2-Mar-2015: Determining g

PURPOSE

The purpose of this experiment was to determine the acceleration due to the force of gravity through analysis of the collected data.

PROCEDURES

Figure 1: The column lined with spark-sensitive
paper that will be used to track the free-falling
body's motion

Figure 2: The electromagnet's power supply

Figure 3: Spark generator

We started this experiment by setting up a system consisting of a column (Figure 1), a free-falling body, a spark generator (Figure 3), and an electromagnet. The free-fall body was held at the top of the column with the electromagnet. When the electromagnet's power supply (Figure 2) was turned off, the body was released and dropped down the column. As the body fell down the column, its location was marked on a spark-sensitive tape every 1/60th of a second by the spark generator illustrated in Figure 3. The resulting tape is shown in Figure 4 below.

Figure 4: Measurement of the body's displacement during each time interval

After we acquired the tape, we measured the distance of each dot from the first, which we established as the origin. We recorded these distances in centimeters and the time in seconds at which the free-falling body was at those distances in an Excel file under the column labeled as "x." Then, we created a column named Δx by subtracting these x values from each subsequent x value. For example, Δx1 was found by subtracting x1 from x2. Next, we found the mid-interval time by dividing each t value by two and recorded the values in the fourth column of the Excel file. For the final column, we found the mid-interval. This was done by dividing the displacement by the time interval. For example, the first mid-interval speed was found by dividing 1.4 by 1/60. The resulting spreadsheet is displayed below in Figure 5.

Figure 5: Data spreadsheet

From the data above, we created two graphs that would allow us to derive a value for the acceleration due to gravity. For the first graph, we plotted the fifth column against the fourth column (mid-interval speed vs mid-interval time) using a scattered plot. Then, we applied a linear fit to the graph. We subsequently graphed the second column against the first column (distance vs time), also with a scattered plot. However, for this graph, we implemented a polynomial fit of order two instead of a linear fit. The graphs that were produced from these steps are illustrated below in Figures 6 and 7.

Mid-interval speed vs mid-interval time

Figure 6: Mid-interval speed vs mid-interval time graph

Distance vs time

Figure 7: Distance vs time graph

ANALYSIS

We can learn several things from analyzing these two graphs. First, we can determine that the velocity in the middle of a time interval is the same as the average velocity for that time interval. We can do this by taking the slope of the distance versus time graph for each time interval and comparing it to the corresponding mid-interval speed that we found earlier. Although the distance versus time graph is curved, we can treat the graph to be linear within those small time intervals.

Second, we can deduce the acceleration due to gravity by observing the first graph (Figure 6). We can achieve this by looking at its slope, which describes how much the velocity of the object is increasing with time; this is equivalent to its acceleration. Since the only force acting on the object is gravity, we can safely assume that this acceleration is the acceleration due to gravity. The slope is 953.41. This value is in cm/s², so when you convert it to m/s² (9.5341), the value is relatively close to the accepted value of 9.8 m/s². We will later see just how close the values are by using the standard deviation of the mean of the class' data.

We can also obtain the acceleration due to gravity by looking at the second graph (Figure 7). This is done by taking the derivative of the equation of the graph. When we do this, we get an equation for the object's velocity in terms of time. With this equation, we can either take the derivative again or just look at its slope like we did with the first graph. Regardless of the method that we use, we acquire a value 963.12 cm/s² or 9.6312 cm/s². This is even closer to the accepted value of 9.8 m/s² than the experimental value that we found by analyzing the first graph.

CONCLUSION

Relative Difference = (Experimental Value - Accepted Value)/(Accepted Value) x 100%

= (9.63 m/s² - 9.80 m/s²)/(9.8 m/s²) x 100%

= - 1.72 %

There are a number of assumptions that we made when doing this lab that may have contributed to our experimental value not being exactly what we expected. First, we did not account for the fact that the free-falling body most likely experienced friction while falling down the column. This is probably why the experimental value was smaller than the expected value. Another assumption that we made was that the spark generator was able to accurately record the object's location every 1/60th of a second. One thing that is problematic with this assumption is that the location of the object could have been recorded incorrectly. Another thing that is of concern with this assumption is whether or not the marks were made at exactly 1/60th of a second intervals. However, unlike the first assumption that the friction was negligible, there is no way of knowing what kind of effect this assumption had on the experimental value. Therefore, this assumption is a reasonable one.

In addition to the assumptions mentioned above, there are a number of factors that contribute to the uncertainty that we have with the measurements. For example, there is uncertainty in the distances that we measured because we used a meter stick. The meter stick limits our ability to measure the distances because it only measures to the tenth of a centimeter. This means there is a considerable amount of uncertainty with the measurements that we made since the distances are not large enough for these uncertainties to be negligible.

CLASS DATA

Figure 8: Spreadsheet of the class data, including the class' average value of g

The pattern with the class data is that they are all smaller than the expected value of g with the exception of one group. The class' average value of 9.52 m/s² is pretty close to the accepted value of 9.8 m/s² since its relative difference from the expected quantity is less than three percent and because the value is within one standard deviation from the expected value. The standard deviation is shown in the bottom right corner of Figure 8. However, our group's data was more accurate than the class' average value by more than a percent. This disparity among our class is most likely due to random errors and not systematic errors because systematic errors tend to have a similar effect on everybody performing the lab.

The point of this part of the lab is to see that even when people use the same equipment and perform the same experiment, they will not get the same results. This shows us how significant these sources of error can be. In addition, this part of the lab showed us that there is usually one outlier (the group that got a g value of 10.4 m/s²) whenever data is taken from multiple parties.