course PHY 201
9:52 PMSunday, September 27, 2009
Class 090921
The following conventions will allow your instructor to quickly locate your answers and separate them from the rest of any submitted document, which will significantly increase the quality of the instructor's feedback to you and to other students.
When answering these questions, give your answer to a question before the &&&&. This is different than my previous request to place your answer after the &&&&.
When doing qa's, place your confidence ratings and self-assessment ratings on the same line as the prompt.
If you don't follow these guidelines you may well be asked to edit your document to make the changes before I can respond to it.
Thanks.
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Acceleration vs. ramp slope
If you missed class today you can easily complete this experiment by staying an extra 5 minutes or so next time. It's a familiar situation (ramp supported by a domino lying flat, then on its long side, then its short side). You will be asked to calculate the acceleration of the ball and the slope of the ramp, create a three-point graph, and calculate two graph slopes.
• By lining up dominoes in their proper orientation, as demonstrated in class, determine the slope of your ramp when supported by a domino lying flat, then lying along its longer edge, then along its shorter edge.
• Using your pendulum in the manner you deem most accurate, take observations necessary to find the acceleration of the ball on each ramp. Try to determine the time down the ramp as accurately as possible.
Give your data and an explanation of what they mean:
Ramp length: 30cm
Pendulum used for flat domino: 25cm
Flat domino time interval: 2.5 half-cycles = 1.25 cycles = 1.25 seconds
Flat domino ramp slope: 0.0333
Flat domino v_ave: 24 cm/s
Flat domino a_ave: 38.4 cm/s^2
Pendulum used for side domino: 20cm
Side domino time interval: 2.5 half-cycles = 1.25 cycles = 1.1175 s
Side domino ramp slope: 0.0833
Side domino v_ave: 26.846 cm/s
Side domino a_ave: 47.152 cm/s^2
Pendulum used for tall domino: 20cm
Tall domino time interval: 1.5 half-ycles = 0.75 cycles = 0.671 s
Tall domino ramp slope: 0.167
Tall domino v_ave: 44.709 cm/s
Tall domino a_ave: 133.261 cm/s^2
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Graph acceleration vs. ramp slope and give the three graph points you get as a result of your analysis of the data.
38.4, 0.0333
47.152, 0.0833
133.261, 0.167
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Find the slope between the first and second point on your graph.
slope = rise / run
= (47.152 - 38.4) / (0.0833 - 0.0333)
= 8.752 / 0.05
= 175.04
this would be the slope for vAve vs. ramp slope, not for acceleration vs. ramp slope
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Find the slope between the second and third point on your graph.
slope = rise / run
= (133.261 - 47.152) / (0.167 - 0.0833)
= 86.109 / 0.0837
= 1,028.781
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What do you think is the uncertainty in your time measurements?
+/- 1/8 cycle
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What do you think is the percent uncertainty in each of your time measurements?
100% * 0.125 / 1.25 = 10%
100% * 0.125 / 1.1175 = 11.186 %
100% * 0.125 / 0.671 = 18.629 %
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What do you think is the percent uncertainty in each of your calculated accelerations?
If the time is off by as much as 18.629% then the accelrations would reflect that as well.
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What do you think is the percent uncertainty in each of your slopes?
Given that the run is estimated at 30cm for each ramp, the slope is likely off by a greater percentage with the tall domino than the flat domino, but in any case, probably 1% or less.
100% * .1 / 30 = < 1%
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When you analyze your data for this experiment:
For what object are you calculating the acceleration?
A steel ball bearing.
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What events define the beginning and the end of each time interval you are measuring?
The beginning of the time interval is when the ball and the pendulum are released simultaneously.
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Where is the object at the beginning of the interval and where is it at the end of the interval?
At the beginning, the ball is at the top of the ramp. At the end, the ball is at the bottom of the ramp.
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What is the displacement between those positions?
The length of the ramp. Approx. 30cm
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Does the displacement depend in any way on the length of the pendulum?
No.
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What information do we get from the length of the pendulum?
Clock time and (eventually) time interval. Using the formula period = .2`sqrt(L), where L = pendulum length in cm, we can find the period of the pendulum in seconds, and thus convert our findings to seconds.
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Interpreting Trapezoids
A trapezoidal approximation graph has 'altitudes' 70, 90, 120 and 130 corresponding the 'horizontal' coordinates 120, 150, 190, 220.
We sketch the graph and complete the three corresponding trapezoids. We calculate their slopes, then construct the equal-area rectangles and use them to calculate the areas of the trapezoids.
We easily obtain the following information:
trapezoid # rise run slope ave altitude area
1 20 30 2/3 80 2400
2 30 40 3/4 105 4200
3 10 30 1/3 125 3750
Now we interpret this, assuming that the graph represents velocity in cm/s vs. clock time in s.
The vertical quantity is velocity in cm/s, and the horizontal quantity is clock time in s.
The rise of a trapezoid therefore represents the corresponding change in velocity, designated `dv, in cm/s.
The run of a trapezoid represents the corresponding change in clock time, designated `dt, in seconds.
The slope is rise / run = `dv / `dt, in (cm/s)/s = cm/s^2. This is change in velocity / change in clock time, i.e., average rate of change of velocity with respect to clock time, which is the definition of acceleration.
The average altitude is the average of the two altitudes, which represent velocities. The average altitude therefore represents the approximate average velocity on the interval. We can designate this as vAve.
The area is the product of the average altitude and the run, which represents the product of the average velocity vAve and the change in clock time `dt. The average velocity is the average rate of change of position with respect to clock time, which by definition is change in position / change in clock time, or `ds/`dt. So when we multiply vAve by `dt we are multiplying the approximate value of `ds / `dt by `dt, and the result is the approximate value of `ds, the change in position. Thus the area of the trapezoid represents the approximate change in position during the corresponding interval. In this case the units are cm/s * s = cm.
We conclude that the object whose velocity is represented by the graph has average accelerations 2/3 cm/s^2, 4/3 cm/s^2 and 1/3 cm/s^2 on the three respective intervals, and travels approximate distances of 2400 cm, 4200 cm and 3750 cm.
The instructor will again run an Atwood machine experiment, with you doing the timing. Write down your data carefully.
We can also interpret this data assuming a graph of force in pounds vs. position in feet. To visualize a possible situation, think of the force exerted by a bungee cord as the position of its free end changes (thereby stretching it):
The rise of a trapezoid will represent the change in the force `dF in lbs.
The run will represent the change in position `ds, in ft.
The slope will therefore represent `dF / `ds, i.e., change in force / change in position. By definition of rate this is the average rate of change of force with respect to position for the interval represented by the trapezoid. Its units will be lbs / ft. This tells you how many pounds of force you can expect for each foot you stretch the bungee cord.
The average altitude of a trapezoid will represent the approximate average force F_ave on the corresponding interval, in lbs.
So the area of a trapezoid represents the product of the average force and the change in position, designated F_ave * `ds, in units of ft * lbs. (With the bungee-cord interpretation, if the position is changed in the direction which stretches the cord, this is the work done to stretch the cord. If the cord is an 'ideal' elastic object (which is not completely so), this also represents the potential energy stored in the cord.)
qa assignments
There is a series of over 30 q_a_ exercises, plus an additional set of qa's designed to address areas of difficulty indicated by submitted work. Copy the address into the Address box of your browser:
http://vhcc2.vhcc.edu/ph1fall9/frames_pages/qa_grid.htm
It is recommended that you bookmark this page.
To help you prepare for the upcoming Major Quiz, you should plan to run through qa's #2 - 8 during the upcoming week. If you do one a day you'll be done by the first of next week. The first few should go pretty quickly, and if you've mastered the content to this point of the course none of them should take a lot of time. In any case you'll find that they are worth the time you put into them.
There are at least two ways to run through these documents:
• If you have mastered all or most of the content of a document (as will be the case for most of the class on the earlier documents) then you can simply read each problem, think through in your head how you would solve it, then read the given solution and see if there's anything you didn't think of.
• If you thought of everything, you can just move on to the next question. If you don't insert an answer, confidence rating, self-critique and self-critique rating, it will be taken as an indication that you fully understand the problem and its solution and would be able to use the corresponding ideas to solve similar problems.
• If there's something you didn't think of, then if you are very sure you understand the details you didn't think of, you can move on, but it's recommended you jot down some notes in your notebook. If you're not sure of the details, then you should insert a detailed self-critique in which you demonstrate what you do and do not know, ask questions, etc., according to the instructions for the self-critique process.
• If you're not completely sure, you should stop and insert a detailed self-critique.
• If you have not mastered the content, or if you discover that at some point you aren't sure, then you should at that point begin inserting your solutions, confidence assessments and self-critiques.
Homework:
Your label for this assignment:
ic_class_090921
Copy and paste this label into the form.
Report your results from today's class using the Submit Work Form. Answer the questions posed above.
Note also that you will receive a subsequent document with some alternative materials, and that you will be asked to complete a short portion of that document.
&#This looks good. Let me know if you have any questions. &#