Lab 2: March 1, 2017: Free Fall Lab

Lab 2: Determination of g and some statistics for analyzing data

Amy, Chris, and John

March 1, 2017

Today's lab purpose was to examine the validity of the statement: 

In the absence of all other external forces except gravity, a falling object will accelerate at 9.8 m/s^2.Calculate values of g by group and compare them to other group's findings of g and calculate the standard deviation as well as errors of uncertainty. 

Theory: For this particular lab we had to figure out a way of measuring g, looking at the distance between sparks on a long tape created by the spark generator as an object was in free fall. By measuring the distances of the position as the object was in free fall we could gather a time lapse of how fast the object was falling. We graphed the findings from our times and distances, from the tape with markings, and calculated through a velocity vs. time graph how fast the object was accelerating down or g though the slope of the data we entered. Because gravity is a constant downward acceleration we hope our slope is linear. 

Apparatus and Procedure: For today's lab we used a spark generator. When the free fall object, held at the top by an electromagnet, is released then marks are made at the intervals of 1/60th of a second on the tape attached to the column for us to measure the distance covered by the free falling object in order to measure acceleration. 

Spark generator

Tape with spark markings

Measured Data:  This table shows the measurements all derived from measuring the distance between the markings on the spark tape. 

Calculated Results and Graphs: Our graph was showing the mid-interval speed and mid-interval time deducted from our excel worksheet. 

Derivations to fill in excel rows D,E, and F

By calculating ∆x and mid-interval time excel calculated mid-interval speed.  

The top graph shows the speed vs. time calculations and the slope of the curve shows the increase of speed (acceleration) the free fall object was experiencing. In our calculations the slope was approximately 969.695 cm/s^2 which converts to 9.69695 m/s^2.

Conclusion: 

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. 

-Taking a piece of the graph for the time interval, from our speed vs. time graph, and calculating its area would provide the same number as the average velocity for that time interval because they are both correlated graphs. 

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

-If you divide velocity/time=acceleration, because our graph is showing us the change of velocity over time it is indicated by the slope from the line y = 969.695x + 65657 then the slope 969.965 cm/s^2 is our acceleration; which converts to 9.69695 m/s^2 which is slightly lower than the accepted value we use in class which is 9.81 m/s^2. 

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

-By taking the derivative of the equation given y = 485.47x^2 + 65.022x + 0.1046 we get y' = 970.94x + 65.022 taking that slope again to be acceleration and convert it to SI units, it becomes 9.7094 m/s^2, it is slightly lower than the value of acceleration 9.81 m/s^2. 

Errors of Uncertainty: Our main error of uncertainty could be narrowed to our measurements of distance from one marking to other. Initially we made a small error mistaking a pencil mark to be our spark mark and it threw off the initial graph. Later looking over the marks more carefully we saw the small mark was not meant to be there. Another simple mistake was also the inputting of numbers into excel, because we tried to be as precise as possible one of our group members accidentally typed the wrong number and maybe we did not continue to have this constant looking over for all distances. Due to the meter stick only going by increments of one we also have an uncertainty of  (+/-)0.01 cm.

Different values of gravity derived by each table, and the standard deviation.

Questions:  

1.What patterns (if any) is there of our values of g?

-Values of g are slightly smaller than 9.81 m/s^2

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

-Our value of g is smaller than the accepted value. On the picture above the group g's, our g is off by 1.17% 

3. What pattern (if any) is there in the class' value of g? 

-All the values are smaller than the accepted value.

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

-Although air resistance may play a small part there is also measurement errors we might have not taken into account when measuring the distance of the markings. These can be categorized as random errors. 

5. Write a paragraph summarizing the point of this part of the lab. 

-Figuring out the errors of uncertainty can make our assumptions much more precise and figuring out how the propagate in our everyday calculations makes us grasps how much we do not take into account when measuring simple things like distance with a yard stick. By narrowing down where the small error propagated we can avoid further mistakes and make a much more precise conclusion. While graphing the bell curve for our values of g we deducted that our values are smaller shifting the bell curve slightly behind the accepted value of g this being a systematic error.