#$&*
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.
** Rubber Band Calibration_labelMessages **
October 3, 2012 @ 6:11 PM
** **
1 hr and 40 minutes
** **
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):
20 cm, 30.2 cm
10.23 cm
With my ruler I started the measurement on 20 cm since the lines were visible and it did not look distorted. I then looked at the other end of the rubber band and saw that the measurement on that side was 30.2 cm. I then took both numbers and subtracted it and got 10.2 then I noticed that the line was closer to .2 than it was .3 so I estimated it to be .03.
The rubber band is marked with a number and the system of the rubber bands is drawn so I knew which one is on top and which one is on the bottom. They are also color coded.
#$&*
Explain the basis for your estimate of the uncertainty of the length of the first rubber band.
Your answer (start in the next line):
The measurement for me was 30.2 and it wasn't close to 30.3 and because of that to me the better estimated was 30.23 because it wasn't even close to .3 and it did not look halfway to .3 either.
So I estimated an uncertainty of .01-03 cm.
#$&*
Measure as accurately as possible the lengths of the remaining rubber bands. Be sure you keep track of which is which. You may move your measuring device from one rubber band to the next.
In the space below enter the ruler markings of the ends of the first rubber band, delimited by commas, in the first line (this will be the same information you entered in the first line of the last space ), the ruler markings of the ends of the second rubber band on the second line, etc., until you have a comma-delimited line for each rubber band.
Then put the word 'End' in the very next line.
Follow this in the very next line by a comma-delimited line containing the numerical distances in cm, each estimated to within .01 cm, of the rubber bands in your chain.
Follow this by a line indicating the markings on the rubber bands.
Finally indicate the uncertainty in your measurements, which should probably be the same as the uncertainty as that given in the preceding space .
Your answer (start in the next line):
20 cm, 30.2 cm
20 cm, 32.02 cm
20 cm, 28.4 cm
20 cm, 36.9 cm
20 cm, 31.4 cm
20 cm, 32.3 cm
End
10.23 cm, 12.02 cm, 8.46 cm, 16.96 cm, 11.47 cm, 12.34 cm
6,5,4,1,3,2
An uncertainty of .01-.03 cm.
#$&*
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):
10.42 cm, 12.23 cm, 8.64 cm, 17.97 cm, 11.65 cm, 12.46 cm
These measurements are from the weight of two dominoes.
#$&*
Continue adding dominoes and measuring until one of the rubber bands exceeds its original length by 30%, or until you run out of dominoes, then stop. To keep the time demands of this experiment within reason, you should beginning at this point adding two dominoes at a time. So you will take measurements for 4, 6, 8, ... dominoes until the 'weakest' of your rubber bands is about to stretch by more than 30% of its original length, or until you run out of dominoes.
If one rubber band reaches its limit while the rest are not all that close to theirs, remove this rubber band from the experiment and modify your previous responses to eliminate reference to the data from this band. However, keep the band and keep your copy of its behavior to this point.
In the space below, enter on the first line the actual lengths in cm of your rubber bands when supporting four dominoes, in comma-delimited format.
Enter in the same order you used previously.
On the second line enter the number 4 to indicate that this result is for four dominoes.
On the third line enter in comma-delimited format the lengths in cm when supporting 6 dominoes.
On the fourth line enter the number 6 to indicate the six dominoes being supported.
Continue in this manner until you have entered all your lengths and numbers of dominoes.
Then on the next line enter 'End'.
You may then enter any brief identifying information or commentary you wish. However since the nature of the information has been defined by previous spaces, this is optional.
If you have reason to believe the uncertainty in your measurements has changed, indicate this also. Otherwise it will be assumed that your previous uncertainty estimates apply.
Your answer (start in the next line):
11.12 cm, 12.47 cm, 9.13 cm, 22.02 cm, 11.96 cm, 13.34 cm
4
11.92 cm, 12.73 cm, 9.76 cm, 29.53 cm, 12.48 cm, 14.37 cm
6
END
I stopped at six dominoes because all the dominoes I had in the package ran out.
#$&*
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.
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 ).
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.
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.
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):
1 domino; .19 newtons
10.23, 12.02, 8.46, 16.96, 11.47, 12.34
2 domino; .39 newtons
10.42, 12.23, 8.64, 17.97, 11.65, 12.46
4 domino; .76 newtons
11.12, 12.47, 9.13, 22.02, 11.96, 13.34
6 dominoes; 1.14 newtons
11.92, 12.73, 9.76, 29.53, 12.48, 14.37
END
The made four graphs for the first four length of rubber bands.
In the first line it contains the amount of dominoes followed by the amount of newtons for the domino.
The second line contains the measure of each rubber band starting with 6,5,4,1,3,2 (the number for each marked rubber band).
#$&*
If you haven't already done so, construct a graph for each rubber band and fit a smooth curve that you think best depicts the actual behavior of that rubber band.
In the space below describe the shape of the curve you drew to approximate the force vs. length behavior of first rubber band.
The curve in the last figure above could be described as 'increasing at a decreasing rate, then increasing at an increasing rate'.
Other possible descriptions might be 'increasing at an increasing rate throughout', 'increasing at a decreasing rate throughout',
'increasing at an increasing rate then increasing at a decreasing rate', etc.).
Then describe the shapes of all six rubber bands. Follow your last description by a line containing the word 'End'. You may if you wish add comments starting on the next line.
Your answer (start in the next line):
I only made four graphs for the first four rubber bands that were on the line.
(6)
The curve for the first rubber band is increasing at an increasing rate.
(5)
Increasing at a decreasing rate
(4)
Increasing at a increasing rate
(1)
Increasing at a decreasing rate
End
#$&*
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.
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):
A 9.8cm rubber band would have an approximate 1.19 newtons.
#$&*
Estimate the length in cm of a rubber band that gives a force of 1.4 Newtons and report the number in the second line.
Your answer (start in the next line):
The rubber band, with an estimatation, would be close to 10.2 cm
That is if the rubber band is the same same as the rubber band that is marked four.
Since each rubber band would be difference according to size.
The longer rubber band which is marked at one would have a different measurement at an approximate 34 cm.
So it varies with the size of the rubber.
#$&*
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):
For the first rubber band I have observed the forces to be.
.25, .37, .79, 1.19 (In newtons)
.06, .02, .03, .05 (in newtons; difference from estimated from curve and actual weight)
With first being with 1 domino then 2 dominoes, 4 dominoes, 6 dominoes..
#$&*
From the curve estimate, using or your first graph, report in comma-delimited format, in the first line, the length corresponding to each of the forces .19 N, .38 N, .76 N, 1.14 N,
etc.. In the second line indicate in comma-delimited format by how much each of these lengths differs from the length you actually observed when the rubber band was resisting this force.
Your answer (start in the next line):
10.3 cm, 10.5 cm, 11.2 cm, 11.98 cm
.07 cm, .08 cm, .08 cm, .06 cm
#$&*
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 curves would give you a better estimate of how much force is being done by each length.
Why? Yeah we could say that only one domino is doing .19 N on the first rubber band, but being a skeptic I would go with the curve giving a better representation of how much each length has in force than saying an exact number of .19 N.
I believe the best uncertainty for each force would be +- .04 N.
Evidence: I saw that for the first rubber band graph the differences were .06, .02, .03, .05 so I think the best choice would be .04 so it would be as close as possible for all numbers.
@&
Very good.
*@
#$&*
If you were to estimate a length for a given force using one of your graphs, what do you think would be the uncertainty in that length (e.g., +- .05 cm, or +- .13 cm, etc.) and what
is your evidence for this estimate?
Your answer (start in the next line):
I would have to choose an uncertainty of .07 because with the first graph I did the differences were .08, .07 and .06. So since .07 is in the middle I would think the best for the graph would be +- .07 cm.
#$&*
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:
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):
2.156
2.141
2.141
2.188
2.125
2.219
2.266
2.281
2.281
2.281
The above data is the time in seconds of 5 samples of a one domino ramp one direction and 5 samples of a one domino ramp in the reverse direction
#$&*
Enter your data for the 2-domino setups in the same format, being sure to include your brief explanation:
On the 'readable' posted version this data will follow the boldfaced heading
'5 trials each way 2 dominoes'
------>>>>>> 2 domino results
Your answer (start in the next line):
1.344
1.406
1.344
1.406
1.328
1.421
1.422
1.422
1.391
1.391
The above data is the time in seconds for a ball to roll down a ramp. The first 5 data are the ramp one way and the last 5 data are the ramp reverse direction.
#$&*
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):
1.125
1.141
1.109
1.125
1.109
1.141
1.172
1.156
1.172
1.172
The above data is time in seconds for a ball to go down a 3 domino ramp. First 5 samples for ramp one direction. Last 5 samples for ramp reverse direction.
#$&*
Calculate mean time down ramp for each setup
In the previous hypothesis testing exercise, you calculated and reported the mean and standard deviation of times down each of the two 1-domino setups, one running right-left and the other left-right.
You may use any results obtained from that analysis (provided you are confident that your results follow correctly from your data), or you may simply recalculate this information, which can be done very quickly and easily using the Data Analysis Program at
http://www.vhcc.edu/dsmith/genInfo/labrynth_created_fall_05/levl1_15\levl2_51/dataProgram.exe\
In any case, calculate as needed and enter the following information, in the order requested, giving one mean and standard deviation per line in comma-delimited format:
• Mean and standard deviation of times down ramp for 1 domino, right-to-left.
• Mean and standard deviation of times down ramp for 1 domino, left-to-right.
• Mean and standard deviation of times down ramp for 2 dominoes, right-to-left.
• Mean and standard deviation of times down ramp for 2 dominoes, left-to-right.
• Mean and standard deviation of times down ramp for 3 dominoes, right-to-left.
• Mean and standard deviation fof times down ramp or 3 dominoes, left-to-right.
On the first subsequent line briefly indicate the meaning of your results and how they were obtained:
------>>>>>> mean, std dev each setup each direction
Your answer (start in the next line):
2.150, 0.02381
2.266, 0.02685
1.366, 0.03745
1.409, 0.01680
1.122, 0.01339
1.163, 0.01392
Above are the mean and standard deviation for the 1, 2 and 3 domino ramps. With 5 samples right-to-left and 5 samples left-to-right for each domino set up. These mean and standard dev were calculated by using the data program.
#$&*
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):
13.023
12.357
20.498
19.872
24.955
24.076
The above numbers are average velocity in cm/s. We were given ‘ds=28cm. Each number was obtained by the definition of average velocity formula vAve=’ds/’dt. For example the data in the first line was calculated as
vAve=’ds/’dt=28cm/2.150s=13.023cm/s
I put this formula in excel and used for all data. I then copied results back to this lab.
:
#$&*
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):
12.115
10.906
30.011
28.208
44.484
41.403
The above data is acceleration of the ball down the ramp in cm/s^2. The definition of average acceleration was used to calculate the data as follows.
aAve=(vf-v0)/’dt
Given that the ball started from rest we use v0=0. Given uniform acceleration and vf=0, then we use vf=2*vAve. We calculated vAve in the section before. For example the data in line 1 above was calculated as follows:
aAve=(vf-v0)/’dt
v0=0
vf=2*vAve=2*13.023cm/s=26.047cm/s
aAve=(26.047cm/s - 0cm/s)/2.15s=12.115cm/s^2
I put the above formula in excel and copied the results back to this lab.
:
#$&*
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):
11.510
29.110
42.943
The data above is the average acceleration in cm/s^2of two calculated accelerations. For example data 1 is (12.115+10.906)/2=11.510cm/s^2. As the ramp height grew higher the acceleration grew higher.
:
#$&*
Find acceleration for each slope based on average of left-right and right-left times
Average the mean time required for the right-to-left run with the mean time for the left-to-right run.
Using this average mean time, recalculate your average rate of velocity change with respect to clock time for the 1-domino trials
Do the same for the 2-domino results, and for the 3-domino results.
Report your results in the box below, giving one average rate of change of velocity with respect to clock time per line, in the order requested. In the subsequent line explain how you obtained your results and what you think they mean.
------>>>>>> left-right, right-left each setup, ave mean times and give ave accel
Your answer (start in the next line):
11.494
29.096
42.916
The above results are acceleration in cm/s^2. The first number is for the one domino setup. The next number for the 2 domino setup and the final number for the 3 domino setup. These were obtained by averaging the mean time of the r-l and l-r means and using this average mean to recalculate the aAve for each trial.
:
#$&*
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 speculate they would be close to the same.
They are nearly the same. The difference between my one domino aAve was .016, between my two domino aAve was .04 and between my 3 domino was .027. Our measurements and methods were not accurate to this degree.
:
#$&*
Associate acceleration with ramp slope
Your results will clearly indicate that, as expected, acceleration increases when ramp slope increases. We want to look further at just how the acceleration changes with ramp slope.
If you set up the ramps according to instructions, then the ramp slopes for 1-, 2- and 3-domino systems should have been approximately equal to .03, .06 and .09 (if you used coins and the 15 cm ramp instead of dominoes and the 30-cm ramp, your ramp slopes will be different; each dime will correspond to a ramp slope of about .007, each penny to a slope of about .010, each quarter to a slope of about .013).
For each slope you have obtained two values for the average rate of change of velocity with respect to clock time on that slope. You may use below the values obtained in the preceding box, or the values you obtained in the box preceding that one. Use the one in which you have more faith.
In the box below, report in the first line the ramp slope and the average rate of change of velocity with respect to clock time for the 1-domino system. Use comma-delimited format.
Using the same format report your results for the 2-domino system in the second line, and for the 3-domino system in the third.
In your fourth line specify the units of these quantities. Ramp slope is a unitless quantity; be sure you report this. Also briefly explain how you got your results and what they tell you about this system:
------>>>>>> ramp slope ave roc of vel each system
Your answer (start in the next line):
.03, 11.510
.06, 29.110
.09, 42.943
The numbers above are the slope first and then aAve, No unit for the slope and the aAve units are cm/s^2
:
#$&*
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):
.01
-5
No units on x axis , this is the slope rise/run and both units were cm. So they canceled. At this x axis intercept the accel is at 0cm/s^2.
The units of the vertical intercept are cm/s^2. At this y intercept the slope is equal to 0. In other words there is no slope.
:
#$&*
@&
Slopes are based on two graph points. If one (or both) happen(s) to be an intercept that doesn't change anything.
A straight line with horizontal intercept .01 and vertical intercept -5 has slope 5 / .01 = 500.
The horizontal quantity is unitless; as you observe the cm are divided by cm when calculating slope, leaving the slope unitless.
The vertical quantities are accelerations in cm/s^2.
So the slope is 5 cm/s^2 / .01 = 500 cm/s^2.
This is within the range usually observed.
*@
Mark the point on your best-fit line which would correspond to a ramp slope of .10. Determine as accurately as you can the rate of velocity change that goes with this point, so that you have both the horizontal and vertical coordinates of the point.
Report the horizontal and vertical coordinates of that point on the first line below, in the specified order, in comma-delimited format. Starting at the second line, explain how you made your estimate and how accurate you think it might have been. Explain, briefly, what your numbers mean and how you got them.
------>>>>>> mark and report best fit line coord for ramp slope .10
Your answer (start in the next line):
.01, 0
I estimated this from my best fit line. I used graph paper to try and be as accurate as possible using my hand. I used a ruler edge for my straight line to be as accurate as possible with my line. I think this is fairly accurate. What these number mean is that at slope of .01 there is no acceleration.
:
#$&*
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):
.02
11
550
I think this is the acceleration per slope unit.
:
#$&*
@&
This isn't exactly the same as what you get from your intercepts, but it's not much difference and is probably more accurate.
With units this slope is 550 cm/s^2.
*@
Assess the uncertainties in your result
The rest of this exercise is optional for Phy 121 and Phy 201 students whose goal is a C grade
Calculate average of mean times and average of standard deviations for 1-domino ramp
Since there is uncertainty in the timing data on which the velocities and rates of velocity change calculated in this experiment have been based, there is uncertainty in the velocities and rates of velocity change.
We first estimate this uncertainty for the 1-domino case.
In the box below, report in the first line the right-to-left mean time, the left-to-right mean time and the average of these two mean times on the 1-domino ramp. This third number, which you also calculated previously, will be called 'the average of the mean times'.
In the second line report the standard deviation of right-to-left times, the standard deviation of left-to-right times and the average of these standard deviations for the 1-domino ramp. This third number will be called 'the average of the standard deviations'.
Starting in the next line give a brief explanation and speculate on the significance of these results.
------>>>>>> 1 dom ramp mean rt-left and left-rt, then std def of both
Your answer (start in the next line):
I regret not going further, but I’m one week behind in Physics work and have to take the hit for not doing the rest of this lab.
:
#$&*
@&
This part is completely option for Phy 121. My statement wasn't clear on this.
So there's no hit on your grade.
You did very well on this exercise.
*@