THE
ACCELERATION OF GRAVITY:
EXPERIMENTS
WITH A FREE FALLING OBJECT
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video of the experiment
RATIONALE:
When you complete this experimental exercise you will have demonstrated that a freely falling body falls with a constant acceleration of about 9.81 m/s2. This, of course, you already know. Our purpose in including this activity in your program of study is much broader.
First, as an experimental activity it is relatively straight forward so that we can concentrate on some of the important technical aspects without getting lost. Second, it illustrates an important analytical method which enables us to make a complete determination of the speed and acceleration histories of a body from a knowledge of its position as a function of time. The procedures involved are readily extended to rotational motion and to motion with non-uniform acceleration.
SPECIFIC OBJECTIVES:
When you have completed this experimental activity you should be able to: (1) given a record of the position of an object at various times, construct an x(t) graph for the object; (2) derive the v(t) graph and the a(t) graph for the object from the x(t) graph; and (3) give the physical interpretation of the slopes and intercepts of the various curves produced.
EXPERIMENTAL ACTIVITY:
A strip of paper with a weight attached to one end will be allowed to drop from rest. The acceleration of the paper strip will be analyzed. The free-fall apparatus consists of a blower and a spark generator. The purpose of the blower is to reduced the friction between the apparatus and the strip of paper. The spark generator will mark on the strip of paper at regular known intervals. The frequency of the spark generator can be varied. The setting for this experiment should be 60 Hz. Verify that the spark generator is set to this setting.
Your instructor will demonstrate the operation of the apparatus at the beginning of the lab period, and answer any questions that you have about the apparatus.
Stretch the data tape out on the work table, select a mark on the trace of the stylus near the beginning and mark it as zero. Do not use any of the marks in the cluster at the beginning of the tape. Record in a data table the position of every mark all the way down the tape. Always measure from the point marked zero.
ANALYSIS OF RESULTS:
Prepare a graph showing the position of the falling object in distance versus time. Plot time in units of 1/60th second. Remember to follow the suggestions for proper graphing that are in the beginning of this manual.
The points for the position - time curve should form a smooth curve (but not a straight line). Now, using a straight edge and a sharp pencil, draw tangent lines at four widely separated points on the curve. (Another way to do this is to import the Excel graph into Paint or another picture editing program, and draw lines with your computer.) The slope of the tangent line shows the instantaneous speed of the body at the time represented by the point of tangency. Therefore, you are able to compute the instantaneous speed at four times during the fall when you compute the slopes of the four tangent lines you have drawn.
Determine the slopes of each of the four tangent lines. Follow the procedure outlined in the introduction of this manual. REMEMBER THAT YOUR TIME UNIT IS 1/60th SECOND. On the face of the graph, neatly show the slope calculation in the form:
slope = D x/D t = 20 cm/(4/60 s) = 300 cm/s
Since this is only an example, your numbers will likely be different. Show the slope calculation for each of the four tangent lines in the same way. Now take a second sheet of graph paper and plot the four speed values as a function of time. The time scale of the second graph should be exactly the same as the first. Remember, the time values for the second graph will be taken from the tangent POINTS, and the speed values will come from the tangent SLOPES.
If your work has been carefully done, the four data points in the second graph will fall nearly on a straight line. If they do not, check your work in the calculation of the slopes, and if necessary, do the computations for a fifth point from the first graph. Now, draw the BEST straight line through the data points of your second graph.
The acceleration of a body moving in a straight line is given by the slope of the speed - time curve. Therefore, to find the acceleration of the object in your experiment, you need to find the slope of the line in your speed vs. time graph. Again show the calculation on the graph, complete with proper units.
A second way of determining acceleration is using the position – time graph. Add a trendline of the form polynomial of order 2 (a quadratic equation), that is, of the form y = ax2 + bx + c. Compare this quadratic equation with the kinematic equation for the position of an accelerating object x = xo + vot+ ˝ at2. It should be clear that the a, b, and c values in the quadratic equation correspond to the physical values of xo, vo, and (˝ a). Use your trendline equation to calculate the acceleration of a freely falling object. It should be near the value you calculated using the speed vs. time graph. Inconsistency in the two values represents inherent error in drawing trendlines and calculating slopes.
The acceleration of a freely falling object should be about 9.81 m/s2. Compare the two values you get from your experimental data with this accepted value by finding the percent error for each value.
FINAL SUMMARY:
In your report summary, include your values for g. How do these value compare with the standard value? What does this comparison indicate about this laboratory procedure? What conclusions can you draw from this lab?
QUESTIONS: