Lab 3: Acceleration of Gravity

Transcription

1 Lab 3: Acceleration of Gravity The objective of this lab exercise is to measure a value for g, the acceleration due to gravity for an object in freefall. For Lab 1 and Lab 2 we used data, from a fictional story of astronauts measuring a rock in freefall, to create a data table and graph. We also found the slope of the best-fit line for the data; the slope represented the acceleration (i.e. slope of a velocity vs. time graph) of the rock in freefall. This acceleration was due to gravity on the distant planet the astronauts visited. For Lab 3, we will repeat the process except we will use data from the motion of a real object in freefall in our classroom. After measuring the data, you will create a data table and corresponding graph, and find the best-fit line. The slope of the best-fit line is a measure of the acceleration of the object, which should be the acceleration due to gravity. The astronauts had a camera that took a picture every ¼ second; they then used the pictures to determine the height of the rock. We would like to replicate the astronauts method, but we have a small problem: the acceleration of our object in freefall will be about six times greater than that of the astronaut s rock (i.e. gravity is stronger on Earth than on that distant planet.) This means we would need to take six pictures every ¼ second, or about 24 pictures per second to create the same amount of data that the astronauts made. We do not have a camera to use for our data collection, but we do have something else: electricity. The electrical system in the United States is alternating current, which means the direction of the electrical current changes direction at a very specific rate. This rate is 60 Hz, or 60 cycles per second. Since the current has to stop for an instant before reversing direction, this essentially means that the current turns off and on 60 times per second. (This also means that conventional light bulbs essentially turn on and off 60 times per second, but this is much to fast for our eyes to detect... so the light intensity seems to remain constant.) How can we make use of this feature of our electrical system? Our freefall apparatus consists of a small tower from which we will drop a object with a metal ring. As the object falls, the ring almost makes contact with a pair of wires that are connected to a high voltage power supply; there is a small gap between the ring and the wires. The power supply essentially turns on and off 60 times per second, so every 1/60 of a second a spark will jump the gap between the ring and the wires. We will cover one wire with a thin strip of waxed paper; each spark created by the ring will burn a small hole through the paper. The paper will have a series of small holes that precisely show the position of the ring every 1/60 of a second. In this way, the positions of the holes can be used to create data similar to what the astronauts collected from their pictures.

2 There are three tasks you will need to complete for this lab: 1. Measure your data with your lab partner. 2. Modify your Excel file from Lab 2, using your new data. 3. Collect results from the class to determine the class average. Measuring Data You and your lab partner will have a strip of tape with dots that were created by the sparks of the falling object. The object took about ½ a second to fall, so there should be about 30 dots (60 per second) on your tape. You will not use every dot for three reasons: 1. Measuring all 30 dots would be tedious. 2. Occasionally the apparatus misfires, failing to spark; a dot could be missing 3. The time between dots, 1/60 second, is an awkward decimal. Instead, you will measure every third dot. This addresses all of the three issues. You will only measure approximately 10 dots. If a dot is missing, there is a 2 out of 3 chance that you wouldn t need it anyway. And the time interval between measured dots is 3/60 second, or 0.05 second. Your tape will look like this: The large dot at the beginning is not part of your data; only the small dots are to be used. The large dot was created when the object was at rest at the top of the tower. Only dots created when the object was in motion are valid for your data. Survey your entire tape. Make sure you can see the entire series of dots. If you suspect any dot is missing, ask me. Circle the first dot and label it y 1 = 0. Circle every third dot, labeling them y 2, y 3 etc. Carefully measure the position of each dot, i.e. the distance from zero. Measure in cm, with precision to the nearest 0.05 cm (i.e. to the nearest half millimeter.) Write your measurements directly on the tape.

3 Note: the smallest divisions (i.e. each tiny line marking) on a meter stick are in millimeters. You can reasonably see if a dot is between lines, i.e. on half-millimeter, or on a line. But you cannot get better resolution than this. So all of your measurements should end with either a zero or a 5 in the second decimal place. After measuring, your tape should look like this: Remember: your measurements are positions on a y-axis. All measurements are the distance from zero to each dot and not the gap between adjacent dots. Create a Worksheet for Lab 3 in your Excel file Your Excel file from last week should include two data tables (one for the original data, one for the data to calculate the slope & intercept of the best-fit line), a graph and a calculations section. If these were completed properly, all of these sections are linked to the original measured data through the equations you entered. Add a new worksheet to your workbook by clicking the tab to the right of the Lab 1 & 2 label (near the bottom left corner of the screen.) Rename the new sheet Lab 3 Copy and paste all of your work from Lab 1 & 2 onto the new sheet labeled Lab 3: o Highlight your entire Lab 1 & 2 sheet by clicking the small blank box in the upper left corner where the row numbers meet the column letters. o Press Ctrl and C, or right click and choose Copy. o Click the Lab 3 tab at the bottom; you should see your blank Lab 3 sheet. o Click the little box in the upper left corner to highlight the entire sheet. o Press Ctrl and V to paste. You should now have an exact copy of your Lab 2 work on the Lab 3 sheet. Delete the measured data (i.e. the first two columns of data) from the first table and replace it with your new data. Note that the time for each dot increases by 0.05 from the previous dot. Change the title in your first data table; change the labels (to cm ) where necessary.

4 Delete the excess rows in the first data table. Reformat your graph: change the title, axis labels and the range on each axis to match your new data. Change the cell references for the graph data to the data on the Lab 3 sheet. Delete the equation of the best-fit line on the graph and then add the equation again when your graph is correct. Delete the excess rows in the second data table and change the labels where necessary. Change the values displayed in the equations in your calculations section. Collect Results from the Class We expect that the slope of your graph, which represents the acceleration of the object in freefall, should be 980 cm/s 2 (Note that we are measuring in cm, so all of our calculations will be in cm.) We will find, however, that everyone in the class will measure something slightly less than this value. Why do we get a result that does not match expectations? There are always two reasons that our results could fail to match expectations: random error and systematic error. Random error is an unavoidable consequence of measuring any data. The simple fact is that no measurement can be exact ; there is always a limit to the precision and accuracy of any measurement. Those limits depend on the measurement process; the limitations of the measuring device is always part of this process. In our lab today, using a meter stick means that our precision is limited to the nearest half-millimeter. The effect of random error can be minimized by taking advantage of the fact that it is random ; that is, we expect an equal number of measurements to be slightly higher and slightly lower than actual. The more measurements we take, the better these random errors will cancel when we average our measurements. Our measurements will always carry with them an uncertainty, but that uncertainty decreases with more measurements. Systematic error is simple in concept but can be very difficult to deal with. If, after accounting for random error (which can be minimized with repeated measurements), our results do not meet expectations, we have to acknowledge that there was a problem with either our expectations or with the system we used to take our measurements. We will deal with random error in today s lab by collecting results from everyone in the room and averaging them for a more accurate class result. The slope of the best-fit line of your graph represents the acceleration of the object. You should find that your data points form a nearly perfect

5 straight line. This indicates the high accuracy with which you have measured the acceleration of the object. But while we expect the acceleration to be 980 m/s 2, you will likely find it is between 970 and 977, i.e. a little slower than expected. You might think this is just random error, since it is a tiny bit less than 980. If random error is an issue, we can reduce it by averaging your value with those of the rest of the class. We can also calculate the standard deviation of the mean, which serves as a measure of uncertainty in the average value. In other words, the SDOM indicates the effect of random error on your average value. Create a small section somewhere on your Excel sheet in which to list the results from the class. Give the section a title. Within that section, label a box Average and calculate the average value next to it. Label another box SDOM and calculate the SDOM next to it. Use the Excel functions AVERAGE and STDEV for these calculations. The SDOM is the STDEV divided by the square root of the number of values used in the average. You will likely find that your average is around 974 and the SDOM is around 1. This indicates that the class, collectively, measured the acceleration of the falling object to be slightly less than the expected value of 980. The only explanation is that a systematic error exists in our work. So now we have to ask... what could have caused our falling object to accelerate slightly slower than expected?