Acceleration of Gravity Introduction: In this experiment, several objects' motion are studied by making several measurements of the objects position (or displacement) at different times. Since the objects will be either dropped or tossed straight up, their resulting 1-D motion should be described by a constant acceleration directed downwards. This exercise actually consists of several distinct parts, allowing for several independent measurements of the acceleration of gravity, and a final challenge to test the students' understanding of constant acceleration as it applies to free fall. The analysis for this lab is primarily graphical: the computer can automatically produce plots of position, velocity and acceleration versus time from measurements of position at various times. It is important to note that only the position and time data are "real" in that actual measurements will be made of positions x 1, x 2,... x n at certain times t 1, t 2,...t n and that the position versus time graphs are made by plotting x i versus t i. Velocity and acceleration estimates are calculated from the position-time data. For example, the velocity estimates are calculations of the average velocity over each interval: v i = x i 1 x i = x i t i 1 t i t i The acceleration estimates (complicated a bit by intervals which may not be constant) would be given by a i = v v i 1 i t i 2 t i /2. Computers are excellent tools for doing repetitive calculations like these. In this lab, such calculations will be performed automatically by the software programs used in the exercises. t 3, x 3 x t 1, x 1 t 2, x 2 v 1 =slope= x 2 x 1 t 2 t 1 Figure 1: Velocity and Accelerations Estimates from Position- Time Data There are a number of ways to extract the value for the acceleration from a set of data. The average of all the acceleration estimates provides a measurement of the acceleration. The slope of velocity graph is the acceleration, and the use of built in "curve fits" 1 (fitting a straight line to the data) in programs can used to obtain a value for acceleration. 1 A "curve fit" takes a model such as a quadratic equation y=ax 2 +Bx+C and a set of data {x i,y i}and "tweaks" the parameters (A, B and C in the case of the quadratic) until the model closely matches the data. One standard approach is to choose parameters so that to sum of the distances from the data points to the model curve is minimized. Many data analysis programs such as spreadsheets and graphing programs have built in curve fitting tools. t v 2 =slope= x 3 x 2 t 3 t 2 a 1 =rate changeof slope = v 2 v 1 t 2 t 1 /2
Activity 1: Measurement of Gravity with a Photogate and Picket Fence start/stop button Figure 2: The Photgate and Picket Fence The apparatus consists of a photogate (mounted to a stand) plugged into port one of the computer interface, the a piece of plastic with alternating opaque and transparent stripes. The acc_g.ds file should be used to start the interface program. The figure below corresponds to the initial setup of the file. Start/Stop Button Graph 1: Position Graph 2: Velocity Graph 3: Acceleration Figure 3: Data Studio Window Each data collection run consists of three actions: press start, drop the picket fence, press stop. The resulting data graphs for position, velocity and acceleration are displayed in three different windows. In each of these windows there are important controls for selecting data sets ( ), performing curve fits ( ) and other statistics ( ) on the data. There is also an autoscale function ( ) which automatically selects the range for the graph axis so that all data is displayed. These should be done in a timely manner so that all three runs will fit nicely on a single graph. After each run, use the autoscale button ( ) to see if the position data looks acceptable. If so, go to the acceleration data window and write down the mean acceleration (found towards the bottom of the last column of the data table). This is your experimental value of g for that run. Repeat for three runs. Put
all three runs of data on the same graph, autoscale and print out that combined graph. Your data for this part therefore consists of one combined plot and three values for g. Compute the average of these three values to compare with the accepted value for g. Activity 2: Measurement of Gravity from Video For this activity, we will be using the Java applet Tracker to analyze video clips. These video clips show objects which are thrown upwards and accelerate under the influence of gravity. Using the Tracker video analysis software, the position of the objects can be tracked. Using standard length objects in the videos (usually a meter stick), the position of the object(s) can be measured as a function of time. Tape Measure player controls Figure 4: The Tracker Window, With a Video Opened for Analysis In order to aid your analysis, maximize Tracker's window. After you have loaded the video file for analysis, you can "zoom" in by right clicking on the video and selecting zoom and then the level of zoom desired. Zooming "to fit" will cause the video to be enlarged to completely take the available space in the window. Select the tape measure icon in menu bar to activate the tape measure. The tape measure consists of a line with an arrow at each end and a number which corresponds to the physical distance between the endpoints. Drag one end of that arrow to the top of the image of the meter stick in the video, the other end to the bottom of the meter stick. After you have carefully aligned the tape measure line with the meter stick, click on the number that is next to the tape measure and edit it so that it has a value of one. This sets the scale for distance for this video. If the appearance of the tape measure in the window creates difficulty for later analysis you can click on the tape measure icon again to hide the tape
measure. To set the Tracker program to track an object in the video, in the menu select Track: New: Point Mass. (Repeat this as many times as is necessary if you need to track more than one object.) Use the player controls to set the video to the first frame that has the object to be tracked in plain view (free of its launcher). With the mouse pointer over the video, hold down the shift button (this brings up a special mouse pointer), carefully line up the cursor with the object to be tracked and click the left mouse button. This will cause the Tracker program to record where on the screen you clicked and to automatically advance the video to the next frame. Repeat the recording of the object's location for the entire free trajectory of the objects flight (do not record the objects position at landing!). At this point the raw data has been recorded, and analysis can begin. From the menu select Window: Right View which will open the (up till now) hidden panel on the right which consists of a graph on the top and a menu on the bottom. Figure 5: Using Tracker's Graphing and Data Analysis Within the window containing the graph, click on the vertical axis label x and select y for a plot of vertical position versus time. Once the graph is displaying y versus t correctly you can double click on the graph to bring up the data set tool.
Figure 6: Tracker's Data Set Tool One you have the Dataset Tool window open, select the Fits check box at the top of the window, select Parabola from the list at the bottom left part of the window and record the curve fit parameters a, b and c. Our model for constant acceleration under the influence of gravity is given by y= y 0 v 0 t 1 2 g t 2 and so by equating the quadratic terms of the fit and the model the acceleration of gravity is related to the fit parameters by g = 2a. Right click on the graph in the dataset window and select Snapshot. This will open a snapshot of the graph in a new window from which the graph can be printed. You results for this activity will include the analysis of two videos (as indicated by the instructor) and your results should include a value of the acceleration of gravity as obtained from each video as well as a printout of the y versus t graph. Activity 3: What's Wrong with This Motion? The final activity for this lab is also a video analysis exercise using tracker. However, in this exercise you will be analyzing an "artificial" video containing four objects. Only one of the four objects has motion that corresponds to a constant acceleration of gravity, the other three have aspects of their motion which does not "fit". Your task is to identify which of the four objects obeys the law of gravity. You need to specify (and document with printout of appropriate graphs) what is wrong with each of the other three objects.