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.
Apparatus: The apparatus shown to the right consists of a heavy tripod
with leveling screws, a free-fall body, spark paper, and an electromagnet. In addition, we also used a spark generator (not pictured). The electromagnet is seen at the top of the apparatus and it is used to hold the free-fall body. The spark paper is setup along the spine of the apparatus and it used to record the distance traveled by the free-fall body. The spark generator is essential because it will mark the location of the free-fall body on the paper. It will appear as a dot.
In order to fully understand how the apparatus work, please select the following link: https://www.youtube.com/watch?v=J6lqgJsA6Xk
Skip to the 4:40 mark to see the apparatus in action.
Procedure: Due to time constraints, our group did not use the apparatus. Instead a demo of the apparatus was given to the entire class and we were handed a spark paper from a previous experiment. Our job was to establish an origin on the spark paper and to record the distance from that point to the origin. The setup is seen below. The direction of fall is noticeable because the points begin to grow in distance.
The picture above shows the position of the falling mass at every 1/60th of a second.
We were able to record the distance of 20 points and then inputted the data into a spreadsheet.
The first column shows our time intervals, which increase at 1/60th of a second.
The second column shows the distances we measured for the falling mass.
The third column shows the change in distance over a time interval.
The fourth column shows the mid-interval time and increases at a rate of 1/120th of a second.
The fifth column shows the average velocity of the falling mass.
In constant, acceleration, the velocity in the middle of a time interval is the same as the average velocity for that time interval.
For example:
- Mid-Interval Speed =66
Average Velocity = Change in x/ Change in t
- V_av = 2.1-1/0.03333333-.01666667 = 66
We then used our data and created the following graphs:
Position vs. Time
Equation of line: y=477.96x^2 +41.348x+.1541
This equation strongly resembles that of X=Xo+Vot+(1/2)at^2
Therefore, in order to find the acceleration you would simply multiply 477.96 by 2, which equals 956. The acceleration for the P vs T graph is 9.56 m/s^2 This differs from the accepted value of 9.81m/s^2
Velocity vs. Time
Equation of line: y=954x+42.529
In order to find the acceleration of our falling mass, we simply take the slope of our V vs T graph.
From here, we can see that our V vs T graph resembles a straight line. This is appropriate because acceleration is constant during free fall. We recognize g as 9.81m/s^2 and based on our graph our value of g is 9.54m/s^2.
Errors:
There are a couple errors in the lab because our calculated acceleration does not match 9.81m/s/s.
In order to calculate the percentage error we used the following formula:
[(Experimented Value-Accepted Value)/Accepted Value]x100%
Calculated Percentage Error: -2.75%
Furthermore, there was error in two recorded measurements, as seen in the v vs t graph. The ruler we used to measure the distances on the spark tape had a piece of tape that prevented us from seeing the correct measurement.
Class Data: After conducting the experiment on our own, we collected and analyzed the other groups' values of g.
The first column shows the different values of g and the calculated class average.
The second column shows the deviation of a group's value of g from the class average.
In the third column, the deviation was squared and an average was calculated.
The fourth column shows the standard deviation of the mean, which is the square root of the average from the third column.
The standard deviation of the mean for the class data is (20.12). This means that we are:
- 68% confident that the real value of g falls between 956 +/- 1 Dev. Range:936 to 976
- 95% confident that the real value of g falls between 956 +/- 2 Dev. Range:916 to 996
An ideal value would be 980 +/- 02 with 95% confidence
The percentages were taken from Pg.7 of the Lab Packet.
We can be certain this data is correct unless our experiment had a systematic error; such as not accounting for the friction in the apparatus as the free-fall object dropped. If there were friction, it would make our calculated values smaller.
Final Thoughts:
- As far as I can tell, I do not see any major patterns in our values of g. However, there are three values in the 960 range.
- Our average value of 956 differs from the accepted value of 981.
- The class' values of g range from 926 to 992, with the majority being close to 950.
- There might be a small difference between the average value of our measurements and those of the class because the ruler we used to measure the distances on the spark tape had a piece of tape that prevented us from seeing the correct measurement. This can be classified as a random error. The apparatus can cause a systematic error because it may not have the precision required for this lab. A systematic error can also be caused by inputting the wrong data into the spreadsheet.
- At the end of every lab, we amass data and analyze the results. The point of this part of the lab was to learn how to analyze data properly. In order to analyze the data properly, we increased our sample size and compared our data to the accepted value of g. In doing so, we learned some essential skills on Excel. Ultimately, we learned that not all experiments will be a success but when given data; our job is to know how to interpret it properly and find its accuracy.
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