PH1 Query 0

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course PHY 201

5/28/2012 5:30

Samantha RogersPHY 201

PH1 Query 0

ph1 query 0

Most 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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The discrepancies in timing are on the order of 0.1 second.

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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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The numbers are all within 0.2 seconds of each other which is good, but if they were closer that would be better, so more precision would give us more accuracy. It doesn't seem that a high percentage of discrepancies in timing are due to this factor, but with more rolls it could become more of an issue.

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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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Human triggering most likely is not a huge discrepancies in timing because a human's reflexes can be very fast. If a computer where triggering the timer that would be more reliable but error could still be involved too.

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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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The discrepancies for this most likely have to do with object itself. For example, what if the object went down the slope slightly crooked? That could change the timing of the object. Or what if there was a slight imperfection on the surface of the slope itself? These both would be discrepancies in timing.

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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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The percent of discrepancies in timing due to the differences in positioning the object prior to release is most likely very low. At this initial point, the object does not have that much velocity therefore perhaps the slightest difference might not make the biggest difference overall in the timing (as long as the difference is not huge).

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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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The percent of discrepancies in timing for when the object reached the end of the incline would be much higher. This is because the object has a high velocity and a slight error of timing could have a greater effect.

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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.

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I would be about 60% confident. After fixing all of the discrepancies, the timings for all of the intervals would most likely shift. That being said the overall mean of the given intervals would shift as well. What is unknown to me is how much this would effect the shift.

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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 be 10% confident. This level of confidence even seems a little bit high .01 is as precise as the timer program is in this example. Basically to get the mean within .01 second would mean the mean would have to be exact.

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

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Once again, I would not be very confident.

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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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Yes, because the uncertainty is around 4%, which gives the estimated mean a greater range to be within the actual mean.

No, because 0.01 second is the smallest precision that the program can make, therefore the estimated mean would have to be exactly the actual mean.

No, I could not be 90% confident. Once again the chances of the actual mean being within the +/- range of 0.03 seems slim to none.

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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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The only thing that you could do about the lack of precision of the TIMER program is maybe run the experiment multiple times and then average those times to get a more accurate measurement. Other than that, the computer program is already set.

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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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To help the uncertainty of human triggering, the human could practice multiple times. Once again, taking the average of the times will help as well.

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

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We do not have much control over this factor. Averaging the measurements will help eliminate uncertainties about the time required for the object to travel then same distance.

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· Differences in positioning the object prior to release.

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To eliminate this uncertainty a release system could help. This would ensure that the object is starting at the exact same place and by the exact same way every time.

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

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Just like having a starting system, we could construct a stopping system too. This stopping system could trigger a sound that is very easy to hear. Once the observer hears this sound, they would stop the timer.

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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 find the object's average speed, I would take the distance traveled and divide it by the time it took to travel the distance. velocity = distance / time

confidence rating #$&*: 3

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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:

40cm/5sec= 8 cm / sec

This is connected to my experience because I had a total distance that the object had to travel and I timed how long it took for the object to travel this distance. So I also too the distance the object traveled and divided it my the time it took to get an average velocity on the incline.

confidence rating #$&*: 3

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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:

40cm/2=20cm (half way point)

20cm/3sec= 6.67cm/sec average velocity on the first half

20cm/2sec= 10cm/sec average velocity on the second half

First I had to split the incline in have so i did 40/2 to get 20 cm distance traveled for the object on each half. Then I kew that it took the object 3 seconds to travel the first half, so I subtracted 5(total time)-3(time required for the first half)=2.

Confidence Assessment: 3

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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:

Then According to the results of my pendulum experiment, doubling the length of the pendulum resulted in more than half the frequency of the first pendulum.

Confidence Assessment: 3

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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:

This is because when a point is on the x axis there is no vertical positioning therefore it has a y-value of zero. When a point is on the y axis there is no horizontal positioning therefore it has a x-value of zero.

Confidence Assessment: 3

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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:

If the line were to intersect the vertical axis it would mean that the length of the pendulum is somehow negative (therefore nonexistent), but according to the graph it would still have a very high frequency.

Confidence Assessment: 2

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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:

If the line were to intersect the horizontal axis it would mean that the length of the pendulum is very long but the frequency is negative, therefore the pendulum would not complete a full cycle.

Confidence Assessment: 2

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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:

These two points are 30 cm apart. The average velocity is 6cm/sec and it takes 5 sec.. 6cm/sec * 5sec = 30 cm.

Confidence Assessment: 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:

My solution was correct.

confidence rating #$&*: 3

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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:

I understand the concept of percent uncertainty (questions 5 and 6) but question 11 confused me with having to do with volume of the sphere. I know the formula for finding the volume of a sphere and I know how to find the percent uncertainty for straight measurements. It says +/- 0.09. So when do I use this? After I've found the volume of the sphere????

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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^2I 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.

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