted the graph with a linear fit, as the graph seems to be a straight line. We also graph the Distance vs. Time in the same way and curve fitted it as a polynomial fit of order 2. Here is what the two graphs look like:

From the mid-interval speed vs. mid-interval time plot, we can determine the acceleration due to gravity from the slope of the graph. From our graph, we determined that from the experimental data, our calculated acceleration due to gravity is 946.15 cm/s^2.

Conclusion Part 1: In retrospect, our data was very off from the accepted value of 9.8 m/s^2. The data seems to be lower than the accepted value by approximately 0.34 m/s^2. This experimental error could have resulted from the friction from the paper that was applied to the object falling from the electromagnet. This friction could have slowed down the velocity slightly and lead to the lower acceleration that we got from calculations and the graph. The percentage error of this experiment is approximately 3.47%. Not too bad of an error, if I do say so myself.

1.) As mentioned in my procedure portion of the blog, the mid-interval speed is the same as the average velocity over the entire interval. Graphically, due to the constant acceleration, which, by definition, means the rate at which velocity changes over time. the velocity is rising at a constant rate. Thus, we can take the final velocity at the final time plus the initial velocity at the initial time, all over 2, in order to get the average velocity. The equation looks like this:

Vaverage = (Vf + Vi) / 2

On the graph, this value is the same value as the mid-interval speed value.

2.) From the velocity/time graph, I can get the acceleration through looking at the slope of the line. Because acceleration means the change in velocity over time, the slope is the equivalent to the value of acceleration. Also, another way is to take the derivative of the line equation and find the acceleration through that. The derivative is essentially finding the equation of the rate at which the line's values changes, and thus this method accomplishes the same task as just looking at the slope.
The slope of the graph is 946.16, meaning the acceleration is 9.4616 m/s^2, which is slightly off from the accepted value of 9.8 m/s^2 due to the external factors such as friction and air resistance that affected the data.

3.) To get the acceleration from the position vs. time graph, I can take the derivative of the given equation twice. This would take me to the velocity vs. time equation in the first derivative, and then the acceleration vs. time equation in the second derivative. Through this, I can see the value of the acceleration and compare my result with the accepted value of the acceleration due to gravity. My result by taking the derivative twice is 9.3326 m/s^2 and this is slightly off from the velocity vs. time graph because the data spread is far more curved and could have made the numbers slightly off. Also, it seems to be further from the accepted value of 9.8 m/s^2 than when I took the slope of the velocity vs. time graph. It is possible that by looking at the position vs. time graph the uncertainty is accentuated since it is essentially the raw data without any alterations from equations, sig figs, or other number manipulating methods.

Part 2: Analyzing the Class' Data for g

Purpose: The goal of this part of the lab is to analyze the data spread of the multiple trials run separately by other groups who participated in the same experiment. This would allow us to see whether our methods were considered viable in determining the validity of the statement that the accepted value of g is 9.8 m/s^2. Also, if there is a wide data spread, it would allow us to reconsider our methods and detect where the uncertainty lies exactly. Though there is already a good idea of where it lies just by looking at one trial, more trials strengthens the assertion that the uncertainty could lie at the friction of the paper with the falling object.

Procedure: By taking the data of each group and making a table that gives us the standard deviation, we could see how far the data spreads from each other. To find the standard deviation, we take each group's value for g and subtract it from the average value of the entire class' value of g. This gives us the number of how much a certain group deviates from the average and whether it deviates higher or lower from the average. Then, we take that number and square it. We repeated process for every group. Then, we add up all the group's deviation squared and divided by the total number of groups. And finally, we square rooted the value and that would leave us with the standard deviation. Here is a table that gives the values while going through the process of finding the standard deviation of the class' data:

In this case, the standard deviation of the entire class' result of g is 10.0147569. This is quite a huge spread of data, and the data are all under the accepted value of 9.8 m/s^2. Our data has a systematic error, which means that our method, indeed, has something that needs to be corrected, and I think that there is a high probability that the friction of the paper may have contributed to the lower values.

Conclusion Part 2: As stated before, the possibility that the friction contributed to the lowered class data is high. And it is evident that we had a systematic error. Perhaps changing the type of paper to one that has lower friction could help the data be closer to the accepted value of 9.8 m/s^s, or another way of measuring the position of the falling object by video recording with high frames per second. Overall, though the data was lower than expected, it still is not too far off from the accepted value, and we could fix this by simply making some slight adjustments to our method.

1.) There is a pattern that our data seems to be lower than the accepted value of g. This is possibly due to the existence of friction with the paper, and it likely reduced the falling speed.

2.) The average value is smaller than the accepted value, and this is, again, due to the presence of uncertainties in our method.

3.) The overall class' data of g is also smaller than the accepted value of g, meaning that our data is not the only one being affected, and that the uncertainty is in our procedure.

4.) Comparing our data with the class' data, we think that our deviation from the average value could be due to the existence of random error that is caused by external factors such as air flow which leads to some air resistance, and the existence of a systematic error from the friction of the paper. The friction from the paper is a systematic error because it affects the entire class' data and is the fault of the procedure. By changing our methods, we could remove such errors, and in this case by possibly changing the paper type or data recording methods by video recording.

5.) The point of this part of the lab is to practice analyzing data through standard deviation and analyzing the data spread. We came up with the conclusion that there was something wrong with our procedure through looking at the data and observing that the data was lower than the accepted value. This led to our looking for some possible factors that led to such a deviation and came up with the possibility that friction was the leading cause. As such, this process of finding error and looking at our procedure brings up the key idea of how we, as scientists, conduct experiments. We can improve upon our ways of experimenting and summarize our data in tables and analyze them. It is through such an effective outlook of experimenting that scientists can expand and improve older experiments and conjure more accurate data, and why science is an ever so growing field of observing the world around us.