07-March-2015: Free Fall Lab

The purpose of this lab was to determine the value of the gravitational acceleration constant "g."

Procedure:
In order to calculate "g", we would simply need to let some type of object free fall and determine the acceleration at which it did so. In order to measure this, an apparatus was set up. This apparatus consisted of a 1.86 m column that would allow an object to free fall 1.5 m. A long paper strip was attached to the apparatus, and a wooden cylinder with a metal ring around it would free fall next to the paper. As it fell, a spark going at 60 Hz would hit the metal ring as it went down. The paper strip would record the markings between the spark and metal ring. Since we know the time interval between each marking was 1/60th of a second, we could calculate the rising velocity of the wooden cylinder as it fell.

Fig. 1
Using a ruler, we began
measuring from one dot
and continued to measure
relative to that dot.

For this lab, however, we were already given a strip of paper which had markings. The marking were obtained using the same procedure as stated above. To determine the acceleration of the falling wooden cylinder, we needed to find the distance between each of the markings. Since acceleration should be constant, the starting dot we chose was irrelevant. So we began recording the distance from one certain dot, to 15 dots away from that one. We recorded our data in an Excel spreadsheet. (see Fig. 1) We measured the distance relative to the starting point. Once we had all our data, we then input columns for time, change in position, mid interval time, and mid interval velocity. The time change was 1/60th of a second, the change in position was the difference of the distance between each point, and the mid interval velocity was the change in position over the time interval. The mid interval time was used to graph the velocity graph, since the velocity we found was simply the average velocity. We filled these columns down to create a spreadsheet that we could graph. (see Fig. 2).

Fig. 2
This spreadsheet served as an organized data table that Excel could use to graph and give us equations for position and velocity.

Our first graph was a position versus time graph. We plotted the distance column vs the time column.(see Fig. 3) The next graph was the velocity vs time graph, We plotted the Mid Interval Velocity column vs Mid Interval Time column to form this graph. (see Fig. 4)

Fig. 3
Using Excel, we were able to construct a graph that could
illustrate the relationship between velocity and time, and thus
lead us to find acceleration.

Fig. 4
Using Excel, we were able to construct a graph of position
vs time that would also allow us to find acceleration through
derivation.

Questions/ Analysis:

1. For constant acceleration, the velocity in the middle of a time interval will always be the same as the average velocity of that time interval. This is because the relationship between velocity and time is linear if acceleration is constant. The slope is continuous and straight. The average velocity between two time intervals will be the same if you follow the slope of that graph to the mid time interval between those two. (see Fig. 5)      

   

   Fig. 5

   
   

                             Since the velocity graph is linear, the average of two points can also be found at their midpoint.                                                                                                                                                       

2. Finding acceleration through a velocity vs time graph is very simple. Since we know that acceleration is a function of velocity/time, we can assume that the slope of a velocity graph is acceleration. Thus, if we have an equation for the line created by a velocity graph, the slope of that line is the acceleration.
3. Finding the acceleration through a position graph requires some derivation. If we have a constant acceleration, a position graph will look parabolic. Given the equation for position, the first derivative of this function would be the velocity equation, and the second derivative would be our acceleration value.

Our Value for "G":

From second derivative of our position graph, we calculated the acceleration to be 930.08 cm/s^2. From the slope of our velocity graph, we found the acceleration to be 930.46 cm/s^2. Taking the average of these two values, our final calculated acceleration for gravity was 937.77 cm/s^2. Or 9.38 m/s^2 for us physicists. Our percent error was 4.31%, which can be mostly accounted for our lack of proper equipment. (using a cm ruler) For calculations, see Fig. 6

Fig. 6
Calculations that led us to our value for gravitational acceleration.