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Class: R @ 9:30 a.m

Lecture Instructor: Dr. Biswas

Lab Instructor: Dr. Michael Zelin

Dates:

Lab performed: 09/29/16

Report submitted: 10/05/16

PHYSICS

Lab 4: Free Fall

Laboratory Manual & Lab Report Template

by Mackenzie Tate Krista Cortez

RaShelle Goldman

Content Page Assessment

points: basic max novelty max 1. Abstract 1 _ 10 2. Objective 2 3. Introduction 2 _ 10 _ 20 4. Apparatus & Materials 2 _ 20 _ 20 5. Procedure & Results 3 _ 30 _ 20 6. Analysis & Summary 5 _ 10 _ 10 7. Questions 10 _ 5 _ 10 8. References 10 _ 10 _ 10 9. Acknowledgments & Credits 10. Appendices 12 _ 5 _ 10

Abstract

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Arkansas State University Chemistry & Physics Department

1. Objective: Investigation of a free fall to measure gravitational acceleration. 2. Introduction The International Committee on Weights and Measures has adopted as a standard value for the acceleration of a body freely falling in a vacuum g = 9.80665 m/s2 [1]. The actual value of g varies as a function of elevation, h (in meters), and latitude, f, in SI units, m/s2, according to the Helmert’s equation:

g = 9.80616 - 0.025928 cos(2 f) + 0.000069 cos2(f) - 3.086x 10-4 h

The strength of the gravitational force on the standard kilogram at 42o latitude is 9.80 N/kg , and the acceleration due to gravity at the sea level is therefore g = 9.80349 m/s2 for all objects. At the equator, g = 9.78 m/s2 and at the poles g = 9.83 m/s2. (This is because the radius of the Earth is larger at the equator than it is at the poles by about 26.5 km , and because the Earth rotates at 2 radians per day introducing an apparent repulsive force that flattens the spherical shape) [1].

Various techniques have been developed to measure gravitational acceleration [1 – 7] since the time when Galileo Galilei conducted his famous experiments (see Appendix I). In this Lab, we will use a Picket Fence Free Fall through a photogate technique [6] and use these kinematic formulas for velocity, v, and position, y: v(t) = gt + vo and y(t) = ½gt2 + vot + yo

3. Apparatus and Materials: transparent ruler that has opaque stripes spaced 5 cm apart, photogate from Pasco, Inc, LabQuest Mini data acquisition box (plug in the LabQuest Mini to the USB port on the computer and the photogate into the LabQuest Mini) (Fig. 1).

Figure 1 Experimental setup

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4. Procedure and Results:

To measure gravity, we will be dropping through a photogate a transparent ruler that has opaque stripes spaced 5.0 cm apart. As each stripe passes through the gate, the computer will note the time of the event. These photogates work by passing an infrared beam across an opening to a receiver unit. If something crosses the opening, it stops the beam from reaching the receiver, which causes the photogate to send a signal to the computer to signify that an object has entered the area. When the object leaves the opening, the beam is once again picked up by the receiver, which causes the photogate to send another signal to the computer to let it know that the object has left the area. In this manner, the photogate is able to communicate the timing of the movements of an object, which the computer is able to time and track. At the conclusion of the experiment, it will display these arrival times, along with the position of each stripe on the ruler, velocity, and acceleration.

1. Run the Logger Pro software from the icon on your desktop and click on the small clock icon in the row of buttons near the top of the program. Set the samples/millisecond to 30…100 and the duration to 1500…2000 millisecond (Fig. 2) – depends on the original distance of the picket fence from the photogate.

2. Press the Start Data Collection button on the LabQuest unit. After ~0.5…1s, drop the picket fence through the gate. Make sure that the fence does not hit the gate as it falls.

3. Make sure that the position, velocity, and acceleration versus time graphs are on the screen. If the data appears to be correct (the picket fence fell through the photogate without hitting the side or falling off to the side), then take screenshots for your report (Fig. 3).

4. Copy the data into an Excel spreadsheet. Convert time from milliseconds to seconds and velocity from m/ms to m/s. Plot velocity as a function of time. Add a linear treadline and display the regression equation and R2 values - take screenshots for your report (Fig. 4)

5. Find acceleration by using three methods you learned in the Lab 2. Kinematics (Fig. 4):

i. average acceleration - a ratio of change in velocity to the corresponding time interval ii. mean acceleration - sum of acceleration values from LoggerPro divided by the number

of measurements iii. acceleration from the regression analysis (treadline) - a coefficient in front of the

variable (time)

Figure 2 Logger Pro home screen

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6. Choose the measured value, gmes, which is the closest to the expected, gexp =9.8 m/s2, and calculate the percent error: PE=|gmes - gexp|/gexp x100%

7. Repeat steps 1-4 four more times with the ruler starting at different heights above the photogate.

Figure 3 Logger Pro data: position, velocity, and acceleration as a function of time

Figure 4 Excel data and velocity as a function of time graph

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8. Calculate the mean and the standard deviation of the acceleration values by using data from different runs: fill Table 1.

Table 1 Gravity acceleration measurements

Run 1 2 3 4 5 Mean Standard Deviation

Acceleration, m/s2

9.962 m/s^2

9.475 m/s^2

10.028 m/s^2

9.258 m/s^2

9.641 m/s^2

9.673 m/s^2

0.32479

*See Appendix I table 1 for calculations

5. Analysis and Summary

The following screen shots are from our trials 1-5. The first is of our logger pro data and the second is a

graph representing our logger pro data in an excel file. We did many trials; however, picked our best 5 to

be displayed in the report.

Logger Pro Trial 1: We started the trial 26.5in above the table.

Excel File Trial 1:

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We copy/pasted our logger pro data form trial 1 into excel and created a graph. Then we fitted our graph with a line of best fit along with the R^2 value and the equation of the line. We then calculated the average acceleration, the mean acceleration, and reported the acceleration from the graph. Then we found the percent error. See Appendix I trial 1 for calculations.

Logger Pro Trial 2: We started the trial 27in above the table.

Excel File Trial 2: We copy/pasted our logger pro data from trial 2 into excel and created a graph. Then we fitted our graph with a line of best fit along with the R^2 value and the equation of the line. We then

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calculated the average acceleration, the mean acceleration, and reported the acceleration from the graph. Then we found the percent error. See Appendix I trial 2 for calculations.

Logger Pro File Trial 3: We started the trial 29in above the table.

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Excel File Trial 3: We copy/pasted our logger pro data from trial 3 into excel and created a graph. Then we fitted our graph with a line of best fit along with the R^2 value and the equation of the line. We then calculated the average acceleration, the mean acceleration, and reported the acceleration from the graph. Then we found the percent error. See Appendix I trial 3 for calculations.

Logger Pro File Trial 4: We started this trial 30in above the table.

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Excel File Trial 4: We copy/pasted our logger pro data from trial 4 into excel and created a graph. Then we fitted our graph with a line of best fit along with the R^2 value and the equation of the line. We then calculated the average acceleration, the mean acceleration, and reported the acceleration from the graph. Then we found the percent error. See Appendix I trial 4 for calculations.

Logger Pro File Trial 5: We started the trial 31in above the table.

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Excel File Trial 5: We copy/pasted our logger pro data from trial 4 into excel and created a graph. Then we fitted our graph with a line of best fit along with the R^2 value and the equation of the line. We then calculated the average acceleration, the mean acceleration, and reported the acceleration from the graph. Then we found the percent error. See Appendix I Trial 5 for calculations.

Dr. Zelin also asked us to include the following video to show that he is improving in communicating with his students. He asked every table nicely if they needed any help with their lab: https://youtu.be/M7UNn_iJGYM

6. Questions (Replaced #1 with our own):

1.How high is a building if it takes 5 seconds for a ball to hit the floor?

Pretend the object was dropped from the rooftop and a man was holding the ball at a height of 1 meter before letting it go g= 9.8 m/s^2.

2. A professor teaching similar Lab at the University of South Alabama found that in Mobile,

AL, g = 9.793394 m/s2 [6 ]. Can you find g value in Joneseboro, AR, if its elevation is

d=(gxt^2)/2d=(9.8m/s^2*(5)^2)/2d=122.5m

122.5m-1m=121.5m

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79 m and the latitude is 35.8422222°?

fromtravelmath.com

3. Similar Labs are conducted at many schools [1-6]: what question(s) would you ask your peers?

• Their opinion on the most accurate way to drop the stick • How accurate of a result they got • Timed dropping the stick and clicking start correctly

7. References

1. MIT, Falling Object Lab

2. University of Rochester, Acceleration of Gravity Lab

3. Carl Martikean, Illinois Institute of Technology, Free Falling Lab

4. University of Southern California, Falling Body Experiment

5. University of Tennessee, Acceleration of Gravity

6. University of South Alabama: Freely Falling Bodies Lab

7. Vernier Software & Technology, Picket Fence Free Fall

8. MoonConnection.com, http://www.moonconnection.com/moon_gravity.phtml

8. Acknowledgments: This Lab is based on original materials developed by Dr. Bruce Johnson in 1999 and updated by Steven Hoke in 2014. Dr. Koushik Biswas, Dr. Bin Zhang, and Dr. Brent Ross Carroll contributed to the discussion.

9. Credits: Numerous materials developed at various schools and organizations, including MIT [1], University of Rochester [2], etc. were used to leverage the best resources on the topic. Students are strongly encouraged to submit their materials and get citation credits: names will be mentioned in the final Lab manual, which you can reference on your resume.

9.79781 Calculated using:

http://www.sensorsone.com/local-gravity-calculator/

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Appendix I. Historical Background

”What goes up, must come down.” You have probably heard this phrase at some point in your life (and if you have not up until this point, then you have now). People use it to describe situations in which something popular becomes unpopular or someone with wealth becoming poorer. It originated from the

observation that any object thrown into the air eventually falls back to the ground. It is not a new saying, nor is it a new observation. The earliest humans noticed this. They just did not know why or how it happened. One of the more famous explanations as to why and how things fell was given to the world by the Greek scholar Aristotle (384-322 B.C.). He believed in the concept of teleology, which is the philosophy that the explanation of an occurrence was found in its final causes. Applied to the motion of objects, Aristotle believed that object always wanted to return to their beginnings where they could be with like objects, i.e. a rock that was from the ground wanted to return to the ground to be amongst other rocks. At the time, this was as good of an explanation as any other that had been given, and since it emanated from the mind of Aristotle, who was held in such regard, it was accepted as the correct explanation for over a thousand years in the Western

world. Aristotle did not stop at just giving a reason for why things fall down. He also went on to describe the motion. In particular, he stated that heavier object fall faster than lighter objects.

For instance, he posited that an object that was twice as heavy as another object would fall twice as fast. If you dropped the two at exactly the same time, then the heavier object would hit the ground at the same time that the lighter object had only fallen half of the distance. This, too, was accepted by the public, based somewhat on the authority of Aristotle and somewhat on observations. For instance, if you drop a light piece of paper and a heavy rock at the same time, the rock hits the ground well before the paper. Today, we know that the cause of this phenomenon is air resistance. However, it was not until the early 1600’s that Galileo Galilei showed that this theory of heavier objects falling faster was wrong. He did this by running experiments wherein the weight of the object did not affect the rate at which the object moved. Legend has it that he dropped two different sizes of cannonball from the Leaning Tower of Pisa (Galileo was a professor at the University of Pisa), which both hit the ground at the same time. Unfortunately, while this story has a certain amount of romance to it, there is no evidence that Galileo ever did any such thing, although he does discuss theoretically dropping cannonballs to disprove Aristotle.

The experiments that Galileo performed to study gravity were done with a ball rolling down an inclined plane, rather than a ball falling through the air. The reason for this was quite simple: Galileo wanted to time how long it took the ball to move a given distance, and the clocks of his time did not allow for accurate measurements on such short time scales as a second or so. By having the balls roll down an inclined plane, Galileo lengthened the time over which the balls moved. Using the most accurate pendulum and water clocks of his time, he was able to make fairly accurate measurements of the elapsed time. These measurements allowed Galileo to do more than to show that Aristotle was wrong. He was able to test and discern mathematical relationships between various factors, which led to the first accurate equations about the motions of objects. What Galileo found was that, under the influence of gravity, the distance that a ball travels from rest is proportional to the square of the amount of time that the ball is allowed to travel.

Aristotle

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This picture painted in 1841 by G. Bezzuoli, attempts to reconstruct an experiment Galileo is alleged to have made during his time as lecturer at Pisa. Off to the left and right are men of ill will: the blasé Prince Giovanni de Medici (Galileo had shown a dredging-machine invented by the prince to be unusable) and Galileo's scientific opponents. These were leading men of the universities; they are shown here bending over a book of Aristotle, where it is written in black and white that bodies of unequal weight fall with different speeds. Galileo, the tallest figure left of center in the picture, is surrounded by a group of students and followers [3].

Table 1 Calculations:

Standard deviation was calculated using an online calculator located at: https://www.easycalculation.com/statistics/standard-deviation.php

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Trial 1 Percent Error Calculation:

Trial 2 Percent Error Calculation:

Trial 3 Percent Error Calculation:

Trial 4 Percent Error Calculation:

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Trial 5 Percent Error Calculation:

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Appendix II. Supplemental Materials

Figure A1 Equipment used