## II. Measuring g, Using Sparky

We will study the motion of a freely falling body and, in particular, measure the acceleration due to gravity. With the apparatus supplied, an object is allowed to fall freely and its positions at the ends of successive time intervals are recorded on a strip of paper by means of electric sparks. The experiment will first be demonstrated for the class.

The falling body B is initially suspended by the electromagnet M. When the body is released, it falls parallel to two vertical wires W, one of which, W1, is covered with a strip of paper P as shown in the top view to the left.. A spark timer provides a large electrical voltage across the wires every 1/60th of a second. The voltage difference is not sufficient to allow a spark to jump the full space between the wires, but a spark can jump from W2 through the conducting ring on the body B, and through the paper P to wire W1. Such a spark will leave a mark on the paper strip and thus record the position of the falling body every 1/60th of a second.

Data and Analysis:

1. Create a spreadsheet for the data. Remember to include a title for your spreadsheet and labels for each column of numbers. Place the paper strip on a flat surface. Circle each of the spots with a pencil and number them 0, 1, 2, 3, etc. Enter these spot numbers in the first column of the spreadsheet. The spot numbers are the number of 1/60th second time intervals which have elapsed since the initial spot was made.
2. Place a meter stick on the paper strip such that the graduated edge lies along the line of spots. The spot you have numbered 0 does not need to be located at the end of the meter stick. Keeping the meter stick stationary, read the position of each spot (distance from spot 0) and record the value in the second column. Remember to report your measurements to 1/10th of the smallest scale division.

Following the instructions on the Excel Hints handout, or using the online help, create a graph of position versus spot number. (This is equivalent to position versus time.) As expected, your graph is not a straight line. But we do expect a linear relationship between speed and time.

3. Compute the average speed of the falling object during each 1/60th second time interval:
4. average speed = displacement / time interval.

Note that the displacement is not the same as position, it is the displacement for each interval of time. In the third column, at the cell in the same row with the spot numbered 0, enter the formula

= (y1 - y0) / (1/60),

where y1 and y0 are the addresses of the cells containing the positions of spot numbers 1 and 0, respectively. Then use the Copy command to complete the corresponding calculations for the entire column. Don't use the cell in the same row with your last numbered spot.

5. The time we associate with an average speed is at the midpoint of the time interval. Calculate time in the fourth column using the formula
6. = 1/120 + 1/60*(spot number),

where (spot number) is the appropriate cell designation.

7. Create a graph of average speed versus time. Does your graph approximate a straight line? If air resistance is negligible, a falling object would experience a constant acceleration due to gravity and, therefore, its instantaneous speed versus time graph would have a constant slope. (If the speed increases linearly with time, the average speed over the time interval corresponds to the instantaneous speed at the midpoint of the interval.)
8. To determine the acceleration due to gravity we need to find the slope of the line which best fits the data in this graph. Use the spreadsheet command linest: select time as the independent variable and average speed as the dependent variable. This command and the output are described in the Excel Hints handout.
9. The y-intercept b is in this case the average speed at time t = 0 and the slope m is the acceleration. Do you see why?

How does your slope compare with the accepted value for the gravitational acceleration, g = 9.80 m/s2?

To display the best-fit line on the graph you first need to make another column in the spreadsheet which contains points that lie on the line using the trend command. Plot this column of data on the graph as a connected line with no symbols. Now you can see how closely the data points lie to the best-fit line. In fact, the scatter of the data points around the best-fit line is due to the presence of random error. How could you use this to report a 90% confidence interval for the slope or the y-intercept? This will be one of the topics we will discuss in next week's lab.

Title your graph, label the axes, and print it.

10. Print the spreadsheet for your notebook.
11. What are some possible sources of error in this experiment?