The purpose of this lab is to measure the gravitational acceleration constant *g* by measuring the rate at which a falling object increases its speed.

In this lab, you will drop a ruler through a photo gate. From this we can infer the rate at which the ruler will accelerate due to the earth’s gravitational force. The clear plastic ruler will be marked at regular intervals with a masking tape that will block the light beam of the photo gate and turn on and off the timer which sends the results to the computer. The times are recorded by the computer and displayed on the monitor. Using the distance between successive pieces of masking tape and these times, the computer will perform the calculation of the average velocity of the ruler in these intervals during its fall. The results can be displayed graphically on the monitor in various instructive ways.

We need to get the interval *d* between two successive leading edges of the tape pieces, i.e. one piece of tape and one blank spot.

To be more accurate, measure the distance *D* from the first leading edge to the last one on the ruler and divide it by the number of intervals d between them.

As well as measuring *D*, we need to make an estimate in our error for our measurement of *D*. Things you should take into account are the accuracy of your measurement tool, and how straight your masking tape is. Fill in the table on your worksheet with your values for *D,d* and their respective errors.

Connect the photo gate output to the interface box by plugging its cable into the first socket of the interface box. Turn on the computer and check the system by following the following instructions: in the main menu use the up-down keys to highlight “Motion Timer” on the screen and then press “Enter”. Push the “ SET” button on the interface box to permit the flow of data to the computer.

Test the photo gate: block the photogate beam with your finger and see the red light on the cross bar of the photogate turn on.

Now hold the ruler just right (with the first masking tape edge close to the light beam) above the photo gate and then drop it through the photo gate so that the tape on the ruler interrupts the light path of the photo gate. Make sure that the ruler is oriented perpendicular to the light beam during its fall. Push the “STOP” button on the interface box after the ruler has fallen. A table of time intervals will appear on the computer screen after you hit “Enter” on the keyboard. Make sure that there are ~8 measurements of time in your table. If there isn’t, then repeat the ruler dropping until you obtain a table with ~8 time measurements. Hit “Enter” again and select “Graph Data” from the menu. Then select “Velocity vs Time” from the new menu. Then select “Other (specify the length)” and input your value of *d*. Leave the menu “Select Graph Style” by hitting “Enter” and select “Automatic Scaling, Axis Starts at 0” from the new menu for both horizontal and vertical scale. You see the velocity vs time graph on the screen. Leave it by hitting “Enter” and select from the new menu “Display Table of Values”. Copy the values in to you lab notebook.

On the measurement computer, you will now plot distance , velocity and acceleration vs time and sketch these graphs into your lab notebook. Hit “Enter” to exit the table and select “Return to Data Analysis Menu”. Then choose “Graph Data” again. Obtain the following three plots from the computer and sketch them roughly in your lab notebook:

- distance vs. time
- velocity vs. time
- acceleration vs. time

In case you have a suppressed zero for the acceleration plot, do the following: hit “Enter“ to exit the plot, then choose “Change Scale and Regraph”, hit “Enter” to exit from the “SELECT GRAPH STYLE” screen. Select, as you did above, “Automatic Scaling, Axis Starts at 0” for both horizontal and vertical scale.

For your notebook copies of the three plots above, draw a smooth “best-fit” line through the data in each of the plots.

We are now going to make a plot of *v* vs. *t* to determine the value of *g*. In this plot we need to have estimates for our error in *v*. The error in *v* comes from the error in our measurement of *d* and you should convince your self that the **relative** errors in *d* and *v* are the same and use this to calculate the error in *v*. We will consider that errors in *t* are too small to worry about (the computer is very accurate, or at least much more accurate than you or I are at measuring a length with a ruler!). This means you do not need to enter anything in to the *x* error box and you should select the option that there are errors only in *y*. Do you think this graph should go through (0,0) (or in other words at *t=0*, should we have *v=0*)?

Make your plot of *v* vs. *t* using the plotting tool. The slope of the graph should be equal to *g*. How does the value you obtain compare to accepted value of 9.81 m/s^{2}?

In the approach we took above we estimated the error in *g* based upon the propagation of what we expected to be the most significant source of error in our experiment. Another approach we can take is to make several measurements, and we will do this using the built in fitting tool in the measurement program. This tool, unlike the one you just used, will **not** take into account the uncertainty in your input values. We will instead estimate the error in our measurement using Equation E.5b from the error manual to estimate the error in an average.

You are going to drop the ruler through the photo gate as you have done earlier. You have already one value of *g*. In order to get this value from the computer you go to “Velocity vs. Time”. From the new menu select “Other (specify the length)” and check whether your value of d is still correctly stored. Hit “Enter” and from the “SELECT GRAPH STYLE” menu choose “Statistics” and turn it ON by hitting the space bar. Hit “Enter” and choose “Automatic Scaling, Axis Starts at 0” for both horizontal and vertical scale, hit “Enter” to get the graph. At the top of the screen read the value of “M” and record it in your workbook. After each drop, you only need to record this computer-generated value of the acceleration. Perform 4 more drops of the ruler so that you can obtain a total of five values for g.

Find the average and error of the average of these values and compare both with your previous experimental value and the established value of *g*.

Hold the ruler a few inches above the photo gate and then drop it through the photo gate. Then look at the computer generated plot of velocity vs. time and record the initial velocity *v _{0}* of the graph. Is your initial velocity greater or less than the one you recorded earlier?

Although you have now completed the required part of the lab you may be interested in using another approach for measuring the acceleration due to gravity on a free falling object.

In this experiment we produced a specialized object so that we could measure it's acceleration using a photogate. It is also possible to use sound to measure the time taken for objects to fall, by having weights tied at defined distance on a string. This approach is nicely illustrated at Physclips (Section 2.2)

However, if we want to measure the acceleration of an arbitrary object we can take a video of it and process it so that we can see the position of the object in each frame (which at 30 frames per second is every 0.033 seconds). The tool to do this is here. (Note: Currently this tool will only work if you are using the Firefox browser.) If you use the plotting program on that page it will automatically fit the data to a quadratic relationship, the x^{2} coefficient of this fit should be equal to $\frac{1}{2}$*g*.

How does this value compare to the value you got from the ruler drop? If you do this part of the lab don't forget to paste your plot in to you lab notebook!

Note: This tool can be used with a variety of videos. The main restriction is that it helps if the background to the moving object is not changing and the video is taken in good light. If you have a video of something moving that you think would be interesting to analyze please let me know!