Free Fall Lab

Objective: 
         This purpose of the lab is to determine the value of g, learn a bit about Excel, and explain the error in doing lab using uncertainty.

The Set-Up:

An 1.5m height electromagnet is set up and hold a free-falling body on the top. The electromagnet is connected to the power and ready to be used. When the free-falling body is released from the top, its fall is precisely recorded by a spark generator. A spark-sensitive tape also attaches to the electromagnet to record the marks created by the spark generator. These marks made at different intervals give us the permanent record of the fall. The apparatus also comes with a heavy tripod base with leveling screws and a weighted clip to anchor the spark paper. Below is the apparatus for this experiment (Figure 1).

Figure 1: The apparatus of the experiment. The spark-sensitive tape attaches to the electromagnet, the tripod at the bottom holds the electromagnet, and the electromagnet is plugged into the power.

Data Collection and Analysis:

        We then turned on the power and let the body fall down freely. As I explained before, we expected to get a series of dots on the tape due to the spark generator. As a result, we did get a tape with a series of dots, seen in figure 2. Then we tried to find the distance of the dots relative to the origin. We lied the tape down below a 2m stick and then measured the distance.

Figure 2: The tape is lied down below the 2m stick in order to measure the distance of these dots relative to the origin.

       When we are done measuring the distance of these dots, we started to put them into Excel. We created five columns: the first one was time, the second one was distance, the third one was ∆x, the fourth one was mid-interval time, and the last one was mid-interval speed. In this step, we learned a bit about Excel. We started name cell A1 Time, enter 0 into cell A2, then used an equation for cell A3=A2+1/60. Then we filled it down up to the number of data we got (we got 20 results for distance, that means we need to fill down up to cell A21). For cell B1, enter Distance, then enter 0 for cell B2, and continued enter data until we reached our last distance. After that, we went on to cell C1, enter ∆x, and establish a formula B3-B2 for cell C2, then fill it down again. For the last two column, we set up equations to calculate the mid-interval time and mid-interval speed. Respectively, we enter D2=A2+1/120 and E2=C2/(1/60). If you do exactly what is explained above, you should get a table like this:

Figure 3: What we set up in Excel with the order from left to right: time, distance, X, mid-interval time, and mid-interval speed. Calculation is also done and the data is ready for the next step.

       Since we already done the calculation, we would plot them to examine the relationship. We first analyze the relationship between mid-interval time and mid-interval speed by plotting them on a scatter graph. We highlighted column D and E, then went to Chart tab, and choose Scatter. The result we got is a straight line like this (Figure 4).

Figure 4: The graph shows the relationship between mid-interval time and mid-interval speed. Based on the graph, we could say that mid-interval sped has a proportional relationship with mid-interval time.

        Going through the same process how to plot a graph, we also examined the relationship between time and distance. Below is our graph time vs distance (Figure 5).

     

Figure 5: The graph shows the relationship between time and distance. We could see that as time increases, the distance increases as well.

        The reason why we plotted two graphs was because we would like to attain the value of acceleration based on two different slope. For example, for the first graph (mid-interval time vs mid-interval speed), the slope m=939.74 divided by 100 gave us the value of acceleration g=9.3974m/s^2. On the other side, the graph of time vs distance gave us the half value of acceleration. The slope of the graph m=474.14 time 2 then divide by 100 give us the value of acceleration (g=9.4828m/s^2).

          After plotting two graphs and finding out the relationship between mid-inteval time and mid-interval speed, as well as time and distance, we then moved on to examine the effect of experimental uncertainty. We all know that our result has an uncertainty. The uncertainty affected our result. It is shown in this equation

Measurement = best estimate + or - uncertainty.

          Then we learned two ways to calculate the difference between expected and experimental values. One was absolute difference and the other was relative difference. How calculation done is shown in the figure 6.

Figure 6: Two methods to calculate the difference between experimental value and accepted value.

          We go further to find out the standard deviation of the mean, which is a good way to express of the uncertainty of the result. We then were introduced to the 68-95-99.7 rule for the standard deviation. Basically, the rule states that the smaller the date spreads out, the more accurate the result is. That means that the result is more accurate within the first standard deviation (68%), while the accuracy reduces within two standard deviation (95%) or three standard deviation (99.7%).
           Analyzing the class’ data for g is the last thing we did in the lab. We opened up a new Excel window and asked other groups to get their value for g. We then entered the value of acceleration into column A. In column B, we put "deviation from mean" and column C is the square of deviation. 
           Deviation from mean is calculated by using this equation:

Deviation = the instantaneous value of acceleration - the average value of acceleration.

Deviation square = deviation from mean * deviation from mean

          After we calculated value of deviation from mean and dev^2, we then filled it down like the way we did before. The results we got was this table (Figure 7)

Figure 7: The class data for acceleration and the calculation is done to show the standard deviation of the mean of the class's data.

Conclusion:
           In this lab: free fall body, we examined the value of acceleration using the Excel program, as well as analyzing the class's data for acceleration. Before the experiment, we did make some assumptions such as no friction or sparks are exactly 1/60th second apart in order for the experiment to run smoothly. For the first part, we ended up getting two value of acceleration based on the slope of graph time vs distance and mid-interval time vs mid-interval speed. Then we moved on to find out the effect of experimental uncertainty. We came up with two methods to find the differences between expected and experimental values. One was absolute difference and the other was relative difference. Also the standard deviation of the mean and the rule 68-95-99.7 were also introduced. We finally analyzed the class's data for acceleration and ended up with the standard deviation of the mean of the class's data.