#$&*
PHY 121
Your 'rubber band calibration' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
#$&* Your initial comment (if any): **
2/21/12 2:00 PM
#$&* first line ruler markings, distance in actual cm between ends, how obtained: **
1h30m
#$&* The basis for your uncertainty estimate: **
Copy this document, from this point to the end, into a word processor or text editor.
Follow the instructions, fill in your data and the results of your analysis in the given format.
Regularly save your document to your computer as you work.
When you have completed your work:
Copy the document into a text editor (e.g., Notepad; but NOT into a word processor or html editor, e.g., NOT into Word or FrontPage).
Highlight the contents of the text editor, and copy and paste those contents into the indicated box at the end of this form.
Click the Submit button and save your form confirmation.
In this experiment you 'calibrate' six rubber bands by measuring their lengths when stretched by varying forces. You will obtain for each rubber band a table of force vs. length, and you will construct force vs. length graphs for four of the six bands. These rubber bands will be used in subsequent experiments.
Most students report that this experiment takes between 2 and 3 hours; some report times of less than 1 hour, some report times in excess of 4 hours. This version of the experiment defers analysis of two of the six bands and should require about 15% less time than the version on which these reports are based.
Taking Data for Calibration:
Note: You should not stretch any of the marked rubber bands more than 35% beyonds its maximum unstretched length. If you stretch a rubber band beyond this length you will permanently distort it. This means, for example, that if a rubber band is 8 cm long you should not stretch it by more than 2.8 cm, to a maximum length of 10.8 cm.
Important: Throughout the course you will be using the rubber bands and the calibration graphs you make here, so be sure you keep the rubber bands and the graphs in a place where you can locate them, and be sure the graphs are clearly labeled so you know which one goes with which rubber band.
For this experiment you will use one of the plastic bags that came with your lab materials and the dominoes from the packet, along with a ruler, paper clips and marked rubber bands.
You have a bundle of thin rubber bands and a pack of over 100 thicker rubber bands. You will use rubber bands from the pack.
Pick at random six of these rubber bands from your lab kit. If any of the selected rubber bands have obvious flaws, discard then and replace with other randomly selected bands. Preferably using a permanent marker, put 1, 2, 3, 4, 5 and 6 marks on the respective rubber bands, so you can easily identify them later.
Using paperclips bent into the shape of hooks, form a 'chain' of all six of your marked rubber bands (a chain of two rubber bands is shown below). Be sure you observe which is which, and when you record data make sure that the individual rubber bands are clearly identified by the number of marks.
Hang the plastic bag from the chain.
Place one domino in the bag.
Measure as accurately as possible the length of the topmost of your rubber bands. Be sure you keep track of which is which.
Measure from one end of each rubber band to the other. You will therefore be recording the positions of both ends of each rubber band. Be sure you measure the end-to-end distance, from the point where one end of the rubber band ceases and the air beyond the end begins, to the similar point at the other end.
You should not attempt to align the end of your measuring device with either of the positions you are recording. Rather align one of the markings (e.g., the 10.0 cm marking) on your measuring device with one end of the rubber band, see what marking corresponds to the other end, and record both markings.
To get the most precise measurement possible you should use a reduced copy of a ruler. To make sure the measurement is also accurate, you should take into account any tendency toward distortion in the corresponding part of that copy. You can choose whichever level of reduction you think will give you the most accurate and precise measurement.
In the box below, indicate in the first line the ruler markings of both ends of the first rubber band, entering two numbers in comma-delimited format.
In the second line indicate the distance in actual centimeters between the ends, to an estimated precision of .01 cm..
In the third line explain how you obtained the numbers in the second line, and what the meaning of those numbers is. Also indicate how this rubber band is marked, and the limits within which you think your measurement is accurate (e.g., +- .03 cm, indicating that you believe the actual measurement to be between .03 cm less and .03 cm greater than the reported result).
Your answer (start in the next line):
55.5cm,129.8cm
74.3cm
I subtracted one endpoint from the other to obtain a total length of the rubber band. I believe my data is accurate to within +-.05cm
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Explain the basis for your estimate of the uncertainty of the length of the first rubber band.
Your answer (start in the next line):
I think I can accurately estimate to within half of 1/10 based on the markings of my ruler.
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Measure as accurately as possible the lengths of the remaining rubber bands. Be sure you keep track of which is which. You may move your measuring device from one rubber band to the next.
In the space below enter the ruler markings of the ends of the first rubber band, delimited by commas, in the first line (this will be the same information you entered in the first line of the last space ), the ruler markings of the ends of the second rubber band on the second line, etc., until you have a comma-delimited line for each rubber band.
Then put the word 'End' in the very next line.
Follow this in the very next line by a comma-delimited line containing the numerical distances in cm, each estimated to within .01 cm, of the rubber bands in your chain.
Follow this by a line indicating the markings on the rubber bands.
Finally indicate the uncertainty in your measurements, which should probably be the same as the uncertainty as that given in the preceding space .
Your answer (start in the next line):
55.5,129.8
172.0,244.8
288.0,361.5
405.5,481
523,596.5
639.5,712
End
74.3,72.8,73.5,75.5,73.5,72.5
1,2,3,4,5,6
I still believe my uncertainty to be +-.05cm
#$&*
Add another domino to the bag and repeat your measurements. The positions of the ends should be recorded in your lab book, and should be backed up electronically in a way you can easily interpret at any future date (a comma-delimited text file or a spreadsheet file would be good; a tab-delimited file would also work but tabs can be variable and invisible so if you are going to use a text file, a comma-delimited is probably the better choice).
You won't enter the endpoint information here, but as cautioned above be sure you have it so if the information reported here has any anomalies, you can go back to your raw data and correct them.
Determine the distances in centimeters between the ends of each rubber band, and enter them in the space below, in the same order you entered them in the preceding space . Use one line and use comma-delimited format.
In the second line indicate that these results were from the weight of two dominoes.
Your answer (start in the next line):
75,73.3,74.2,77,75.5,73.5
Lenghts for two dominoes
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Continue adding dominoes and measuring until one of the rubber bands exceeds its original length by 30%, or until you run out of dominoes, then stop. To keep the time demands of this experiment within reason, you should beginning at this point adding two dominoes at a time. So you will take measurements for 4, 6, 8, ... dominoes until the 'weakest' of your rubber bands is about to stretch by more than 30% of its original length, or until you run out of dominoes.
If one rubber band reaches its limit while the rest are not all that close to theirs, remove this rubber band from the experiment and modify your previous responses to eliminate reference to the data from this band. However, keep the band and keep your copy of its behavior to this point.
In the space below, enter on the first line the actual lengths in cm of your rubber bands when supporting four dominoes, in comma-delimited format. Enter in the same order you used previously.
On the second line enter the number 4 to indicate that this result is for four dominoes.
On the third line enter in comma-delimited format the lengths in cm when supporting 6 dominoes.
On the fourth line enter the number 6 to indicate the six dominoes being supported.
Continue in this manner until you have entered all your lengths and numbers of dominoes.
Then on the next line enter 'End'.
You may then enter any brief identifying information or commentary you wish. However since the nature of the information has been defined by previous spaces, this is optional.
If you have reason to believe the uncertainty in your measurements has changed, indicate this also. Otherwise it will be assumed that your previous uncertainty estimates apply.
Your answer (start in the next line):
77,75.5,77,79.5,76,74
4
79,77,78.5,76,79,76
6
81,78,79,81,79.5,78.5
8
82.5,80.5,81,81.5,80.5,80.5
10
End
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Compiling and Graphing your Data
Each domino is pulled downward by the Earth's gravitational field. Each rubber band resists this force by stretching out, which creates a tension equal and opposite to the force exerted by the Earth (each rubber band also supports the rubber bands below it, but the rubber bands don't weigh much so we neglect that weight). The force exerted by the Earth on each domino is about .19 Newtons.
Make a table of the force exerted by each of the first four rubber bands vs. the length of the rubber band. You do not need to do this with all six, but you should retain the last two rubber bands and your data for those two, in case you have need of them in later experiments.
Make a force vs. length table for each of these four bands. The length will go in the first column, the force in the second. Your graph will be of the type shown below, but you probably won't have quite as many data points; your forces will also differ from the forces indicated by this graph.
wpe3.jpg (9057 bytes)
There is a tendency for students at the beginning of a physics course to connect graphs point-to-point. This is a usually a very bad idea in physics, since there are experimental uncertainties in our data and we learn nothing by following those uncertainties around. The graph below is an example of this Bad Idea.
Note also the REALLY bad idea, which is to treat the 'origin' as if it is a data point. In this example, we never measured the force at the 8 cm length, and there is no justification at all for using the 'origin' as a data point (actually the point where the axes come together in this graph is not the origin, it's the point (8 cm, 0); the origin would be (0 cm, 0) and is well off the scale of this graph ).
wpe7.jpg (12151 bytes)
It is a good idea to add a smooth curve to the data. This is because we expect that force will change smoothly with rubber band length. However we acknowledge that errors might occur in our data, so we never attempt to make the smooth curve pass through the actual data points, though we don't try to avoid them either.
In the example below the curve wobbles around from point to point instead of smoothly following the trend of the points.
wpe8.jpg (11743 bytes)
In the next example the curve doesn't try to 'hit' each data point, but rather to follow the pattern of the actual force vs. length. It passes among the data points, remaining as smooth as possible and coming as close as possible to the data points without making unsightly 'wobbles' in an attempt to pass through specific data points.
wpe9.jpg (13268 bytes)
In the space below give your table in a series of lines.
The first line will contain, in the previous order, the lengths the rubber bands supporting 1 domino, separated by commas, followed by the downward force exerted by gravity on 1 domino ( i.e., the number, indicating .19 Newtons). You can copy most of this information (all except the .19) from a previous space .
The second line will contain, in the previous order, the lengths the rubber bands supporting 2 dominoes, separated by commas, followed by the downward force exerted by gravity on 2 dominoes. Again you can copy most of this from a previous space .
Continue in this manner until you have all the lengths and downward forces, in the same comma-delimited syntax described above.
Follow your data with a line containing the word 'End'.
In subsequent lines specify the meaning of each column of your table, the units and the quantity measured in each.
Your answer (start in the next line):
74.3,72.8,73.5,75.5,73.5,72.5,.19
75,73.3,74.2,77,75.5,73.5,.38
77,75.5,77,79.5,76,74,..76
79,77,78.5,79.5,79,76,1.14
81,78,79,81,79.5,78.5,1.52
82.5,80.5,81,81.5,80.5,80.5,1.9
End
Columns 1-6 are the lenghts in mm of the rubber bands for 1,2,4,6,8, and 10 dominoes. The final column is the force in Newtons exerted on the rubber bands by the dominoes
#$&*
If you haven't already done so, construct a graph for each rubber band and fit a smooth curve that you think best depicts the actual behavior of that rubber band.
In the space below describe the shape of the curve you drew to approximate the force vs. length behavior of first rubber band. The curve in the last figure above could be described as 'increasing at a decreasing rate, then increasing at an increasing rate'. Other possible descriptions might be 'increasing at an increasing rate throughout', 'increasing at a decreasing rate throughout', 'increasing at an increasing rate then increasing at a decreasing rate', etc.).
Then describe the shapes of all six rubber bands. Follow your last description by a line containing the word 'End'. You may if you wish add comments starting on the next line.
Your answer (start in the next line):
increasing at a decreasing rate and then increasing at an increasing rate
increasing at a decreasign rate and then increasing at an increasing rate
increasing at a increasing rate, then increasing at a decreasing rate
increasing at a increasing rate, then constant, then increasing at an increasing rate
increasing at an increasing rate, then increasing at a decreasing rate, then increasing at an increasing rate
increasing at an increasing rate, then increasing at a decreasing rate
End
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Estimating Forces
We can now use our curve to estimate the force at a given length, or to estimate the length that will give us a specified force.
In the figure below we estimate the force for the 9.5 cm length.
From the data point it might appear that the force corresponding to 9.5 cm is about 1.5 Newtons. However we're going to put our trust in the curve.
We project a line from the L = 9.5 point on the horizontal axis, straight up to the curve, then straight over to the F axis.
Reading the point on the y axis as F = 2.6 or maybe F = 2.7 we see that the curve gives us a force between 2.6 and 2.7 Newtons.
If our curve has been drawn carefully and if it appears to make good sense then we believe that the curve is more reliable than our data points, and we will tend to believe this estimate more than our data point.
Similarly we use the curve to estimate the length that gives us a force of 2 Newtons.
We project a horizontal line from the F = 2 point on the vertical axis to the curve, then from this point we project vertically downward to the horizontal axis.
We read a length of about 10.4 cm. Again we use the curve, which 'averages out' the characteristics of several data points, to estimate the required length.
wpeA.jpg (9880 bytes)
If you haven't already done so, include in your report a table of your data for force vs. length for each of the four selected rubber bands.
Now for the first rubber band, sketch your best smooth curve, the one you believe best shows the real force vs. length behavior of a rubber band. Describe your curve and describe your thinking about how to construct the curve.
Use your curve for the first rubber band (the one with 1 mark) to do the following:
Estimate the force in Newtons corresponding to a length of 9.8 cm and report the number in the first line of the space below.
Your answer (start in the next line):
3.6
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Estimate the length in cm of a rubber band that gives a force of 1.4 Newtons and report the number in the second line.
Your answer (start in the next line):
8.8
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From the curve estimate the force in Newtons corresponding to each of the lengths you actually observed. For example, if you observed lengths of 8.7, 8.9, 9.3, 9.8, 10.1 cm with 1, 2, 4, 6 and 8 dominoes, what forces would be predicted by the curve for each of these lengths? Give your estimates in the first line, using comma-delimited format. In the second line indicate by how much the estimate of the curve differs from the actual weight supported.
Your answer (start in the next line):
.15,.34,.8,1.21,1.6,1.82
-.04,-.04,+.04,+.07,+.08,-.08
#$&*
From the curve estimate, using or your first graph, report in comma-delimited format, in the first line, the length corresponding to each of the forces .19 N, .38 N, .76 N, 1.14 N, etc.. In the second line indicate in comma-delimited format by how much each of these lengths differs from the length you actually observed when the rubber band was resisting this force.
Your answer (start in the next line):
74.5,75.3,76.8,78.5,80.6,83
+.2,+.3,-.2,-.5,-.4,+.5
#$&*
Which do you have more faith in, the values from the curve you just created or the values you reported in your table, and why?
If you were to estimate a force for a given length using one of your graphs, what do you think would be the uncertainty in that force (e.g., +- .12 N, or +- .03 N, etc.) and what is your evidence for this estimate?
Your answer (start in the next line):
I think the values from the table are more accurate because they were measured with a fairly accurate ruler and not estimated from a handful of data points and a hand drawn best fit curve.
+- .06N. I think these falls in line with the inaccuracies from what I measured to my graph.
#$&*
If you were to estimate a length for a given force using one of your graphs, what do you think would be the uncertainty in that length (e.g., +- .05 cm, or +- .13 cm, etc.) and what is your evidence for this estimate?
Your answer (start in the next line):
I believe it would be with +-.5mm. My data and my graph were within this threshold for all measurements.
#$&*
Your instructor is trying to gauge the typical time spent by students on these experiments. Please answer the following question as accurately as you can, understanding that your answer will be used only for the stated purpose and has no bearing on your grades:
Approximately how long did it take you to complete this experiment?
1h30m
Please copy your completed document into the box below and submit the form:
Copy this document, from this point down, into a word processor or text editor.
Follow the instructions, fill in your data and the results of your analysis in the given format.
Regularly save your document to your computer as you work.
When you have completed your work:
Copy the document into a text editor (e.g., Notepad; but NOT into a word processor or html editor, e.g., NOT into Word or FrontPage).
Highlight the contents of the text editor, and copy and paste those contents into the indicated box at the end of this form.
Click the Submit button and save your form confirmation.
This lab exercise is based on the observations you previously made of a ball rolling down ramps of various slopes. We further investigate the relationship between ramp slope and acceleration.
The mean time reported to complete this exercise is 2 hours. The most frequently reported times range from 1 hour to 3 hours, with some reports of shorter or longer times.
Note that there are a number of repetitive calculations in this exercise. You are encouraged to use a spreadsheet as appropriate to save you time, but be sure your results check out with a handwritten analysis of at least a few representative trials.
Document your data
For ramps supported by 1, 2 and 3 dominoes, in a previous exercise you reported time intervals for 5 trials of the ball rolling from right to left down a single ramp, and 5 trials for the ball rolling from left to right.
If in that experiment you were not instructed to take data for all three setups in both directions, report only the data you were instructed to obtain.
(Note: If you did the experiment using the short ramp and coins, specify which type of coin you used. In the instructions below you would substitute the word 'coins' for 'dominoes').
Go to your original data or to the 'readable' version that should have been posted to your access page, and copy your data as indicated in the boxes below:
Copy the 10 trials for the 1-domino setups, which you should have entered into your original lab submission in the format specified by the instruction
'In the box below, give the time interval for each trial, rounded to the nearest .001 second. Give 1 trial on each line, and give the 5 trials for the first system, then the 5 trials for the second system. You will therefore give 10 numbers on 10 lines.'
In the 'readable' posted version this data will follow the boldfaced heading
'5 trials each way 1 domino'
Enter your 10 numbers on 10 lines below, and on the first subsequent line briefly indicate the meaning of the data:
------>>>>>> ten trials for 1-domino setups
Your answer (start in the next line):
2.188
2.184
2.090
2.070
2.109
2.121
2.152
2.144
2.070
2.109
The first five lines are from the intitial trial and the second five are after reversing the ramp
#$&*
Enter your data for the 2-domino setups in the same format, being sure to include your brief explanation:
On the 'readable' posted version this data will follow the boldfaced heading
'5 trials each way 2 dominoes'
------>>>>>> 2 domino results
Your answer (start in the next line):
1.422
1.496
1.383
1.469
1.355
1.449
1.465
1.480
1.469
1.453
The first five lines are from the intitial two domino trial and the second five are after reversing the ramp
#$&*
Enter your data for the 3-domino setups in the same format, including brief explanation.
On the 'readable' posted version this data will follow the boldfaced heading
'5 trials each way 3 dominoes'
------>>>>>> 3 domino results
Your answer (start in the next line):
1.230
1.184
1.184
1.121
1.152
1.219
1.137
1.188
1.219
1.188
#$&*
Calculate mean time down ramp for each setup
In the previous hypothesis testing exercise, you calculated and reported the mean and standard deviation of times down each of the two 1-domino setups, one running right-left and the other left-right.
You may use any results obtained from that analysis (provided you are confident that your results follow correctly from your data), or you may simply recalculate this information, which can be done very quickly and easily using the Data Analysis Program at
http://www.vhcc.edu/dsmith/genInfo/labrynth_created_fall_05/levl1_15\levl2_51/dataProgram.exe\
In any case, calculate as needed and enter the following information, in the order requested, giving one mean and standard deviation per line in comma-delimited format:
Mean and standard deviation of times down ramp for 1 domino, right-to-left.
Mean and standard deviation of times down ramp for 1 domino, left-to-right.
Mean and standard deviation of times down ramp for 2 dominoes, right-to-left.
Mean and standard deviation of times down ramp for 2 dominoes, left-to-right.
Mean and standard deviation of times down ramp for 3 dominoes, right-to-left.
Mean and standard deviation fof times down ramp or 3 dominoes, left-to-right.
On the first subsequent line briefly indicate the meaning of your results and how they were obtained:
------>>>>>> mean, std dev each setup each direction
Your answer (start in the next line):
2.128,0.05455
2.119,0.004219
1.425,0.01365
1.463,0.01249
1.174,0.04071
1.190,0.03354
Lines 1 and 2 are the mean and standard deviation for the one domino setup. Lines 2 and 3 indicate the same data for the two domino setup. The same data for lines 4 and 5 is indicated for a 3 domino setup.
#$&*
Calculate average ball velocity for each setup
Assuming that the ball traveled 28 cm from release until the time it struck the bracket, determine each of the following, using the mean time required for the ball to travel down the ramp:
Average ball velocity for 1 domino, right-to-left.
Average ball velocity for 1 domino, left-to-right.
Average ball velocity for 2 dominoes, right-to-left.
Average ball velocity for 2 dominoes, left-to-right.
Average ball velocity for 3 dominoes, right-to-left.
Average ball velocity for 3 dominoes, left-to-right.
Report your six results in the box below, one result per line, in the order requested above.
Starting in the seventh line explain how you obtained your results, giving the details of how you obtained at least one of your results. These details should include the definition of the average velocity, and should explain how you used the mean time and the distance down the ramp to arrive at your result, and should show the numbers used and the numbers obtained in each step.
------>>>>>> ave velocities each of six setups
Your answer (start in the next line):
13.158
13.214
19.649
19.139
23.850
23.529
These are the vAve computed as `ds/`dt or 28cm/`dt. The `dts are indicated from the mean times calculated from the six different trials.
vAve = 28cm/2.128sec = 13.158cm/sec
28cm/2.119sec = 13.214cm/sec
28cm/1.425sec = 19.649cm/sec
28cm/1.463sec = 19.139cm/sec
28cm/1.174sec = 23.850cm/sec
28cm/1.190sec = 23.529cm/sec
#$&*
Calculate average ball acceleration for each setup
Assuming that the velocity of the ball changed at a constant rate in each trial, use the mean time interval and the 28 cm distance to determine the average rate of change of velocity with respect to clock time. You will determine your results in the following order:
Average rate of change of ball velocity with respect to clock time for 1 domino, right-to-left.
Average rate of change of ball velocity with respect to clock time for 1 domino, left-to-right.
Average rate of change of ball velocity with respect to clock time for 2 dominoes, right-to-left.
Average rate of change of ball velocity with respect to clock time for 2 dominoes, left-to-right.
Average rate of change of ball velocity with respect to clock time for 3 dominoes, right-to-left.
Average rate of change of ball velocity with respect to clock time for 3 dominoes, left-to-right.
Report your six results in the box below, one result per line, in the order requested above.
Starting in the seventh line explain how you obtained your results, giving the details of how you obtained at least one of your results. These details should include the definition of the average rate of change of velocity with respect to clock time and should explain, step by step, how you used the mean time and the distance down the ramp to arrive at your result, and should show the numbers used and the numbers obtained in each step.
------>>>>>> ave roc of vel each of six setups
Your answer (start in the next line):
12.367
26.428
39.298
38.278
47.7
47.058
aAve = `dv/`dt
Since v0 is 0cm/sec, we know that vf must be equal to 2*vAve. vf/`dt = a
26.316cm/sec/2.128sec = 12.367cm/sec^2
26.428cm/sec/2.119sec=12.472cm/sec^2
39.298cm/sec/1.425sec=27.578cm/sec^2
38.278cm/sec/1.463sec=26.164cm/sec^2
47.7cm/sec/1.174sec=40.63cm/sec^2
47.058cm/sec/1.190sec=39.545cm/sec^2
#$&*
Average left-right and right-left velocities for each slope
For the 1-domino system you have obtained two values for the average rate of change of velocity with respect to clock time, one for the right-left setup and one for the left-right. Average those two values and note your result.
For the 2-domino system you have also obtained two values for the average rate of change of velocity with respect to clock time. Average those two values and note your result.
For the 3-domino system you have also obtained two values for the average rate of change of velocity with respect to clock time. Average those two values and note your result.
Report your results in the box below, giving one average rate of change of velocity with respect to clock time per line, in the order requested. Starting the the first subsequent line, briefly indicate how you obtained your results and what you think they mean.
------>>>>>> ave of right-left, left-right each slope
Your answer (start in the next line):
12.4195
26.871
40.0875
These are the average accelerations for the 1,2 and 3 domino setups. They are the result of averaging the initial ramp setup as well as the reversed setup.
#$&*
Find acceleration for each slope based on average of left-right and right-left times
Average the mean time required for the right-to-left run with the mean time for the left-to-right run.
Using this average mean time, recalculate your average rate of velocity change with respect to clock time for the 1-domino trials
Do the same for the 2-domino results, and for the 3-domino results.
Report your results in the box below, giving one average rate of change of velocity with respect to clock time per line, in the order requested. In the subsequent line explain how you obtained your results and what you think they mean.
------>>>>>> left-right, right-left each setup, ave mean times and give ave accel
Your answer (start in the next line):
12.393cm/sec^2
27.214m/sec^2
39.812cm/sec^2
These are the aAve for a mean clock time using the original and reveresed `dt. They were calculated by dividing the `dv by the mean `dt.
26.316cm/sec/2.1235sec = 12.393cm/sec^2
39.298cm/sec/1.444sec=27.214m/sec^2
47.058cm/sec/1.190sec=39.812cm/sec^2
#$&*
Compare acceleration results for the two different methods
You obtained data for three basic setups, each with a different slope. Each basic setup was done with a right-left and a left-right version.
You previously calculated a single average rate of change of velocity with respect to clock time for each slope, by averaging the right-left rate with the left-right rate.
You have now calculated a single average rate of change of velocity with respect to clock time for each slope, but this time by using the average of the mean times for the right-left and left-right versions.
Answer the following questions in the box below:
Since both methods give a single average rate of change of velocity with respect to clock time, would you therefore expect these two results to be the same for each slope?
Are the results you reported here, based on the average of the two mean times, the same as those you obtained previously by average the two rates? Are they nearly the same?
Why would you expect that they would be the same or nearly the same?
If they are not exactly the same, can you explain why?
------>>>>>> ave of mean vel, ave based on mean of `dt same, different, why
Your answer (start in the next line):
I would not expect the exact same results for each method of calculation, though I would expect them to be very similar. The data backs this up, particularly in the one domino setup with a very slight difference in the two aAve. The other two setups were not quite as similar, but still very close together. I would imagine the differneces to be within the percent error since we were using two different mean values to calculate the aAve.
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Associate acceleration with ramp slope
Your results will clearly indicate that, as expected, acceleration increases when ramp slope increases. We want to look further at just how the acceleration changes with ramp slope.
If you set up the ramps according to instructions, then the ramp slopes for 1-, 2- and 3-domino systems should have been approximately equal to .03, .06 and .09 (if you used coins and the 15 cm ramp instead of dominoes and the 30-cm ramp, your ramp slopes will be different; each dime will correspond to a ramp slope of about .007, each penny to a slope of about .010, each quarter to a slope of about .013).
For each slope you have obtained two values for the average rate of change of velocity with respect to clock time on that slope. You may use below the values obtained in the preceding box, or the values you obtained in the box preceding that one. Use the one in which you have more faith.
In the box below, report in the first line the ramp slope and the average rate of change of velocity with respect to clock time for the 1-domino system. Use comma-delimited format.
Using the same format report your results for the 2-domino system in the second line, and for the 3-domino system in the third.
In your fourth line specify the units of these quantities. Ramp slope is a unitless quantity; be sure you report this. Also briefly explain how you got your results and what they tell you about this system:
------>>>>>> ramp slope ave roc of vel each system
Your answer (start in the next line):
.03,12.393cm/sec^2
.06,27.214m/sec^2
.09,39.812cm/sec^2
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Graph acceleration vs. ramp slope
A graph of acceleration vs. ramp slope will contain three data points. The graph will visually represent the way acceleration changes with ramp slope. A straight line through your three data points will have a slope and a y-intercept, each of which has a very significant meaning.
Your results constitute a table with three rows and two columns, representing rate of velocity change vs. ramp slope.
Sketch in your lab notebook a graph of the table you have just entered. The graph will be of rate of change of velocity with respect to clock time vs. ramp slope. Be sure to follow the y vs. x convention to put the right quantities on the horizontal and vertical axes (if it's y vs. x, then y is on the vertical, x on the horizontal axis).
Your graph might look something like the following. Note, however, that this graph is a little too long for its height. On a good graph the region occupied by the data points should be about as high as it is wide. To save space on the page, graphs depicted here are often not high enough for their width
Sketch the best possible straight line through your 3 data points. Unless the points lie perfectly along a straight line, which due to experimental uncertainty is very unlikely, the best possible line will not actually pass through any of these points. The best-fit line can be constructed reasonably well by sketching the line which passes as close as possible, on the average, to the 3 points.
For reference, other examples of 3-point graphs and best-fit lines are shown below.
Describe your best-fit line by giving the following:
On the first line, the horizontal intercept of your best-fit line. The horizontal intercept will be specified here by a single number, which will be the coordinate at which the line passes through the horizontal axis of your graph.
On the second line, the vertical intercept of your best-fit line. The horizontal intercept will be specified here by a single number, which will be the coordinate at which the line passes through the vertical axis of your graph.
On the third line, give the units of your horizontal intercept and the meaning of that intercept.
On the fourth line, give the units of your vertical intercept and the meaning of that intercept.
Starting in the fifth line, give a brief written description of your graph and an explanation of what you think it might tell you about the system:
------>>>>>> horiz int, vert int, units and meaning of horiz, then vert int
Your answer (start in the next line):
0.005
-1
cm
cm^2/sec^2
The horizontal intercept indicates the point where the slope would be so small the ball would not accelerate down the track on its own. It also proves that with every increase in slope, acceleration increases.
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Mark the point on your best-fit line which would correspond to a ramp slope of .10. Determine as accurately as you can the rate of velocity change that goes with this point, so that you have both the horizontal and vertical coordinates of the point.
Report the horizontal and vertical coordinates of that point on the first line below, in the specified order, in comma-delimited format. Starting at the second line, explain how you made your estimate and how accurate you think it might have been. Explain, briefly, what your numbers mean and how you got them.
------>>>>>> mark and report best fit line coord for ramp slope .10
Your answer (start in the next line):
.1,45
I traced a line from .1 across to the vertical axis and estimated the aAve at this point in cm/sec^2. I believe my best fit line to be fairly accurate as well as this estimate.
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Determine the slope of the best-fit line
We defined rise, run and slope between graph points:
The 'run' from one graph point to another is the change in the horizontal coordinate, from the first point to the second.
The 'rise' from one graph point to another is the change in the vertical coordinate, from the first point to the second.
The slope between the two graph points is the rise-to-run ratio, calculated as slope = rise / run.
As our first point we will use the horizontal intercept of your best-fit line, the point where that line goes through the horizontal axis.
As our second point we will use the point on that line corresponding to ramp slope .10.
In the box below give on the first line the run from the first point to the second.
On the second line give the rise from the first point to the second.
On the third line give the slope of your best-fit straight line.
Starting in the fourth line, give a brief explanation and an indication of what you think the slope might tell you about the system.
------>>>>>> slope of graph based on horiz int, ramp slope .10 point
Your answer (start in the next line):
.095
45
473.684
If you multiply the slope by .09, you get a value fairly close to the aAve of the 3 domino system. It would seem the slope is fairly close to my data for that system.
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Assess the uncertainties in your result
The rest of this exercise is optional for Phy 121 and Phy 201 students whose goal is a C grade
Calculate average of mean times and average of standard deviations for 1-domino ramp
Since there is uncertainty in the timing data on which the velocities and rates of velocity change calculated in this experiment have been based, there is uncertainty in the velocities and rates of velocity change.
We first estimate this uncertainty for the 1-domino case.
In the box below, report in the first line the right-to-left mean time, the left-to-right mean time and the average of these two mean times on the 1-domino ramp. This third number, which you also calculated previously, will be called 'the average of the mean times'.
In the second line report the standard deviation of right-to-left times, the standard deviation of left-to-right times and the average of these standard deviations for the 1-domino ramp. This third number will be called 'the average of the standard deviations'.
Starting in the next line give a brief explanation and speculate on the significance of these results.
------>>>>>> 1 dom ramp mean rt-left and left-rt, then std def of both
Your answer (start in the next line):
2.128,2.119,2.1235
0.05455,0.004219,0.0293845
The data is arranged as the mean of the clock times from the right to left, left to right and the average of those two clock times. The second line is the standard deviations in the same format.
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Use average time and standard deviation to estimate minimum and maximum possible velocity and acceleration for first ramp
We will use the average of the mean times and the average of the standard deviations to estimate our error in the average velocity and in the acceleration on the 1-domino ramp.
We will assume that the actual time down the ramp is within in the interval defined by mean +- std dev, where 'mean' is in this case the average of the mean times, and 'std dev' is the average of the standard deviations.
Using these values for mean and std dev:
Sketch a number line and sketch the interval from mean - std dev to mean + std dev. The interval will be centered at the average of the mean times as you reported it in the previous box, and will extend a distance equal to the average of the standard deviations (as also reported in the previous box) on either side.
So for example if the average of the mean times was 1.93 seconds and the standard deviation .11 second, the interval would extend from 1.93 sec - .11 sec = 1.82 sec to 1.93 sec + .11 sec to 2.04 sec.. This interval would be bounded on the left by 1.82 sec and on the right by 2.04 sec..
Report in the first line of the box below the left and right boundaries of your interval. Starting in the second line explain briefly, in your own words, what these numbers represent.
------>>>>>> boundaries of intervals rt-left, left-rt
Your answer (start in the next line):
2.0941155,2.1528845
The first number is the mean - the stand. dev. The second number is the mean + the stand. dev.
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Instead of 'rate of velocity change with respect to clock time' we will now begin to use the word 'acceleration'. So 'average acceleration' means exactly the same thing as 'average rate of velocity change with respect to clock time', and vice versa.
Since we are assuming here that acceleration is constant on a straight ramp, in this context we can simply say 'acceleration' rather than 'average acceleration'.
Using this terminology:
If the time down the ramp is equal to that of the left-hand boundary of the interval you just sketched, then what would be the average velocity and the acceleration of the ball? Report in comma-delimited format on the first line below.
Find the same quantities for the right-hand boundary of your interval, and report in similar format on the second line.
In the third line report the resulting minimum and maximum possible values of acceleration on this interval, using comma-delimited format. Your results will just be a repeat of the results you just obtained.
Starting on the fourth line, explain what your numbers represent and why it is likely that the actual acceleration of the ball on a 1-domino ramp, if set up carefully so that right-left symmetry is assured, would be between the two results you have given.
------>>>>>> 1-dom vel and accel left boundary of interval, rt boundary, min and max possible accel
Your answer (start in the next line):
13.371,12.77
13.006,12.082
12.77,12.082
The first number in the first two sets is vAve, the second is aAve. The third set is teh min and max aAve. Since the right and left limits are presumed to be accurate, we can assume that any imperfections would lie between the standard deviations.
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Repeat for 2- and 3-domino ramps
Do the same for the 2-domino data, and report in identical format, including explanations:
------>>>>>> 2-dom vel and accel left boundary of interval, rt boundary, min and max possible accel
Your answer (start in the next line):
27.350,19.568
26.377,19.217
19.568,19.217
The first number in the first two sets is vAve, the second is aAve. The third set is the min and max aAve.
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Do the same for the 3-domino data, and report in identical format, including explanations:
------>>>>>> 3-dom vel and accel left boundary of interval, rt boundary, min and max possible accel
Your answer (start in the next line):
24.457,42.724
22.967,37.678
42.724,37.678
The first number in the first two sets is vAve, the second is aAve. The third set is the min and max aAve.
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Now make a table of your results, as follows.
You will recall that slopes of .03, .06 and .09 correspond to the 1-, 2- and 3-domino ramps.
In the first line report the slope and the lower limit on acceleration for the 1-domino ramp.
In the second line report the slope and the lower limit on acceleration for the 2-domino ramp.
In the third line report the slope and the lower limit on acceleration for the 3-domino ramp.
In the fourth line report the slope and the upper limit on acceleration for the 1-domino ramp.
In the fifth line report the slope and the upper limit on acceleration for the 2-domino ramp.
In the sixth line report the slope and the upper limit on acceleration for the 3-domino ramp.
Starting in the seventh line give a brief explanation, in your own words, of what these numbers mean and what they tell you about the system:
------>>>>>> slope and lower limit 1, 2, 3 dom; slope and upper limit 1, 2, 3 dom
Your answer (start in the next line):
.03,12.77
.06,19.568
.09,42.724
.03,12.082
.06,19.217
.09,37.678
These are the slopes and corresponding lower and upper limits to acceleration for each of the ramp setups. Each limit should indicate the acceleration for the greatest and least acceleration possible for their respective slopes. The real aAve for each slope probably resides somewhere in between the upper and lower limits.
#$&*
Plot acceleration vs. ramp slope using vertical segments to represent velocity ranges
On your graph of acceleration vs. ramp slope, plot the points specified by this table.
When you are done you will have three points lying directly above the .03 label of your horizontal axis. Connect these three points with a line segment running vertically from the lowest to the highest.
You will also have three points above the .06 label, which you will similarly connect with a segment, and three points above the .09 label, which you will also connect.
Your graph will now contain the best-fit straight line you made earlier, and the three short vertical line segments you have just drawn. Your graph will look something like the one below, though your short vertical line segments will probably be a little thinner than the ones shown here, and unlike yours the graph shown here does not contain the best-fit line. And of course your points won't be the same as those used in constructing this graph:
Does your graph fit this description?
Does your best-fit straight line pass through the three short vertical segments?
Give your answer and be sure to include a couple of sentences of explanation.
------>>>>>>
best-fit line thru error bars?
Your answer (start in the next line):
My graph does fit the description and my best fit line does pass through the vertical segments. However, my first segment for the .03 slope points is very short and the three points are clustered tightly. My second and third line segments are however considerably larger, which I assumed would be the case from the standard deviation data. While my best fit line did pass through these segments, it is clear the data is not as reliable as the .03 slope data.
#$&*
Determine max and min possible slopes of acceleration vs. ramp slope graph
It should be possible to draw a number of straight lines which pass through all three vertical segments. Some of these lines will have greater slopes than others. For example note that the figures below show two lines which pass through all three vertical segments, with the line in the second graph being steeper than the line in the first.
Draw the steepest possible straight line which passes through all three vertical segments on your graph.
Using the x-intercept of this line and the point on this line corresponding to ramp slope .10, determine the slope of the line.
In the first line below report the rise, run and slope of your new line. Use comma-delimited format.
Starting in the second line give a brief statement of what your numbers mean, including an explanation of how you obtained your slope. Be sure to include the coordinates of the two points you used and the resulting rise and run.
------>>>>>>
max possible graph slope
Your answer (start in the next line):
38
.105
361.905
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Now draw the least-steep possible straight line which passes through all three vertical segments.
Follow the same instructions as before, and report your results for this line in the same way, including a brief explanation:
------>>>>>> min possible graph slope
Your answer (start in the next line):
48
.998
48.096
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Your instructor is trying to gauge the typical time spent by students on these experiments. Please answer the following question as accurately as you can, understanding that your answer will be used only for the stated purpose and has no bearing on your grades:
Approximately how long did it take you to complete this experiment?
------>>>>>>
Your answer (start in the next line):
2h
#$&*
Please copy your completed document into the box below and submit the form:
#$&* self-critique
*#&!*#&!
@&
There appear to be two documents here.
Assuming both are yours, can you separate them and resubmit?
*@