Determining the Acceleration Due to Gravity

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1 Chabot College Physics Lab Scott Hildreth Determining the Acceleration Due to Gravity Introduction In this experiment, you ll determine the acceleration due to earth s gravitational force with three different free-fall methods. We say an object is in free fall when the only force acting on it is the earth s gravitational force. No other forces can be acting; in particular, air resistance must be either absent or so small as to be ignored. When the object in free fall is near the surface of the earth, the gravitational force on it is nearly constant. As a result, an object in free fall accelerates downward at a constant rate. This acceleration is usually represented with the symbol g. Your goals for the exercise are: - Explore the use of graphs and equations for distance vs. time, and velocity vs. time, to determine g, the earth s gravitational acceleration. - Master the difference between precision - measuring carefully and accounting for uncertainty in your measurement - and accuracy - measuring correctly and coming close to the right answer. - Work together in a group setting, dividing up the work fairly and double-checking one another. You will need at least two people to take data, two people to draw graphs using Excel, someone to keep the team on task, and someone to ensure that the write up is completed and turned in on-time. Write up Expectations Turn in one full write up for the team, with all of the expected elements of a physics lab report, on or before two weeks after the lab. Refer to our online lab report guidelines for the elements of a lab report at and in particular, please include: a) the abstract that summarizes your results and error, and the names and lab duties of each of the participants in the introduction. You don t need to include much about the background material or procedure, but do if you end up doing anything different than this handout directs. b) organized and labeled data tables for all experiments (don t recopy if they are clearly done in the experiment itself) c) graphs and calculations of the slope of the velocity vs. time curve that determine the value of g d) analysis of sources of errors and uncertainties in your work. THIS IS THE MOST IMPORTANT SECTION FOR THIS LAB. You are encouraged to work together using our on-line forum, You may do these experiments in any order, and the value of g (actual) to use for comparison and determination of error is 9.81 meters/sec 2

2 Chabot College Physics Hildreth Determining the Acceleration of Gravity Experiment 1: Simple Free-Fall Procedure: 1. Measure the height the ball falls from the edge of the table to the sensor pad on the floor. (Be sure to limit your measurement to the proper number of significant digits, and include your uncertainty as +/- ; for example, cm +/- 0.2 cm) 2. Determine the % Uncertainty in your measurement. Percentage Uncertainty is given for any measurement by: [Uncertainty]/[Measured Value] x 100% For the example: [0.2 cm] / [100.2 cm] x 100% = 0.2% 3. Practice dropping the ball to the sensor pad with the photogate system. You ll have to reset the timer after each drop. After a few practice drops, estimate a reasonable uncertainty in your measurement of the time based on the photogate display AND your reaction time. In other words, what level of precision can you assume for the timer in this experiment? Would it be seconds? Or 0.01 seconds, or even 0.1 seconds?) 4. Now conduct the experiment at least 10 times. Record every value of the time, even those that don t work as well. Calculate the relative uncertainty % in time for each single measurement using the uncertainty in time you estimated for step 3 above. 5. Compute the average time, and the average % uncertainty for this time in this experiment, which will be about the average % uncertainty for any one measurement divided by the square root of n-1 where n = the number of trials. (So if you had 10 trials, with an average of 3% uncertainty, the *average* of those trials would be uncertain by just (1/ 9) *3% = 1% This is the reason you take multiple measurements in an experiment to minimize overall uncertainty, and increase your precision. Note that if you are taking the measurements incorrectly, or if the apparatus you are using is incorrectly calibrated, you still might get the wrong answer! So taking multiple measurements may not increase your accuracy! 6. Determine the value of g. How? What You Know: (a) the distance the ball falls (displacement), (b) the time it takes, and (c) the initial velocity of the ball when it starts; v o = 0. If you know 3 things, you can find the fourth the acceleration! What equation can you use?

3 7. Determine and record the overall percentage uncertainty in your experimental value for g. The overall uncertainty ( g ) in your measurement of g, using the percentage uncertainty in the measurements of time and distance, depends upon both of the uncertainties you estimated in time and distance: Relative uncertainty in acceleration of gravity g/g) = 2 t/t) + ( y/y) Note that the time element s uncertainty appears multiplied by (2) because the equation for acceleration depends upon the SQUARE of the time. So the time variable appears twice, and each time its uncertainty matters. The percentage uncertainty in g will then be an estimate of your overall precision for the experiment: % Uncertainty in g = 100% x g/g = 2 x (% uncertainty in time) + (% uncertainty in distance) Example: If you had a 2% uncertainty in time, and a 3 % uncertainty in distance, your overall experimental value for g would be uncertainty by 7% (2 x 2% + 3%) 8. Calculate the actual percentage error in your results; this is a measure of how accurately your experimental value of g compares with the actual value: [ g (actual) g (experimental)] x 100% / ( g actual) Example: Your experimental value of g was 8.90 m/sec 2 ; the actual is 9.81, and your actual percentage error would be: [ ] x 100% / (9.81) = 9.3% Experiment 1 Questions to be answered in your lab write-up, in the Analysis & Results section. a. When you measure the distance of fall, do you measure from the bottom of the ball to the top of the pad, or from the middle of the ball where the clamp holds it? Why? b. What effect do you think the clamp has on the time of free fall? How can you estimate the significance of this effect, if any? c. What effect do you think human reaction time has in this experiment? d. Discuss the difference between precision and accuracy in this experiment. Compare your actual percentage error with the percentage uncertainty in your measurements.

4 Experiment 1 Sample Data Table Drop Distance: (Height ball falls from edge of table to sensor pad, +/- uncertainty. What units? +/-? Relative % uncertainty in drop distance: % Estimated Uncertainty in Individual Drop Time: t seconds T1: +/- % T6: +/-% Actual Drop Times for each trial, +/- relative % uncertainty in each trial T2: +/- T7: +/- T3: +/- T8: +/- T4: +/- T9: +/- T5: +/- T10: +/- Average Drop Time: % Uncertainty in Average Drop Time: [Average relative uncertainty/ (n-1)] Average g % Uncertainty in Average g : Final result: g (experimental) (include units!) +/- % Experimental % Error

5 Chabot College Physics Hildreth Determining the Acceleration of Gravity Experiment 2: Vernier Data Collection Experiment 2 uses the automated data collection system created by Vernier Corporation, using LoggerPro software to analyze the data. I ll walk you through how the system works, and how you can quickly generate velocity vs. time graphs for a falling object, analyze those graphs for the slope, and establish an experimental value for the acceleration of gravity. In this experiment, you will have the advantage of using a very precise timer connected to the computer and a Photogate. The Photogate has a beam of infrared light that travels from one side to the other. It can detect whenever this beam is blocked. You will drop a piece of clear plastic with evenly spaced black bars on it, called a Picket Fence. As the Picket Fence passes through the Photogate, the computer will measure the time from the leading edge of one bar blocking the beam until the leading edge of the next bar blocks the beam. This timing continues as all eight bars pass through the Photogate. From these measured times, the program will calculate the velocities and accelerations for this motion and graphs will be plotted. Picket fence 1. Observe the reading in the status bar of Logger Pro at the bottom of the screen. Block the Photogate with your hand; note that the Photogate is shown as blocked. Remove your hand and the display should change to unblocked. 2. Click to prepare the Photogate. Hold the top of the Picket Fence and drop it through the Photogate, releasing it from your grasp completely before it enters the Photogate. Be sure you are dropping the fence directly into the bucket below, cushioned with a towel to ensure the fence is not damaged. Be careful when releasing the Picket Fence. It must not touch the sides of the Photogate as it falls and it needs to remain vertical. Click to end data collection.

6 3. Examine your graphs. The slope of a velocity vs. time graph is a measure of acceleration. If the velocity graph is approximately a straight line of constant slope, the acceleration is constant. If the acceleration of your Picket Fence appears constant, fit a straight line to your data. To do this, click on the velocity graph once to select it, then click to fit the line y = mx + b to the data. Record the slope in the data table below. 4. To establish the reliability (and improve the precision) of your slope measurement, repeat Steps 2 and 3 at least five more times. Do not use drops in which the Picket Fence hits or misses the Photogate. Record the slope values in the data table. 5. One easy, but rather rough, way to estimate the uncertainty in the slope (and therefore the uncertainty in g ) is to use the spread in your data from the multiple measurements. Determine the difference between the average and maximum or minimum values, and use that as your measurement of +/- g. A better statistical way to estimate the uncertainty in the result would be to take the standard deviation of the data around the average. (Statistically, for large enough samples, about 95% of the data values will lie within 2 standard deviations from the average.) If you repeated this experiment at least 10 more times (for about 16 data points in all), that statistical estimate would have more applicability. IF you have time, and the inclination, try that, and compare your standard deviation to the range indicated by the maximum and minimum values. Experiment 2 Data Table Trial Slope (m/s 2 ) Acceleration (m/s 2 ) Minimum Maximum Average Acceleration due to gravity, g and estimated uncertainty based on the max/min bracket values. m/s 2 Percentage Error % Experiment 2 Questions e. From your six trials, determine the minimum, maximum, and average values for the acceleration of the Picket Fence. Record them in the data table. f. Compare the precision and accuracy of this experiment with the first, Simple Free Fall. Overall which experiment produced the most precise data? Was that also the most accurate? Explain.

7 Chabot College Physics Hildreth Determining the Acceleration of Gravity Experiment 3: Timed Free-Fall You ll use a spark timer to make marks on a tape as an object falls. The timer will produce a spark every 1/60 th of a second (+/ seconds). Method 3A uses the displacements between each mark as the object falls, and the known time interval to estimate an instantaneous velocity for that interval, that will continue to increase as the object is accelerated downwards by the force of gravity. If you graph this instantaneous velocity at each point vs. time, you can estimate the slope of the line and therefore, the acceleration of gravity. A suggested data table for this experiment is attached, but you are free and encouraged to create your own. Method 3A Procedure: Finding acceleration using INSTANTANEOUS velocities 1. Each team will create their own free-fall data strip using the spark timer. The sparks will mark the tape at constant intervals of 1/60 th of a second. Pick one spot at the top of the tape, label that spot top, and then mark the tape, numbering the spots from the first spot beneath the top to the bottom (from t 1 t i ). 2. Starting near the top-most mark on your tape, measure and record the total displacement from the top mark to each successive mark, and also the interval displacements between each spot ( y i = y i y i-1 ) in a table in Excel (or on paper). For example, measure the displacement between the top and point 1 (y 1 - y 0 ), then from top to point 2 (y 2 - y 0 ), AND measure the displacement from point 1 to point 2, y 2 = (y 2 - y 1 ) etc. Estimate the uncertainty in your displacement interval measurements, and calculate the % uncertainty of each relative to the interval measurement. 3. Graph the instantaneous interval displacement versus elapsed time. Does the slope of this graph (which will be velocity!) change? If it does, the object is accelerating! 4. Calculate the instantaneous average velocity of the falling object at each point, and record in your table: v i = y i / t where t is 1/60 th of a second Graph these instantaneous velocities of the falling object vs. time, and determine the slope of the curve. The value of the slope will approximate the value of g. You can use Excel s statistics functions to determine the value of the slope.

8 5. The uncertainty in your value of g will be influenced both by the uncertainty in time, the average uncertainty in measurements of distances travelled, and the number n of data points you recorded. The uncertainty in time is used twice (once to get velocities, and again to get acceleration); the uncertainty in distance is used once. Since you took a number of distances (n) to get the instantaneous velocities, your average uncertainty in distance is improved by the number of trials, meaning you can be more precise, and will be divided by the square root of (n-1). Here, the percentage uncertainty in your estimate of the value of g from the slope, g 3A will depend heavily upon the uncertainty in your time measurements (because of t 2 ) AND the uncertainty in your distance measurements for each interval. % g A = [2 x % Uncertainty in time measurement] + [(Average % uncertainty in distance)/ (n-1)] Example: IF your relative uncertainty in time was 1%, and the average relative uncertainty in distance was 5%, and you had 26 data points, then the uncertainty in your slope (and derived value of g would be % g A = [2 x 1%] + [5%/(/ (26-1)] = 2% + 5%/5 = 2% + 1% = 3% overall Record both your value of the slope and its uncertainty as g 3A +/- g 3A %. Experiment 3 Question (the MOST important one for the entire lab!) g. Compare your results from this spark time experiment with the photogate, and the freely-falling ball. Which produced more precise estimates of g? Why? Which was most accurate? Why? What effect do you think human uncertainty has in this experiment?

9 EXTRA CREDIT! IF you have time, and desire, you can also analyze your data for part 3 using a different method to determine acceleration. Method 3B uses the total distance fallen and the total elapsed time to estimate an average velocity in each time interval, which can be used to create a graph of velocity vs. time, and from that, acceleration. Method 3B Procedure: Finding acceleration using AVERAGE velocities 6. Measure the total displacement from the first spot to each spot, y i. In other words, go from the top spot to the first spot, and measure the distance (y 1 ). Then go from the top spot to the second spot, and measure the distance (y 2 ). Then from the top to the third spot down, and measure the distance, etc. Record these displacements in a second table within Excel or on paper. 7. Calculate and record the overall average velocity of the falling block at each point, which depends upon the total distance traveled and the total time elapsed to fall that distance: Since v (avg) = ½ (v i + v 0 ) and v avg = (total displacement)/time = (y i )/ t i then v i y i / t i ) - v 0 If you assume you start at rest, v 0 is zero; here, the initial point is made when the bob is already falling, so v 0 won t be zero, but you can still get the slope from a graph! 8. Determine the value of g by graphing these average velocities of the falling block vs. elapsed time, and measuring the slope. The value of the slope will approximate the value of g. You can use Excel s statistics functions to determine the value of the slope. Again, as n increases, your overall uncertainty decreases by the square root of (n-1). But here you have less uncertainty in time because the intervals are getting longer and longer (as the distances get bigger and bigger!) % ga = [(2 x % Average Uncertainty in Time) + (Average % uncertainty in distance)]/ (n-1) Record both the value for gravity and your uncertainty. g A +/- g A %. 8. Calculate the actual percentage error in your results to determine the accuracy of your experiments: [ g (actual) g (experimental)] x 100% / [ g actual] Experiment 3 Extra Credit Question: g. What is the difference in results for g from Method A and Method B? Which is more precise? Which is more accurate? Why?

10 Chabot College Physics Scott Hildreth Free-Fall Tables for Experiment 3 Lab Data Spot # Total Distance from top spot on Tape y i Measurement Data Instantaneous Displacement Between Adjacent Points y i = y i y i-1 Elapsed Time t i Instantaneous Velocity v i = y i / (1/60 th sec) Calculated Extra Credit ONLY Avg. Velocity v i (avg) = y i / t i cm cm sec cm/sec cm/sec +/- cm +/- cm +/- sec Displacement/ Time Interval Distance/ Elapsed Time

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