Asst_0 Query

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course Phy 121

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ph1 query 0Most queries in this course will ask you questions about class notes, readings, text problems and experiments. Since the first two assignments have been lab-related, the first two queries are related to the

those exercises. While the remaining queries in this course are in question-answer format, the first two will be in the form of open-ended questions. Interpret these questions and answer them as best you

can.

Different first-semester courses address the issues of experimental precision, experimental error, reporting of results and analysis in different ways and at different levels. One purpose of these initial lab

exercises is to familiarize your instructor with your work and you with the instructor 's expectations.

Comment on your experience with the three lab exercises you encountered in this assignment or in recent assignments.

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Question: This question, related to the use of the TIMER program in an experimental situation, is posed in terms of a familiar first-semester system.

Suppose you use a computer timer to time a steel ball 1 inch in diameter rolling down a straight wooden incline about 50 cm long. If the computer timer indicates that on five trials the times of an object

down an incline are 2.42sec, 2.56 sec, 2.38 sec, 2.47 sec and 2.31 sec, then:

Are the discrepancies in timing on the order of 0.1 second, 0.01 second, or 0.001 second?

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If I use the first trial time as the standard, all discrepancies in timing are on the order of 0.1 second. 2.42 - 2.56 = -0.14.

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To what extent do you think the discrepancies in the time intervals could be explained by each of the following:

· The lack of precision of the TIMER program. Base your answer on the precision of the TIMER program as you have experienced it. What percent of the discrepancies in timing do you think are due to

this factor, and why do you think so?

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Based on my experimentation with the TIMER program, I have seen the intervals as precise as 0.01 of each other. I believe this shows the program is fairly accurate and so would not effect the times

above as much as human error in starting and stopping the program. I would guess based on my use of the program to time an object rolling down a controlled ramp that a large percentage of

discrepancy in timing of objects is going to be simultaneously starting the timer and object and stopping the time at the end of the object's travel, both of which are performed by human not machine.

Perhaps less than 10% discrepancy is due to Timer error.

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· The uncertainty associated with human triggering (uncertainty associated with an actual human finger on a computer mouse). What percent of the discrepancies in timing do you think are due to this

factor, and why do you think so?

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As stated in my response above, I believe most discrepancy is caused by human error/factors. Perhaps as much as 35% or more.

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· Actual differences in the time required for the object to travel the same distance. What percent of the discrepancies in timing do you think are due to this factor, and why do you think so?

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Without a controlled environment to conuct the experiment such as a wind tunnel or vacuum, I would waiger actual time discrepancies, while controlled as much as possible, would also play into the timer

results. Air movement would change even in a relatively still room. Ambient air moisture/humity could fluctuate to a point that the discrepancies vary even if minimally from one trial to the next.

Perhaps 15% of the discrepancy would stem from this.

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· Differences in positioning the object prior to release. What percent of the discrepancies in timing do you think are due to this factor, and why do you think so?

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As it would be humanly impossible to set the object in the same exact path each trial, this seems to be a valid source of discrepancy-the wooden ramp may be slightly smoother in one starting place than

another leading to a longer period of 'drag' experienced by the ball. Perhaps 7% of discrepancy would be caused by this factor as the ball could be marked and aligned with a mark on the ramp to

minimize discrepancy.

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· Human uncertainty in observing exactly when the object reached the end of the incline. What percent of the discrepancies in timing do you think are due to this factor, and why do you think so?

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Again this to me is a large source of dicrepancy. I would say at least 33% is due to having the exact time the ball reaches the end of the incline.

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Question: If you had carefully timed the ball and obtained the results given above, how confident would you be that the mean of those five intervals was within 0.1 seconds of the actual mean? (Note

that the mean of the given intervals is 2.43 seconds, as rounded to three significant figures)? Briefly explain your thinking.

****2.42sec, 2.56 sec, 2.38 sec, 2.47 sec and 2.31

I would feel fairly confident that the results were within one tenth of a second because even the most careful operator could be .13 (2.56) above the mean (2.43) by staring the timer slightly ahead of the

ball and stopping slightly after the ball ended. Likewise the opposite could be true and the Timer could have been started slightly after the ball was released and stopped slightly after and the resulting

time would be shorter than the mean ball time (2.31).

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How confident would you be that the 2.43 second mean is within .01 second? Briefly explain your thinking.

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I would feel less confident the above times could be within .01 second of 2.43, though 2.42 is. Human error such as muscle resonse time seems to create too much discrepancy to allow for .01 second.

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How confident would you be that the 2.43 second mean is within .03 second?

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Again, I would be less confident of the trials being within .03 seconds of 2.43, though this is more probable than being within .01 second.

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At what level do you think you can be confident of the various degrees of uncertainty?

Do you think you could be 90% confident that the 2.43 second mean is within 0.1 second of the actual mean?

Do you think you could be 90% confident that the 2.43 second mean is within 0.01 second of the actual mean?

Do you think you could be 90% confident that the 2.43 second mean is within 0.03 second of the actual mean?

Give your three answers and briefly explain your thinking:

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I could be 90% confident that 2.43 is within 0.1of the actual mean. With great care to control as many variables as possible, I believe 2.43 could be within the actual mean 90% of the time.

I don't believe 90% accuracy is possible for either 0.01 or 0.03 of the actual mean for human error with muscle reaction time discrepancy seems to be too great to obtain such results.

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Question: What, if anything, could you do about the uncertainty due to each of the following? Address each specifically.

· The lack of precision of the TIMER program.

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If the timer program could be reprogrammed to a higher level of precision, say to .00001sec. then most uncerntainty of the TIMER program itself would be removed.

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· The uncertain precision of human triggering (uncertainty associated with an actual human finger on a computer mouse)

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Mechanize the starting of the program with pressure sensing probe that starts the TIMER when the ball moves off. Without resorting to science-fiction, use of a trained assistant to start the TIMER with one

mouse when the ball commenced travel and a second assistant with a second mouse to stop the TIMER when the ball reached the end of the run.

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· Actual differences in the time required for the object to travel the same distance.

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Marking a lane or groove the ramp so the ball traveled down the same path each time.

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· Differences in positioning the object prior to release.

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Similar to above, if the path was groove, and a sort of 'starting gate' was constructed, the differences in position prior release would be minimized.

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· Human uncertainty in observing exactly when the object reached the end of the incline.

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As stated before, if a second assistant we able to solely watch the instant the ball rolled past the finish mark, and was able to stop the TIMER by second mouse, the discrepancies would reduce.

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Question: If, as in the object-down-an-incline experiment, you know the distance an object rolls down an incline and the time required, explain how you will use this information to find the object 's

average speed on the incline.

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Your solution:

To get the total avg speed on the incline, I would take the total length of the inline and divide that distance by the time required to travel the distance.

confidence rating #$&*:

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Question: If an object travels 40 centimeters down an incline in 5 seconds then what is its average velocity on the incline? Explain how your answer is connected to your experience.

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Your solution:

vAvg = 40cm / 5sec = 8cm/sec

confidence rating #$&*:

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Question: If the same object requires 3 second to reach the halfway point, what is its average velocity on the first half of the incline and what is its average velocity on the second half?

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Your solution:

1/2 of 40cm = 20cm

20cm/3sec=6.67cm/sec.

The difference in the total amount of time required (5 sec) and the time required for the first 20cm (3sec) is 2sec. The vAvg for the second half is 20cm / 2seconds = 10 cm/second.

confidence rating #$&*:

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Question: `qAccording to the results of your introductory pendulum experiment, do you think doubling the length of the pendulum will result in half the frequency (frequency can be thought of as the

number of cycles per minute), more than half or less than half?

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Your solution:

From my Pendulum Experiment data, the first pendulum length was 9cm and resulted in 85 cycles in one minute. When this length was doubled to 18 cm (the third pendulum), the resulting frequency was

66 cycles/min.

85-66=19 less cycles in a minute for the doubled pendulum length. 66 is more than half of 85, which is 42.5 cycles.

confidence rating #$&*:

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Question: `qNote that for a graph of y vs. x, a point on the x axis has y coordinate zero and a point on the y axis has x coordinate zero. In your own words explain why this is so.

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Your solution:

The point at which the y- and x-axes intersect is (0,0) (x, y) values respectively. If the point is moved one unit to the right along the x-axis but zero units up or down, along the y-axis, then the new point

would be located at (1, 0)- no change in the y-value leads to a new point because the x-value has increased by 1, however the point is still located on the x-axis because the y-value neither increased or

decreased.

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Question: `qOn a graph of frequency vs. pendulum length (where frequency is on the vertical axis and length on the horizontal), what would it mean for the graph to intersect the vertical axis (i.e., what

would it mean, in terms of the pendulum and its behavior, if the line or curve representing frequency vs. length goes through the vertical axis)? What would this tell you about the length and frequency of

the pendulum?

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Your solution:

For the line of the freq. vs. length graph to cross the vertical axis, the length of the pendulum would have to be less than 0cm. This would I believe an impossibility as the pendulum would have to be a

negative length in order to cross the vertical axis.

confidence rating #$&*:3

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Question: `qOn a graph of frequency vs. pendulum length, what would it mean for the graph to intersect the horizontal axis (i.e., what would it mean, in terms of the pendulum and its behavior, if the line

or curve representing frequency vs. length goes through the horizontal axis)? What would this tell you about the length and frequency of the pendulum?

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Your solution:

In order for the graph to cross the horizontal axis, the length of the pendulum would be so long as to prohibit the frequency of the pendulum to reach any cycles within a minute or whatever unit the period

is measured by.

confidence rating #$&*:3

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Question: `qIf a ball rolls between two points with an average velocity of 6 cm / sec, and if it takes 5 sec between the points, then how far apart are the points?

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Your solution:

d = r*t so, d=6 * 5= 30cm.

confidence rating #$&*:3

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Given Solution:

`aOn the average the ball moves 6 centimeters every second, so in 5 seconds it will move 30 cm.

The formal calculation goes like this:

We know that vAve = `ds / `dt, where vAve is ave velocity, `ds is displacement and `dt is the time interval.

It follows by algebraic rearrangement that `ds = vAve * `dt.

We are told that vAve = 6 cm / sec and `dt = 5 sec. It therefore follows that

`ds = 6 cm / sec * 5 sec = 30 (cm / sec) * sec = 30 cm.

The details of the algebraic rearrangement are as follows:

vAve = `ds / `dt. We multiply both sides of the equation by `dt:

vAve * `dt = `ds / `dt * `dt. We simplify to obtain

vAve * `dt = `ds, which we then write as{}`ds = vAve *`dt

Be sure to address anything you do not fully understand in your self-critique.

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Your solution:

Aside from the typewriter notation for the formulae above be new to me, I understand the alegbraic formula involved in this problem.

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Question: `qYou were asked to read the text and some of the problems at the end of the section. Tell your instructor about something in the text you understood up to a point but didn't understand fully.

Explain what you did understand, and ask the best question you can about what you didn't understand.

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Your solution:

The concept of uncertainty was a new one for me, at least in studying it. I understand the difference between Precision and Accuracy- Precision being the closeness a range of values for a given object

measured and Accuracy being the closeness of those values to the object's 'true value'. I have been using the two interchangeably and will have to catch myself from doing this as there is a difference

between the two terms, even if a technical term.

As far as uncertainty is concerned, if I measure a board's width with a tape measure with increments of cm and mm, and I notice the actual measurment falls between two mm say the measurement at one

end of the board is 15.35cm and the other is 15.75, is the uncertainty the .05 that I am guessing because the smallest increment being measured is a mm? In short do you always set aside the

measurement beyond the last smallest increment?

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STUDENT QUESTION

I understand that we cannot measure to exact precision, but when we are dealing with estimated uncertainty, do we always

just increment our lowest unit by one and that is our uncertainty? Is there a standard that is used to figure out this?

INSTRUCTOR RESPONSE

The standard answer is that we assume an uncertainty of +- 1 of our smallest unit of precision. However, depending on how well we can 'see' that smallest unit, we can get pretty close to +- 1/2 of a unit.

A more sophisticated answer can be given in terms of the statistics of the normal distribution, but in this course we're not going to go into a whole lot of depth with that. A calculus background would be just

about required to understand the analysis well enough to apply it meaningfully.

STUDENT QUESTION

I fully understand how to calculate uncertainty, but what if the uncertainty isn’t given? For example,

problem 6 asks us for the uncertainty of 1.67. Do we just use .01 as the uncertainty?

INSTRUCTOR RESPONSE

Depending on the nature of the instrument and the observation, +- .01 might be necessary, but we could go to +-.005 if can regard 1.67 as an accurate roundoff.

Without very good reason, though, +-.01 would be the safer assumption.

STUDENT QUESTION

I had trouble grasping the uncertainty. I understand the bit about significant figures, but I’m not sure how that applies

to the uncertainty. Is it just the last digit of the significant figure that could be wrong?

INSTRUCTOR RESPONSE

Any measurement is uncertain to some degree.

On some of the initial videos, despite the fact that the ruler was marked in inches and subdivided to eighths of an inch, the resolution of the image was poor and it wasn't possible to observe its position

within eighths of an inch. Had the videos been very sharp (and taken from a distance sufficient to remove the effects of parallax), it might have been possible to make a good estimate of position to within

a sixteenth of an inch or better.

So for the videos, the uncertainty in position was probably at least +- 1/4 inch, very possibly +- 1/2 inch. But had we used a better camera, we might well have been able to observe positions to within

+-1/16 inch.

The video camera is one instrument, and each camera (and each setup) introduces its own unique uncertainties into the process of observation.

The same can be said of any setup and any instrument or combination of instruments.

STUDENT QUESTION: I understood the portion discussing the nature of science and felt familiar with much of the measurement. What I did not fully understand was how do you know when to write an

answer using the powers of 10 or to leave it alone? Several of the tables had values in powers of 10 for metric prefixes such as centi and mili.

INSTRUCTOR RESPONSE

Whether you use scientific notation or not depends a lot on the context of the situation.

As a rule of thumb, I would recommend going to scientific notation for numbers greater than a million (10^6) and less than a millionth (10^-6). When numbers outside this range are involved in an analysis

it's a good idea to put everything into scientific notation.

And when you know that scientific notation is or is not expected by your audience, write your numbers accordingly.

QUESTION RELATED TO UNIVERSITY PHYSICS (relevant only to University Physics students)

I don’t fully understand the dot product rule

INSTRUCTOR RESPONSE

The dot product of vectors A = a_1 i + a_2 j + a_3 k and B = b_1 i + b_2 j + b_3 k is a_1 * b_1 + a_2 * b_2 + a_3 * b_3. The dot product is simply a number.

The magnitude of A is | A | = sqrt( a_1 ^ 2 + a^2 ^ 2 + a_3 ^ 2); the magnitude of B is found in a similar manner.

The dot product is equal to | A | * | B | * cos(theta), where theta is the angle between the two vectors.

If you have the coefficients of the i, j and k vectors, it is easy to calculate the dot product, and it's easy to calculate the magnitudes of the two vectors. Setting the two expressions for the dot product equal

to one another, we can easily solve for cos(theta), which we can then use to find theta.

More importantly for physics, we can find the projection of one vector on another. The projection of A on B is just the component of A in the direction of B, equal to | A | cos(theta). The projection of one

vector on another is important in a number of situations (e.g., the projection of the force vector on the displacement, multiplied by the displacement, is the work done by the force on the interval

corresponding to the displacement).

Dot products are a standard precalculus concept. Check the documents at the links below for an introduction to vectors and dot products. You are welcome to complete these documents, in whole or in

part, and submit your work. If you aren't familiar with dot products, it is recommended you do so.

http://vhcc2.vhcc.edu/dsmith/genInfo/qa_query_etc/pc2/pc2_qa_09.htm

http://vhcc2.vhcc.edu/dsmith/genInfo/qa_query_etc/pc2/pc2_qa_10.htm

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SOME ADDITIONAL COMMON QUESTIONS:

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QUESTION: I didn’t understand how to calculate uncertainty for a number such as 1.34. When given examples we had problems such as 1.34 ±0.5 and with that we had a formula (0.5/1.34)*100. So I

do not understand how to compute uncertainty when no estimated uncertainty is given.

INSTRUCTOR RESPONSE:

The +- number is the uncertainty in the measurement.

The percent uncertainty is the uncertainty, expressed as a percent of the number being observed.

So the question in this case is simply, 'what percent of 1.34 is 0.5?'.

0.5 / 1.34 = .037, approximately. So 0.5 is .037 of 1.34.

.037 is the same as 3.7%.

I recommend understanding the principles of ratio, proportion and percent as opposed to using a formula. These principles are part of the standard school curriculum, though it does not appear that these

concepts have been well mastered by the majority of students who have completed the curriculum. However most students who have the prerequisites for this course do fine with these ideas, after a little

review. It will in the long run save you time to do so.

There are numerous Web resources available for understanding these concepts. You should check out these resources and let me know if you have questions.

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QUESTION: I understood the main points of changing the different units, but I’m not sure when in the problem I should change the number to 10 raised to a certain power. In example 1-8 I did not

understand why they changed 70 beats/min to 2 x 10^9 s.

2 * 10^9 is about the number of seconds in 70 years.
70 beats / min were not changed to 2 * 10^9 seconds; in changing the beats / minute to beats in a lifetime, there was a step where it was necessary to multiply by 2 * 10^9 seconds.
The example actually used 80 beats / min as a basis for the solution. This was converted to beats / second by the calculation
80 beats / min * 1 minute / (60 seconds), which would yield about 1.33 beats / second.
This was then multiplied by 2 * 10^9 seconds to get the number of beats in a lifetime:
2 * 10^9 seconds * 1.33 beats / second = 3 * 10^9 beats.
In the given solution 80 beats / min * 1 minute / (60 seconds) was not actually calculated; instead 80 beats / min * 1 minute / (60 seconds) was multiplied by 2 * 10^9 seconds in one step
80 beats / min * 1 minute / (60 seconds) * 2 * 10^9 seconds = 3 * 10^9 beats.
In your instructor's opinion the unit 'beats' should have been left in the result; the text expressed the result simply as 3 * 10^9, apparently ignoring the fact that the unit 'beats' was included in the quantities
on the left-hand side.
Also the text identified this number as 3 trillion. In the British terminology this would be correct; in American terminology this number would be 3 billion, not 3 trillion.

COMMENT:

I thought that these problems were pretty basic and felt that I understood them well. However, when I got to questions 14 (determine your own mass in kg) and 15 (determining how many meters away

the Sun is from the Earth), I did not understand how to complete these. I know my weight in pounds, but how can that be converted to mass in kilograms? I can look up how to convert miles to meters, but is

this something I should already know?

INSTRUCTOR RESPONSE:

Both of these questions could be answered knowing that an object with a mass of 1 kg has a weight of 2.2 lb, and that an inch is 2.54 centimeters. This assumes that you know how many feet in a mile,

and that the Sun is 93 million miles away. All these things should be common knowledge, but it doesn't appear to be so.

For my own weight I would reason as follows:

I weigh 170 lb and every kg of my mass weighs 2.2 lb. I'll have fewer kg of mass than I will pounds of weight, so it's reasonable to conclude that my mass is 170 / 2.2 kg, or about 78 kg.

More formally 170 lb * (1 kg / (2.2 lb) ) = 170 / 2.2 kg = 78 kg, approx.. (technical point: this isn't really right because pounds and kilograms don't measure the same thing--pounds measure force and

kg measure mass--but we'll worry about that later in the course).

Converting 93 million miles to kilometers:

93 million miles * (5280 feet / mile) * (12 inches / foot) * (2.54 cm / inch) * (1 meter / (100 cm) ) = 160 billion meters (approx.) or 160 million kilometers.

QUESTION

What proved to be most tricky in the problems portion was the scientific notation. I am somewhat familiar with this from

past math classes, but had trouble when dealing with using the powers of 10. I had trouble dealing with which way to move my decimal according to the problems that were written as 10^-3 versus

10^3. Which way do you move the decimal when dealing with negative or positive powers of 10?

INSTRUCTOR RESPONSE

Using your numbers, 10^3 means 10 * 10 * 10 = 1000.

When you multiply a number by 1000 you move the decimal accordingly. For example 3.5 * 1000 = 3500.

10^-3 means 1 / 10^3 = 1 / (10 * 10 * 10) = 1 / 1000.

When you multiply by 10^-3 you are therefore multiplying by 1 / 1000, which is the same as dividing by 1000, or multiplying by .001.

For example 3.5 * 10^-3 = 3.5 * .001 = .0035.

As another example 5 700 000 * 10^-3 would be 5 700 000 * (1 / 1000) = 5 700.

From these examples you should be able to infer how the decimal point moves.

You can also search the Web under 'laws of exponents', 'arithmetic in scientific notation', and other keywords.

There isn't a single site I can recommend, and if I did find a good one its URL might change by the time you try to locate it. In any case it's best to let you judge the available materials yourself.

When searching under 'arithmetic in scientific notation' using Google, the following appear as additional suggested search phrases:

scientific notation

exponents

scientific notation metric prefixes

significant digits

multiply with scientific notation

scientific notation decimal

scientific notation lessons

addition and subtraction with scientific notation

scientific notation metric system

'scientific notation lessons' might be a good place to look.

QUESTIONS AND RESPONSES

1)In the text question five asks for the percent uncertainty of a measurement given 1.57 m^2

I think that we figure this by an uncertainty of .01/1.57m^2 = .6369 or approximately one. ??????Am I correct in how I

calculate this??????? Can I asuume that if the number given was 1.579 then we would calculate it by .001/1.57 = .1 % approximately or am I incorrect?????

You're on the right track.

There are two ways to look at this.

1.57 m^2 represents a quantity which rounds off to 1.57, so presumably lies between 1.565 and 1.575.

This means that the quantity is within .005 of 1.57.

.005 / 1.57 = .003, approx., so the uncertainty is .003 of 1.57, which is the same as 0.3%, of 1.57.

Another way to look at it:

1.57 could be interpreted to mean a number between 1.56 and 1.58. The uncertainty would then be .01, which is .01 / 1.57 = .006, or 6%, of 1.57.

2)In the text question number 11 the book asks what is the percent uncertainty in the volume of a sphere whose radius is

r=2.86 plus or minus .09.

I know that the Volume of a sphere is 4/3 pi r^3, so I calculated the volume to be 4/3 pi (2.86)^3 = 97.99 and to get the

percent uncertainty I tried to divide 0.09/97.99 * 100 =.091846, but the book answer is 9% ??????I am not sure what i am doing wrong here?????????????????

Again there are two ways to approach this.

I believe the book tells you that the uncertainty in the square of a number is double the uncertainty in the number, and the uncertainty in the cube of the number is trip the uncertainty in the number.

An uncertainty of .09 in a measurement of 2.86 is .09 / 2.86 = .03, approx., or about 3%. As you state, you cube the radius to find the volume. When 2.86 is cubed, the resulting number has three times

the uncertainty, or about 9%.

Another approach:

Calculate the volume for r = 2.86.

Then calculate the volume for r = 2.86 - .09 = 2.77.

You will find that the resulting volumes differ by about 9%.

You could just as well have calculated the volume for r = 2.86 + .09 = 2.95. Again you would find that the volume differs from the r = 2.86 volume by about 9%.

STUDENT QUESTION

When reading the section about the scientific notation some of the answers were written in powers of 10 and some were just

written regularly. How do I know when to turn my answer into a power of 10 or to leave my answer as is?

INSTRUCTOR RESPONSE

Good question.

Convenience and readability are the main factors. It's a lot less typing or writing to use 438 000 000 000 000 000 000 than 4.38 * 10^20, and it's easier for the reader to understand what 10^20

means than to count up all the zeros.

For readability any number greater than 100 000 or less than .001 should probably be written in scientific notation.

When scientific notation is first used in a calculation or result, it should be used with all numbers in that step, and in every subsequent step of the solution.

QUESTION

In my problems (I am working from the University Physics text- exercise 1.14) they are asking for the ratio of length to

width of a rectangle based on the fact that both of the measurements have uncertainty. ?????Is there anything special you

have to do when adding or multiplying numbers with uncertainty?????? I know that there are rules with significant figures,

but I don’t understand if the same is true for uncertain measurements.

INSTRUCTOR RESPONSE:

For example:

If there is a 5% uncertainty in length and no significant uncertainty in width, then area will be uncertain by 5%.

If there is a 5% uncertainty in length and a 3% uncertainty in width, then it is possible for the area result to be as much as 1.05 * 1.03 = 1.08 times the actual area, or as little as .95 * .97 = .92 times the

actual area. Thus the area is uncertain by about 8%.

This generalizes. The percent uncertainty in the product or quotient of two quantities is equal to the sum of the percent uncertainties in the individual quantities (assuming the uncertainties are small compared

to the quantities themselves).

(optional addition for University Physics students): The argument is a little abstract for this level, but the proof that it must be so, and the degree to which it actually is so, can be understood in terms of the

product rule (fg) ' = f ' g + g ' f. However we won't go into those details at this point.

QUESTIONs RELATED TO UNIVERSITY PHYSICS (relevant only to University Physics students)

I understand everything but the part on measuring the individual i j k vectors by using cosine.

INSTRUCTOR RESPONSE

It's not completely clear what you are asking, but I suspect it has to do with direction cosines.

The vector A = a_1 i + a_2 j + a_3 k makes angles with the directions of the x axis, the y axis and the z axis.

Let's consider first the x axis.

The direction of the x axis is the same as the direction of the unit vector i.

The projection of A on the x direction is just a_1. This is obvious, but it can also be found by projecting the A vector on the i vector.

This projection is just | A | cos(alpha), where alpha is the angle between A and the x direction.

Now A dot i = A = (a_1 i + a_2 j + a_3 k) dot i = A = a_1 i dot i + a_2 j dot i + a_3 k dot i = a_1 * 1 + a_2 * 0 + a_3 * 0 = a_1.

It's also the case that A dot i = | A | | i | cos(alpha). Since | i | = 1, it follows that A dot i = | A | cos(alpha), so that

cos(alpha) = A dot i / | A | = a_1 / sqrt( a_1 ^ 2 + a_2 ^ 2 + a_3 ^ 2 ).

Making the convention that alpha is the angle made by the vector with the x direction, we say that cos(alpha) is the direction cosine of the vector with the x axis.

If beta and gamma are, respectively, the angles with the y and z axes, reasoning similar to the above tells us that

cos(beta) = a_2 / sqrt( a_1 ^ 2 + a_2 ^ 2 + a_3 ^ 2 ) and

cos(gamma) = a_3 / sqrt( a_1 ^ 2 + a_2 ^ 2 + a_3 ^ 2 ).

cos(alpha), cos(beta) and cos(gamma) are called the 'direction cosines of the vector A' with respect to the three coordinate axes.

Recall that alpha, beta and gamma are the angles made the the vector with the three respective coordinate axes.

If we know the direction cosines and the magnitude of the vector, we can among other things find its projection on any of the coordinate axes.

STUDENT QUESTION (University Physics)

Chapter 1 wasn’t bad of course I had to read in detail the vector section there is little confusion on what is meant by

antiparallel. Does that mean that you wouldn’t displace anything if the magnitude was equal only the direction was different?

Also when handwritten vectors are written above the say A the arrow is only in one direction (to the right) not the direction

traveled?

INSTRUCTOR RESPONSE

I don't have that reference handy, but my understanding of the word 'antiparallel' is two vectors, one of which is in the direction exactly opposite the other.

If two vectors are antiparallel, then their dot product would equal negative of the product of their magnitudes:

The angle theta between antiparallel vectors v and w would be 180 degrees, so v dot w = | v | * | w | * cos(180 deg) = - | v | * | w | .

STUDENT QUESTION

I do not understand the answer to problem 13b. I do not understand why it is not correct to write the total distance covered

by the train as 890,010 meters. I do not understand this because 890 km equals 890,000 meters and if you add the 10 meters

the train overshot the end of the track by, it seems to me the answer should be 890,010 meters. I think the answer has

something to do with uncertainty, but I cannot figure out how to apply it to this problem.

INSTRUCTOR RESPONSE

If the given distance was 890. kilometers instead of 890 km, then the 0 would be significant and it would be appropriate to consider additional distances as small as 1 km.

Had the given distance been 890 000. meters then all the zeros would be significant and additional distances as small as 1 meter would be considered.

As it is only the 8 and the 9 are significant, so that distances less than 10 km would not be considered significant.

Please feel free to include additional comments or questions:

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