Purpose:
The purpose of this experiment was to graphically calculate the value of the acceleration of gravity using data gathered from a sparker, an apparatus which labels the position of a freely falling object every 1/60th of a second on a piece of tape.
Apparatus:
Picture 1: A photograph of a sparker
A sparker was used in this experiment to gather position data required for calculation of the value of gravity. The sparker consists of the metal column, a cylinder with a metal ring that acts as the freely falling object, sparker tape that runs across the inner side of the column, and a sparker generator that generates 60 Hz of current. The sparker works by holding the cylinder at the top of the metal column by electromagnetic forces. When the electromagnet is turned off, the sparker generator creates a spark pulse every 1/60th of a second that is absorbed and released by the metal ring around the cylinder, generating a dot on the sparker tape. This is how the motion of the freely falling wooden cylinder is recorded and monitored.The sparker tape that was used in the experiment is seen in the picture below:
Picture 2: Some sparker tape used in the free fall experiment. The dots can be seen on the closer end of the tape
The sparker tape used in this experiment is about 1.5 m tall, giving a large amount of data. Eighteen of the many dots were chosen to be used as the position data. Some of the chosen dots can be seen in the picture below. As can be seen, the spaces between the dots increase as the tape goes along. This is due to the velocity of the freely falling cylinder increasing due to the acceleration of gravity.
Picture 3: A closer view of the sparker tape used in the free fall experiment
Abstract:
The position of the freely falling cylinder can be measured by setting one of the dots on the sparker tape to be the origin (at time 0 right before the fall) and measuring the distances the dots are from the "origin" dot. Knowing that the distance between each consecutive dot is the distance the cylinder traveled in 1/60th of a second, the average velocity between two dots can be calculated by dividing the distance traveled between the two dots by the time (1/60th of a second). Using the position and velocity data, a graph of position vs. time and velocity vs. time can be constructed, and the acceleration of gravity can be found. The value of gravity can be calculated from the velocity graph by looking at the slope or by taking the derivative of the velocity function. The velocity function is in the form y = mx + b, which is comparable to the kinematic equation V = Vo + at, where a is equal to m and b is equal to Vo. It can also be calculated by taking the second derivative of the position function, which is in the form y = Ax² + Bx + C. This is comparable to the kinematic equation X = Xo + Vo(t) + (1/2)at², where A is equal to (1/2)a.
Procedure:
Once the sparker free fall experiment was performed and the distance between the dots was measured, the data from the tape was recorded into an Excel spreadsheet, with time (in seconds) in one column and distance (in cm) in another. The distance was measured as the distance from each dot to the "origin" dot, and the time was measured proportionally in 1/60th of a second for every dot after the "origin" dot. Delta x was then found by finding the distance between each of the two consecutive dots. The mid-interval time was then recorded; it is the time where the position of the cylinder was right in between the two dots (initial time + 1/120). Then, mid interval speed was calculated by dividing delta x by 1/60. The mid interval speed can be used to find average velocity because acceleration is constant. The area of a rectangle located at half the interval is equal to the area under the curve when acceleration is constant. Then, mid interval speed and time were used to construct the velocity graph seen below, with a linear trendline.
Graph 1: Velocity vs time graph of the freely falling cylinder in the sparker
The distance and time columns were then used to construct the distance vs time graph below, with a polynomial trendline of order 2.
Graph 2: Distance vs time graph of the freely falling sparker object, where the origin is at the top of the sparker
As stated above, the acceleration can be found from the slope of the velocity graph or from multiplying A by 2 in the distance graph. However, the acceleration of gravity in this experiment was calculated from the velocity graph. The value of g was calculated to be 941.5 cm/s², or about 9.42 m/s², a percent difference of about -3.98% from the real value of g (9.81 m/s²). Percent difference can be found by subtracting the experimental value with the real value, then dividing by the real value and multiplying by 100%.
In order to minimize error and uncertainty in this experiment, the class' values of g were pooled and the standard deviation was found. The standard deviation shows the variation from the mean. The average of these values was found to be 957.7 cm/s², or around 9.58 m/s². The standard deviation of the average was then found by subtracting each experimental value of g by the average value of g (called the deviation), then squaring each deviation and taking the average of all of them. The standard deviation was then found by dividing the average squared deviation by the number of "trials" (9 in this case) then taking the square root. The final experimental value of g was found to be 957.7 +/- 18.1 cm/s². The table of these values is found below:
Table 2: Class' pooled values of g, average g, and standard deviation of the average of g
Conclusion:
As can be seen, the real value of g is more than one standard deviation away from the experimental value of g. Therefore, it can be concluded that the sparker is not a reliable apparatus for measuring and calculating the value of g. Two major assumptions done in this experiment that could have greatly affected the results are the neglecting of friction and other outside forces other than gravity as well as the sparks being exactly 1/60th of a second apart, with the first assumption having a much greater effect. The pattern seen in most of the trial values of g and the average value of g is that it is smaller than the real value of g, which is good proof that gravity is not the only force acting on the freely falling object but also other forces such as friction that combat the force of gravity. This is a systematic error, or an error that results in having results that are either too low or too high, but not both. The assumption that the sparks are 1/60th of a second apart does not affect the results of the experiment by much, if at all, since the spark frequency is quite accurate.
Another assumption made that has some effect on the results of the experiment is that the sparker was dropped the same way each time. However, in reality the sparker does not drop the freely falling object in the same way each time, resulting in more or less outside forces on the object each time. This can be seen as random error, or error that results in results that are both too high and too low.
The point of the part of the lab after finding the experimental value was to minimize any sources of error or uncertainty by pooling the class' data and calculating for the average value of g. It was also done to find the variation and spread of the class data through the calculation of the standard deviation. The standard deviation of the mean was also calculated to show the accuracy and precision of the apparatus used, which was found to not be very accurate. It was also used to find the level of confidence in the mean of the data. The level of confidence is another way of showing the accuracy and precision of the apparatus.
(Note to instructor: The rest of the required questions in the handout are answered throughout the post.)
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