25-Feb-2015: Finding the acceleration due to gravity of a free falling body.

     The purpose of this experiment was to figure out if a falling object will accelerate at 9.8 m/s^2 in the absence of all other forces except gravity. In order to accomplish this, an apparatus 1.86 meters tall with a spark generator was used. An object was held at the top of the apparatus by an electromagnet. When released, the fall was recorded by the apparatus by recording a spark every 1/60th of a second on spark-sensitive tape. Once the experiment was over, the tape was removed and placed on a table. 

     To get the data, a meter stick was used to measure the distance between the dots recorded on the spark tape (shown below).

Measuring distances between dots with a meter stick.

     The data was then plotted in excel and graphed (shown below).

     Doing a curve fit of the above graph gives an equation in the form of x=v0t+(1/2)at^2. This shows the acceleration of the falling object was 2 times 476.22, or 952.44 which translates to 9.52 m/s^2. To confirm the data, the mid-interval time and mid-interval speed was plotted in excel and graphed. The mid-interval time was the time between each 1/60th of a second and the mid-interval speed was the change in distance from each recorded point divided by 1/60th of a second (data shown below).

     Above is the graph of the mid-interval speed vs mid-interval time. A linear fit of the graph gives an equation of the best fit line in the form of v=at+vo. This shows that the acceleration of the falling object is 932.73 which translates to 9.32 m/s^2.  

     The results were not quite what were hoped for. The accepted value for the acceleration due to gravity is 9.8 m/s^2. The experimental values from this particular experiment were 9.52 m/s^2 and 9.32 m/s^2, from the distance vs time graph and the mid-interval speed graph, respectively. This is due to error in the experiment. There are two types of error that could be accounted for in an experiment. Systematic error and random error. Systematic error is due to things you can blame on a flaw in your assumptions. In this experiment there were assumptions made such as assuming no air resistance and no drag from the spark generator. Random errors are things we cannot tie down the blame for. Since there were definitely systematic errors in this experiment, data from nine other experiments was collected. This was done to find a range in which a person can be confident the correct value is in. This is called standard deviation. 

     There are several steps to take the standard deviation of data: 1) find the average of all the data (last number in column "g values" in the data below). 2) find the deviation of each data point from the average number found in step one (column "dev from mean"). 3) square each deviation (column "dev^2). 4) find the average of the summation of the square of the deviations (last number in the column "dev^2"). 5) take the square root of the number found in step 4 (number found on the bottom right).

     The table above shows the standard deviation to be about 32. This means we are 68% confident that the true value of the acceleration due to gravity is between 9.20 m/s^2 and 9.84 m/s^2. 

There are uncertainties in every experiment and this one is no exception. As you can see, every data point in the table above was not the accepted value of gravity. This can be due to systematic error, bad measurements with the meter stick, or random error. All that can be done is find better equipment to lessen the uncertainty or find other methods to experiment with.