25-Feb-2015 Sparker Free Fall

PURPOSE:  To verify that the gravity of a free falling object on earth is 9.8 m/s^2.

MATERIALS:

• Electromagnet reader
• Spark generator
• Spark-sensitive tape
• Meter stick

PROCESS:

1)  We first produced a strip of paper that had markings indicating where an object was positioned at every 1/60 of a second, starting from the top of an apparatus of height about 1.86 meters and falling straight down (this was completed by professor and we used previously marked strips):

• Turn dial hooked up to an electromagnet
• Hang wooden cylinder with metal ring up on the electromagnet
• Turn on, hold spark button
• Turn off electromagnet to let the object fall, which will make markings on the strip of paper
• Tear off strip of paper with marked data on it

1a)  Significance of marks on strip:

• Spark generator makes marks on the paper strip every 1/60th of a second
• Dots near the top of strip start off close together and increase distance as object nears the ground

2)  Gathering data from paper strip

• We then lined up a meter stick against the strip, starting 0-cm at a dot near the top of the strip (does not have to be first dot)
• Recorded the position of each dot (in relation to 0-cm) in the downward direction

After collecting this data, we plotted it into excel with the first point corresponding to time = 0, the next as 1/60 s, then 2/60 s, and so on until about 1/4 s (Column "t").  The second column, "x," was the distance of each dot in relation to the 0-cm mark.  The next column, "Δx," was the distance between a point and its following point.  The column, "Mid-interval time" was found by taking the time of a point and adding 1/120, which is the same as the time for the middle of that 1/60 of a second.  Finally, the column "Mid-interval speed" was the speed of the object at the middle of 1/60 of a second, found by change in distance (Δx) over time (1/60 sec).  Our data is shown as follows:

We used this data to plot two graphs: 

•  Mid-interval speed graph
• Gives us velocity vs. time with constant acceleration
• Equation Vf = V0 + at
• Position vs. Time graph
• Gives us position vs. time
• Equation: y = 1/2at^2 + v0t + y0

For this graph of velocity vs time, we chose a linear fit for the points, to show that the slope of the line is the constant acceleration in cm/s^2.  Our result showed that acceleration was 964.2 cm/s^2, or 9.64 m/s^2.

For this graph of distance vs. time, we chose a polynomial fit, to get a kinematic equation of position, initial velocity, time and acceleration.  The coefficient of t^2, which 1/2 acceleration turned out to be 481.56.  We can multiply this by 2 to get an acceleration of 963.12 cm/s^2, or 9.63 m/s^s.

Questions/Analysis:

1.  

2.  As described previously from the velocity vs. time graph, the equation for the line is the kinematic equation for constant acceleration, Vf = V0 + at.  The slope of the line gives us an acceleration due to gravity, 964.2 cm/s^2, or 9.64 m/s^2.  

• Our absolute difference is experimental value - accepted value, or 9.64 m/s^2 - 9.8 m/s^2, which is |-0.16 m/s^2| = 0.16 m/s^2.  
• Our relative difference is (experimental value - accepted value)/accepted value x 100%, or (9.64 m/s^2 - 9.8 m/s^2)/9.8 m/s^2 x 100% = -1.6% 

3.  From the position vs. time graph, the equation of the polynomial is the kinematic equation for position, given initial velocity, y = 1/2at^2 + v0t + y0.  1/2a is 481.56, so acceleration is 963.12 cm/s^2, or 9.63 m/s^2.  

• Our absolute difference is 9.63 m/s^2 - 9.8 m/s^2 = |-0.17 m/s^2| = 0.17 m/s^2.
• Our relative difference is (9.63 m/s^2 - 9.8 m/s^2)/9.8 m/s^2 x 100% = -1.7%.

Class' Standard Deviation of the Mean:

To get a better idea of the uncertainty of the results of this experiment, we looked at the results of all the groups' data from each of our trials.  The first 10 numbers under the "g values" column show the results of each group, while the last value, 952, is the mean value for the acceleration due to gravity.  We subtracted the mean from each group's g value to get the deviation from the mean, and then squared each one to make them positive and added all the squared deviations.  We then divided the result of 10166 by 10 to get the average, and finally took the square root to get 31.884, 

1.  Pattern in values of our values of g:

• All are under the accepted value of g except for 1.

2.  Average value compared with accepted value of g:

• (Avg - Accepted)/Accepted x 100% = (952-981)/981 x 100% = -2.96% relative difference

3.  Pattern in class' values of g:

• Mostly fall under the accepted value of g.

4.  Our measurements vs. class measurements

• Random errors could have occurred such as a small wind push against the falling object.  One of the marks on the paper strip could have gotten smeared, making it harder to take the precise measurement against the meter stick.
• Systematic errors could have occurred such as a tiny lag in the spark generator, or a measurement using the meter stick that was not completely precise.

5.  Key Points:

          For this part of the lab, we were supposed to derive a standard deviation of the mean of the class' data.  We wanted to see how the class' data as a whole compared to the accepted value of acceleration due to gravity, and then speculate why that individual data could have been different.  The value of standard deviation was 31.884, which made one standard deviation be within the values of 920.116 and 983.884 (952 +/- 31.884).  I found that 9 out of the 10 groups' measured values were within this one standard deviation of the mean, which gave a better outcome than the average theory for uncertainty.  Typically at least 68.26% of the values should be within the first standard deviation, but as a class our results were more accurate.  The point of this error part of the lab was also to speculate reasons why our data would be off from the accepted value for acceleration due to gravity.  This could have been from an error in the spark generator, a blunder in taking measurements of the dot distances, or a slight breeze in the room when letting the object fall during our experiment.