02-March-2015: Free Fall Lab

Free Fall Lab

Purpose 

The purpose of this lab is not only to determine the acceleration due to gravity on a free falling object, but also to learn how to analyze data and understand the importance of experimental uncertainty. You will also practice and expand your knowledge about Excel.

Set up

The spark generator was connected to an object attached to a metal rod which allowed for 1.5 m of free falling distance. Spark tape is put parallel to the metal rod along the 1.5 m distance. The falling object was held at rest by an electromagnet before the experiment begins. When the object was released, the object creates sparks which mark the spark tape at a rate of 60 sparks/sec. The spark tape was then removed and taken for measurements.

A meter stick with +/- 0.05 m of uncertainty was used to measure the distance of each spark from the origin. This data was recorded and graphed in excel for analysis.

Part #1 Data colection

What we did was to create a table like the one below in order to make graphs that would display values for acceleration. Since the sparks were produced at a rate of 60 sparks/sec, time intervals were set at 1/60th of a second (1st column). The distance were also recorded per 1/60th of a second (2nd column). The change in distance or Delta X  was then calculated (third column). Time was then split into mid-intervals (4th column) and the speeds during these intervals was also calculated (5th column).

Part # 2 Data Graphing

Velocity was graphed using Time vs Distance in order to verify that the data made sense. This was verified as the graph showed an upwards curve due to a constant acceleration. The value of constant acceleration was then determined by taking the derivative of the y=x^2  function (deriving distance gives you velocity). Thus by taking the derivative of y=484.59 x^2, a value of 969.18 m/s^2 was found. The R^2 value was 0.9999 which showed that the curve had a direct correlation. 

Mid-interval Time vs Mid-interval Velocity was then graphed which gave us acceleration. The acceleration was given by taking the slope of the graph and was determined to be 970.03 cm/s^2 or 9.70 m/s^2. This value was approximately .11 m/s^2 slower than the accepted gravitational acceleration value. The two graphs gave similar accelerations of 9.70 m/s^2 which is -1.12% from the actual value of acceleration. 

Part #3 Data Analysis (how we obtain results from the graphs above)

1. Show that for constant acceleration the velocity in the middle of a time interval is the same as the average velocity for that time interval.

           First you need to look at your Velocity vs Time graph and select a Mid-interval time from your x-axis and                draw a doted line upwards to the trend line and from that point you need to draw a horizontal doted line to            the the y-axis to obtain the corresponding value for velocity.Then you simply solve for velocity using this                values. This is the velocity in the middle of that time interval.  Now to obtain the average velocity for that                time interval you need to find average velocity.  As you can see for constant acceleration, the velocity in                the middle of a time interval is the same as the average velocity for that time interval.

    2. Describe how you can get the acceleration due to gravity from your velocity/time graph.          
        Compare your result with the accepted value.

        To obtain acceleration from a Velocity vs. Time graph you need to find the slope of the line.
        or you can take the derivative of  the velocity function. Comparing your result with the accepted value of g             can be done in two ways

                     Absolute difference = (experimental value - accepted value)

                                                      = (9.7 m/s^2 - 9.8 m/s^2)

                                                      = -0.11 m/s^2

                           

                     Relative Difference =  (experimental value - accepted value)/ accepted value x 100%

                                                      = (9.7 m/s^2 - 9.8 m/s^2)/ 9.8 m/s^2 x 100%

                                                      = -1.12%

    3. Describe how you can get the acceleration due to gravity from your Position vs. Time graph.
        Compare your result with the accepted value.

        From an X vs T graph you obtain a function of velocity. If you then integrate the velocity fuction, you get a             fuction for acceleration. You then compare your result using absolute difference and relative difference.

Part # 4 Analazing the class data for g

The values of gravitational acceleration were taken from the class and a class average was computed. This average was determined to be 9.53 m/s^2 and standard deviation was used to determine a range. The standard deviation had a value of 12.27 m/s^2 thus giving us the +/- value for our results.

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    1.  What pattern is there in the values of our values of g?                                                                                               As expected, the distance traveled by the falling mass increases every period of time. We can  also                        observe that the velocity of the object constantly increases as the mass falls.

    2.  How does our class average value compare with the accepted value of g?

           Absolute difference = (experimental value - accepted value)

                                        = (9.53 m/s^2 - 9.8 m/s^2)

                                        = -0.27 m/s^2

                           

           Relative Difference =  (experimental value - accepted value)/ accepted value x 100%

                                       = (9.56 m/s^2 - 9.8 m/s^2)/ 9.8 m/s^2 x 100%

                                       = -2.75%

        

    3.  What pattern if any is there in the class values of g?    

         As expected there is no pattern. The data is randomly distributed around the average value of the class                value of g

    4.  What might account with any difference between the average value of your measurements and                              those of the class?

          Friction from air resistance could not have been zero because the experiment was not performed in a close           container of something similar. The accuracy of the electromagnet was not very reliable because as we                 noticed on our Velocity vs. Time graph there were two dots  that were completely off from the trend-line.

    5.  Write a paragraph summarizing the point of this part of the lab. What were the key ideas? What were                    you supposed to get out of it?

         

        The point of this part of the lab was to calculate the average g value for the class. From this value we                     calculated the standard deviation of each group's g value. Thus the purpose of this part of the lab was to               analyze experimental data and evaluate experimental uncertainty. Because all of our measurements (For             example, the distance between each dot on the  paper strip) were uncertain by nature, our results were also         uncertain. And so we calculated the error in each of our values and compared it to the accepted value of g           by obtaining the Standard Deviation of the mean.

Error 

The values that we found and the class average were both lower than the actual value. This may be due to friction along the metal rod as the object was in free fall. This friction could have slowed down the object and not allowed for full acceleration. The class average was 2.75% lower than the actual value compared to our experiment's 1.12%. This may be due to inconsistency in the equipment and the nature of the data recording as there were many different people taking data from different strips. This could add up to human error that may have affected the average. 

Conclusion

The experiment was a success as there was only a range of about 3% error between the data. The group acceleration was determined to be 9.70 m/s^2 while the class average was 9.53 m/s^2. These values were compared to the actual value of 9.81 m/s^2. The values were determined using excel and taking the derivative of the position vs time graph or the slope of the velocity vs time graph.