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course Phy 201
9/06 11 Here are the questions and problems for the 9/01 class. Work them out then copy this document into a text editor, insert your answers as indicated, and submit using the Submit Work Form at http://vhcc2.vhcc.edu/dsmith/submit_work.htm . If you have questions you can include them with your answers (if so mark them with ???? before and after), or you can use the Submit Question Form at http://vhcc2.vhcc.edu/dsmith/forms/question_form.htm .
The first two problems deal with projections of one vector onto the line of another vector. The third deals with trapezoids on the v vs. t graph. The fourth deals with the rubber band lab exercise. If you don't understand the projections and/or the trapezoids, you can start with #4 and whatever you do understand about the first three, then wait for class notes before completing those questions.
1. On a coordinate plane sketch the points (5, 9) and (3, 12). You don't need to take a lot of time to meticulously mark off the scale or measure the points with a ruler, just make a reasonable estimate. You should be able to locate the origin and the two points in just a minute or so. Sketch a vector from the origin to the first point and call this vector B. Sketch another vector from the origin to the second point and call it A.
Sketch the projection of the A vector on the B vector, as we did in class. In case you need more detailed instructions:
* Sketch a dotted projection line from the tip of the A vector to the B vector. The projection line must make an angle of 90 degrees with the B vector.
* The 'projected point' is the point where the projection line meets the B vector.
* The projection of A on B is the vector from the origin to 'projected point' .
Estimate the coordinates of the tip of the projection vector and give them in the next line:
(4.5, 7.75)
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What therefore is the length of the projection vector? Give the length and explain how you found it starting in the next line:
Using the Pythagorean theorem, I used the coordinates to solve a^2 +b^2=c^2
4.5^2+7.75^2=c^2
20.25+60=80.25
I then took the square root of 80.25 and the answer is 9.0 (8.95) cm
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2. Repeat the above exercise using the points (4, 2) and (8, 3). You might have to extend the line of the B vector so that the projection line will meet it. Give your results starting in the next line:
The coordinates are (7,4). Again using the Pythagorean Theorem—
7^2+4^2=c^2
49+16=c^2
65=c^2, find the square root of 65 to be 8.1 cm
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3. On a graph of velocity vs. clock time, with velocity in cm/s when clock time is in seconds, what does the point (3, 9) represent?
It equals a velocity of 9cm/s at 3 seconds.
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On this graph of velocity vs. clock time the points (3, 9) and (6, 13) represent the velocities of a moving object at the corresponding clock times. What does the 'rise' between these points represent? What does the 'run' represent? What does the slope of the straight line between (3, 9) and (6, 13) therefore represent?
The rise represents the change in velocity
The run represents the change in clock time
And the slope represents the average rate of change of velocity with respect to clock time, average acceleration.
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If you cut a triangle from this trapezoid, cutting in the horizontal direction starting at the midpoint of the line segment between the two given points, you can simply rotate this triangle 180 degrees and join it to the remaining part of the trapezoid and form a rectangle.
* What is the 'width' of this rectangle and what does it represent?
* What is the 'height' of this rectangle and what does it represent?
* What is the area of this rectangle and what does it therefore represent?
Width represents a change in time— 6-3=3 cm
Height (altitude) is the average velocity—3 cm/s + 6cm/s=9cm/s
(9cm/s)/2=4.5 cm/s
Area= average altitude x height
4.5 cm/s x 3 s= 13.5 cm
It represents the displacement of an object
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4. On a graph of velocity vs. clock time the points (t_0, v_0) and (t_f, v_f) represent the velocity and position of an object at the beginning and the end of some interval. The vertical lines from these points to the horizontal axis form what we will call the 'graph altitudes' of a trapezoid. The line segment from the first point to the second forms another side of the trapezoid, and the segment from t_0 to t_f on the horizontal axis forms the fourth side, which we call the 'graph width/ of the trapezoid. (in correct geometrical language this In terms of the symbols t_0, v_0, t_f and v_f, what are the following:
The average velocity v_Ave.= (v_0 + v_f)/2
The change in velocity `dv.=(v_f – v_0)
The time interval `dt.= t_f-t_0
The displacement `ds.= (v_f + v_0)/2 *(t_f-t_0)
The average rate of change of velocity with respect to clock time.
= (v_f-v_0)/(t_f – t_0)
The average acceleration.= (v_f –v_0)/ (t_f – t_0)
5. For your data in the rubber band experiment, give the endpoints of each rubber band in the first trial, where the rubber bands were just barely beginning to exert a tension force. Report one rubber band in each of the next three lines, giving in each line the x and y coordinates of the first point, followed by the x and y coordinates of the second point, in that order. Separate each number from the next by a comma. You will have four numbers in each of three lines, separated by commas. In the fourth line give the units of your measurements. Start in the next line:
14, 7.5, 12, 16
10.5, 17, 14, 21.5
11, 18, 14, 27.5
cm
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Find the lengths of the three rubber bands, based on the coordinates of the points you reported. Give the lengths in the first line below, separated by commas. Starting in the next line explain how you calculated the lengths:
8.7 cm, 6.0cm, 9.9cm
I used the Pythagorean theorem to solve each set of coordinates by finding the length of the hypotenuse of the right triangle they formed.
In order,
2^2+8.5^2=c^2
4+72.25=c^2
76.25=c^2
8.7=c
4^2+4.5^2=c^2
16+20.25=c^2
36.25=c^2
6.0=c
3^2+9.5^2=c^2
9+90.25=c^2
99.25=c^2
9.96
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Well done, but was that shortest band really only 6 cm long?
Now report the coordinates of the points you observed in the second trial, using the same format as before. Report the three different rubber bands in the same order as before:
16, 5, 10, 19
9, 21, .5, 23
10, 20, 17, 31
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Report the lengths of the three rubber bands, using the same order as before:
13.2 cm , 8.7 cm, 13 cm
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Assuming that each rubber band exerts a force of .5 N for every centimeter of length in excess of its 'barely-exerting-a-force' length, what are the three forces? Report the three forces, separated by commas, in the first line below. In the second line below, explain how you determined your forces.
2.25 N, 1.35N, 1.55N
To determine the forces, we first have to find the difference between our “barely-exerting-a-force” rubber band and exerting force rubber band. After we solve for the difference, we multiply the answer by 0.5 N since we assume that each rubber band exerts a form of .5 N for every centimeter of length in excess. That gives us the force. As shown below:
Rubber band 1—13.2 cm-8.7cm=4.5 cm
4.5 cm * .5 N=2.25N
Rubber band 2—8.7cm-6 cm= 2.7 cm
2.7 cm *.5N= 1.35 N
Rubber band 3—13 cm -9.9 cm=3.1 cm
3.1 cm *.5 N=1.55 N
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Everyone should be able to do the above. At some point below I would expect most College Physics students to get a little lost, so if you are a College Physics student and do get lost, it's OK. We'll go through some further explanation based on the above and you'll be able to answer the questions after the next class.
For your second trial, each rubber band can be represented by a displacement vector, a vector from one of its endpoints to the other. More specifically we will represent each rubber band by the displacement vector from the point closest to the middle to the point furthest from the middle (the middle is the center of that paperclip on which all three rubber bands are pulling).
In the first line below, report the x and y components of the vector representing the first rubber band, using the notation . For example a vector with x and y components 3 cm and -7 cm would be reported as <3 cm, -7 cm>. In the second and third lines, report the components of the vectors representing the second and third rubber bands. Report the rubber bands in the same order as before. Starting in the fourth line give a brief explanation of how you got your results:
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The vectors you reported above are displacement vectors. It should be clear that the magnitude of each is the length of the corresponding rubber band. If you divide both components of a displacement vector by its length, you will get a unit vector which represents the direction of the displacement vector. In the first three lines below, give the x and y components of the three resulting unit vectors. Starting in the fourth line give a brief description of how you obtained your results and what they mean.
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If you now multiply the unit vector for each rubber band by the force it exerts, you get the force vector for the force exerted on the 'middle' paperclip by that rubber band. Do this and report your force vectors below, one to each line, reporting the forces of the three rubber bands in the same order as before. Starting in the fourth line give a brief description of how you obtained your results and what they mean.
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Sketch your force vectors, using an appropriate scale. You should select a scale such that your sketch takes up at least half of a sheet of paper.
* Pick one of the three vectors and sketch the dotted line which extends this vector (this will be the dotted line you will need in order to sketch the projections on this line of the remaining vectors).
* Sketch the projections of the other two vectors onto your selected vector.
* Measure the lengths of your projections.
Give the lengths of your projections in the first line, separated by a comma. In the second line give the length of the vector you picked.
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In the first line below give the forces corresponding to your projections, based on the scale of your sketch. In the second line give the force of the vector you picked previously. In the third line explain how you got the forces from the measured lengths:
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University Physics Students only: Find the magnitude of the projection of each of the force vectors on each of the others, based on the coordinates of the force vectors. (You will use the dot product, as explained in Chapter 1 of your text. If you're in multivariable calculus you'll be familiar with this. If you're not in that course, you might have some questions about this, but take my word for it, it's easy with just a little practice. So if you don't understand it from the text, ask. The magnitude of the projection of vector A on vector B is just the || A || cos(theta), where theta is the angle between the two vectors. Using the dot product it's easy to see that this is just the magnitude of A dot B / || B || ). Using the order in which you have been reporting the three rubber bands, give the projection of the first on the second, the first on the third, the second on the first, the second on the third, the third on the first and the third on the second. In the fourth line give the details of one of your calculations, selecting one that represents the general process:
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&#Your work looks good. See my notes. Let me know if you have any questions. &#
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