In this first chapter of the Physics Companion (The PC) you'll be learning some physics, but you'll be equally concerned with developing your computer skills. Some of these skills are of a general nature - What's a root directory? How do I pull down a menu? Others are more specific - How does a spread sheet work? How do the various feature of The PC work?

You'll learn some of these skills by following directions. You'll see typical computer lingo like File/Print Frame along with an explanation of what this lingo means. You need to learn this lingo in a specific and general way.

File/Print Frame means that you click, with your left mouse button, on the File menu and then move your pointer down to Print Frame and release it. That's the specific part. Now what would Edit/Copy mean.

There are millions of computer literate people in the world today and most of them never had a formal lesson. They all ran into File/Print Frame and said what the heck does that mean. Most asked a friend. When they got to Edit/Copy they had two choices - ask again or look for the pattern and figure it out for themselves. The fact that this has worked so well says a lot about teaching and learning. By learning some of your physics using The PC you'll be greatly enhancing your computer skills and using a technique that seems to be a natural fit for millions of learners.

So when you come to a set of specific instructions, be sure to stand back and see the big picture. You'll be on your own more as we proceed. Good luck.

Graphing & Proportional Relationships

Physics is about relationships. In order to send a rocket to the moon, for example, we need to understand the interrelation among such factors as the thrust of the engines, the forces of gravity and air resistance, and the changing mass of the rocket. We can discover these relationships by analyzing the results of careful experiment. A common sequence of operations is

1. take data
2. graph the data
3. derive equations that describe the graphs

In these activities we want to develop the skills required to use this process. But first we need a little refresher on good graphing technique.

1. Graphs—Who Needs Them?

Have a look at the following graphs and discuss what you think they might be useful for or what you might be able to learn from them. After your discussion, click on the comments icon () below.

Sample Graph #1

This is a graph of the decay of a sample of radioactive material. The rate at which it decays, in counts per second, is plotted against the time over which its decay is observed. Just what does this graph say about this decay? What sort of use do you think might be made of such a graph?

(Note: When you click on the comments icon a new window will appear. When you finish with such a window, be sure to close it. You can do this by clicking in the close box at the top right of the window or use the Close button below the text.)

2. This is a graph of the simultaneous motions of two cars. First try to picture these two motions assuming that the cars were side-by-side at the start. Re-enact these two motions on your desk top.

You probably made some useful observations from the graphs, but you may not have thought about questions like the ones in the follow-up comments. Graphs, for most people, are something you're told to draw, not something you use to answer questions. In Physics we're interested in relationships. Graphs are one of the best ways of organizing information so that we can find these relationships. In the activities that follow you'll learn to recognize some important types (shapes) of graphs. But first we need to look at the basics of drawing graphs.

So far... Be sure to check out the So Far...'s. They summarize each section. Great for test preparation. Click in the frame and Print for a whole chapter's worth. Hint.

2. Data Tables & Graphs

Usually when we're studying some phenomenon we start by taking data and organizing it in a data table. In this table you see high and low temperatures for Lexington, Kentucky for each Monday in 1995, beginning in January. (By the way, this data was easily obtained on the Internet.)
Look at how the table is laid out. The dates are in the first column and the temperatures are in the second (and third). These two variables, the week number, and the temperature are the independent and dependent variables, respectively.

The independent variable is the one that you manipulate or control.

The dependent variable is the one that responds to changes in the independent variable.

We "controlled" the dates in that we chose each Monday as the time to collect data.
By convention, we always put the independent variable in the first column. That way everyone will immediately know which is which when they look at the table.
Now look at the data in the table and see if you notice any trends.

The data is a little hard to analyze since weather is not very well behaved. But you can probably see that there is a general rise and fall over long periods of time. Are there any other trends? A graph would make it a lot easier to see.

To speed up the graph-plotting process, we'll use a piece of software called "Graphical Analysis" hereafter referred to as "GA." It allows you to enter data and let the computer plot it in various ways.

Think about what you'd do if you were doing all this by hand. You'd have written down the data table with title, headings, and units. Next you'd decide on a scale for each axis. Then you'd label the axes with the same names and units used in the data table. Your graph would probably get the same name as the data table. You'd then plot each data point. You might draw a line or curve through the data.

For this activity the data has already been entered and saved on the CD-ROM for you. Watch carefully how GA works. It needs information from you about what goes where, what kind of curve to draw, etc. It does a lot without much work on your part. You'll use GA a lot, so take careful note of how it works.

Leaving your browser running, start GA. Hint.

As you follow the directions below, you can switch back and forth between your browser and GA using two methods:

1. Click on the buttons on the task bar.
2. Using Alt Tab: Hold down the Alt key and then hit the Tab key.

After you start GA, you'll need to go to the File menu and Open the file "hi-low-t.dat". This file is found on your CD-ROM in the directory called "ga-data" which is in the root directory of the CD. Here's how.

When you select File/Open you'll see a menu like this. By default, GA looks in its own, Vernier directory on the C drive for .dat files. You need to change to the CD-ROM drive. This is probably the D drive on your computer. Double click on the [-d-] under directories to switch to the D drive. Double click on [ga-data] to switch to that directory. Then double click on hi-low-t.dat and you're there. (GA may find the ga-data directory on its own.)

You'll see the data table appear on the left and the graph on the right.

Make a statement about the trend in either high or low temperatures over the year.

Here's a tougher assignment. Is there a trend in the weekly difference between the high and low temperatures over the year? That is, does the temperature typically vary during a week by a larger amount in one season than another? It's too hard to tell from the graph. We need a graph of the weekly temperature range (High - Low) vs. time. Think about how you'd create such a graph. We'll let GA do all the work.

Here are the steps in GA.

1. Select DATA/NEW COLUMN/CALCULATED.
   To set up the column using this requester:
   
2. Name the new column TEMP RANGE.
3. Click in the Units box & type F for Fahrenheit.
4. Click in the NEW COLUMN FORMULA box.
5. Click the COLUMNS button. Move the pointer down to HIGH and release. "HIGH" should appear in the formula box.
6. Click - (minus) in the calculator pad. Then choose LOW with the column button. It should now look like the figure. Click OK.
7. Click in the y-axis label on the graph. It probably says Mixed(F). In the requester that appears, click in the Temp Range box to turn it on and in the High and Low boxes to turn in them off. Click OK.

Make a statement about the trend in the daily temperature range over the year. (You won't be able to say much. I wonder if this answer would be different in an El Nino year.)

You've just been using a spread sheet. Notice how its ability to do all that subtracting allowed you to stay with the concepts at hand rather than spending an hour doing repetitive calculations.
I talked you through it this time and I'll give you more help along the way, but you should do what it takes to learn to figure out such procedures using the manual, the Help function and just the logic of the program and its menus. This is now a life skill. Don't panic! You'll catch on.

You might find it useful to go back through the previous steps. Think about our goal - the graph of High - Low. The steps should make perfect sense.

QuickTime Movies

You got the temperature data free. You'll usually have to work for it. One way to do this is to collect data from movies. Let's see how QuickTime movies work. Watch the movie to the left. (If you haven't run a movie since your last computer re-start a requester will pop up here. Just choose Later, or whatever means No, and it will go away.) You can start the movie, and run it forward or backward using the VCR-style controls below it. Cool.

If the movie started running on its own, click on the icon, click "plug-in settings," and then click in the box beside "play movies automatically" to make the check mark go away. Click Save. You should only have to do this once.

• Use , the spacebar, or double click on the movie to start it playing.
• Use , the spacebar, or single click on the movie to stop it.
• Use or the right arrow on your keyboard to move forward one frame.
• Use or the left arrow on your keyboard to move backward one frame.
• Hold down the left or right arrow keys on your keyboard to play the movie backward or forward.
• Hold down or to play the movie very fast in either direction.
• Drag the box-shaped slider in either direction to move forward and backward.
• Hold down the Shift key and double click on the movie to play it backward.

To use the keyboard commands to operate a movie you must click in the movie first. Also, some movies do not play well backward.

Some important details about graphing

The relationship between the weekly high temperature and the date is obviously quite complex. In the study of introductory physics we usually work with much simpler relationships. Let's take some data for such a phenomenon.

The movie you just watched just shows you the basic setup. When a weight is added, the spring stretches by some amount. We want to determine the exact relationship between the amount of weight added and the length or stretch of the spring. Let's take some data. Play this movie.

As you can see, the stretch of a spring-the amount of change in its length-varies as weight is added to its lower end. Initially there is no weight added and no stretch. Is that clear? Initially there is no stretch and no added weight.

We could calculate the added weight and stretch with the following equations where the original values are at the start of the movie.

Stretch = current length - original length

Weight Added = current weight - original weight

Which is the independent variable? (Click somewhere on the little letter/blue semicircle and see what happens.)

  Stretch
  Added Weight

(If you don't see the letters and semicircles next to the answers above, you haven't activated Macromedia Flash. This was explained in "Before You Start" on the initial splash page when you started The PC. See Help for help using buttons.)

To find the stretch, you need to subtract the initial position of the pointer from its stretched position. I'd say the initial position was about 83.4 cm. (Yes, you can read this scale. It's supposed to be fuzzy looking! Does a precision of .1 cm seem ok?) Let's find the first (after the zero) stretch value. To get stretch data, restart the movie and run it again using the VCR controls or keyboard arrows. Just move along full speed or frame by frame, stopping to collect each new piece of data. You can go backward or forward at will. You have complete control.

Start the movie & stop it after the first weight has been added and come to rest. The pointer is at about 85.2 cm. Subtracting 83.4 cm gives us a stretch value of 1.8 cm. Type this value in the second column of the data table. (Click to the right of the last zero and then press RETURN to get a new line.

Each thin, brass mass weighs about .50 N (Newton) (about .11 lb.). The thick ones weigh twice as much. Type .50 under 0.00 in the first column.

2. Continue to add data from the movie to the table provided. The data table should include eight non-zero values.

3. Transfer your data into a GA data table. Here's how: (We'll be doing this a lot so pay attention.)

• Click and drag to select your weight data. In normal language this means: Place your cursor to the left of the top left zero in the weight column. Press the left mouse button, but don't release it. Drag the cursor down and to the right of the last value in the weight column, 4.00. All the data should have been highlighted by this action. Now release your mouse button. Here's what it should look like. Be sure to use the "back to the table button" to return to your data.
• Copy the data using one of these methods:
  • Select EDIT/COPY.
  • Right click on the (still highlighted) data and click Copy from the pop up requester.
• Run GA. (If it was already running and you already have data in the table, choose FILE/NEW to start a new data set. Save first if you need the old data.)
• Click in the graph window, then click on Graph in the menu bar. If there is a check mark beside "connecting lines", click on it to turn off this feature.
• In the data table, select the first blank box under X by clicking in it.
• Select EDIT/PASTE. This will paste the data into this column. (Right clicking won't work in GA.)
  (It's a bit buggy here. If the data doesn't paste, look at the GA tab in the task bar. If it is not in the "depressed state", click on it. This should fix the problem.)
• Repeat this transfer process with the stretch data, putting it in the y-column.
• You'll see that the graph is immediately drawn for you, but the labels are wrong. To make them match the data:
  • Double click on the X heading in the data table. A requester will appear. It will allow you to change the name and units for that column. (Weight (N))
  • Repeat for the Y column. (Stretch (cm))

4. Save the data using the name "Spring1." You'll need a floppy disk for such files. You could use the hard disk, but you'll need to be using the same computer the next time you need to look at this file.

5. Does the data appear to lie approximately on a straight line?

Use your graph, hopefully with a straight line through the points, to answer the next two questions.

If you got the same results as the Chuck Guy, you arrived at these values because you trusted that the stretch increased in a uniform way with the added weight. That is you assumed that the data was along a straight line. We would say that there was a linear relationship between the stretch and the added weight. You used interpolation to find the first value and extrapolation for the second. I hope those terms ring a bell.

8. Incidentally, the graph does not remain linear; eventually the spring will reach its elastic limit. At this point the graph will bend. Do you think it will bend

  toward the vertical?
  toward the horizontal?

9. Could you use such an assumption (of linearity) to determine the high temperature on the third Thursday in January on our High-Low temperature graph?

10. Is our spring data exactly in a straight line?

The fact that our data isn't exactly on a straight line could mean either that the data isn't linear or that our apparatus and measurement technique aren't perfect. Well, the latter is always true. Our data will never be perfect. It's always a judgement call, but I'd say that this data is clearly linear.

Here's a key point. Saying that it's a linear relationship means that the "real" values should be along a straight line. What we call the "line of best" fit is a straight line that passes among the data points showing the ideal, no error case. After we take our data we'll always want to draw this line if our data is linear.

There's a pair of equations that can determine this line for a given set of data. We'll avoid them and either use GA or the skills that we develop by looking at the lines that GA draws. It's really not hard if you pay attention.

11. To produce the line of best fit with GA, click anywhere on the graph to select it, then click on the automatic curve fit icon located near the top of the screen. It will give you several choices of equations that might describe this graph. Choose "y = mx + b". You will see some other options that we'll try later, but for now, just choose OK, then OK, keep fit. The graph will be redrawn with a nice straight line through the data.

12. If possible, print out your graph.

13. Look carefully at how the line sort of averages out the deviations from a straight line. The data points miss by about as much above the line as below it. Also, the earlier points on the left side of the graph miss by about as much as the later ones on the right side. With practice you'll be able to draw a pretty good line of best fit on your hand-drawn graphs.

14. Find a point on your graph where, for a given applied force, the line does not go through the data point for that force value. If you were asked to give your best estimate for the amount of stretch that would occur when that particular force was applied, would you use the value from your data table or the corresponding point on the line of best fit?

data point   point on line

With measurements, we use various methods of averaging. The line of best fit plays that role here. All of your measurements will be just your best attempt for a given situation. By taking many readings and "averaging" them using the line of best fit, you get your best estimate of the true behavior of the spring.

We never use our data points once our line or curve of best fit is drawn! (If you do, your grade will be turned into a pillar of salt.) (Well, if the graph goes exactly through a point...)

Many of the important techniques in graphing were illustrated in the previous exercise. Here is a synopsis of some of those and some others that weren't addressed.

Graphing Technique 1. Always use a pencil! 2. The more data points you plot, the more accurate your graph will be. 3. Determine the independent & dependent variables. the independent comes first in the data table; it's plotted on the abscissa (horizontal). the dependent comes second in the data table; it's plotted on the ordinate (vertical). 4. Noting the range of values to be plotted for each axis, choose a convenient scale for each variable so as to spread the graph out nicely. allow for easy plotting by letting each grid line count say, 2, 5, 10. 5. Label each axis according to what is plotted there. Include units. 6. Draw the line or curve of "best fit." Make no attempt to hit any point, even 0,0 Make no attempt to "connect the dots." 7. Title the graph appropriately. 8. Ignore your data points from here on. Use the line of best fit to represent the data.

16. Before you move on, take some time to see how each of the eight graphing technique items (except #1) is indicated by our final table and graph. Be sure to think about #2.

So far...

3. Linear Relationships & Direct Proportions

We described the data from our spring experiment as being linear. It also fits in a more specific category, it is a direct proportion. The direct proportion is a specific case of the more general linear relationship. Let's see exactly what that means and compare the two relationships.

We found earlier that the spring stretched by about 1.8 cm when the first .50 N was added. Adding this to the original length gives us the length of the spring after stretching. Got it?

Can you see that each value in our stretch table needs the original length added to it to produce a table of length vs. added weight? How will this graph compare to "Spring1?" Let's use GA to plot this for us.

1. Run GA and open your Spring1 file.

2. We need to add a new length column where we'll add the original length to each stretch value. Here's how.

1. From the DATA menu, select NEW COLUMN/CALCULATED. Name this column "Length" and use "cm" as the units.
2. Click in NEW COLUMN FORMULA.
3. In COLUMNS choose Stretch.
4. Using the keypad on the screen add "+12.8"
5. Click OK.
6. Click in the ordinate label in the graph. Check the "Length" box in the requester.
7. Select the Curve Fit icon and select "Data Set 1: Length" in the PERFORM FIT ON selector.
8. Use y = mx + b to draw the line of best fit.
9. Save as "Spring2".

3. Print out your graph. The new line looks similar to the first line except for one simple difference. What is it?

Both of our graphs are linear; that is, the data points are along straight lines. But we can also add that the stretch vs. added weight graph indicates a direct proportion. Let's check out the similarities and differences between these two relationships.

4. Complete the stretch and length columns in this data table. Remember to use points on the line, not data points. A nifty way of doing this is to use GA's examine feature. Click its icon and move your cursor across the graph. Your Length & Stretch values will appear in boxes along with the best fit value. Use the latter.

You can see that the stretch vs. added weight graph is linear from the fact that the stretch increases by equal amounts with equal increments of added weight. That is, with each .50 N of added weight, about 1.66 cm is added to the amount of stretch.

Check the length vs. weight. It should do the same.

When a graph is linear, equal changes in one quantity result in equal changes in the other, like equal size stair steps.

Now, here comes the important distinction.

5. In column 1, the weight is doubled, then tripled as we go from .50 to 1.00 to 1.50 N. Does this doubling and tripling also occur for the stretch data?

6. Does the length data also double and then triple?

We say that the stretch of the spring is directly proportional to the added weight. By this we mean that if you double the weight, you'll double the stretch. The same goes for tripling or even multiplying by 2.7

The length of the spring, however, is not directly proportional to the added weight. Doubling the added weight does not double the length. A little investigation will show you that the graph of a direct proportion is always linear and it always passes through the origin. (Remember, with non-perfect data, approximate results are all we can expect. It will usually just come close to the origin just as the line just comes close to the data points.)

7. Another important point to see about direct proportions is that one variable is always some multiple of the other. In the fourth column, calculate the ratio of the stretch divided by the weight for each of the three weights. As you can see the ratio is pretty constant. If stretch/weight = k, then stretch = k· weight. That is, the stretch is always a constant multiple of the weight. Constants like these, called constants of proportionality usually have special meaning in physics. This one (actually its inverse) is a measure of the stiffness of the spring; it's called the spring constant.

8. What do you think the k's indicate in these equations?

Back to our spring. Since the constant indicates how these two quantities are related, we would like to learn how to accurately determine the constant. You probably got almost exactly the same value from each calculation of this constant. (This is because you got your values from the line of best fit. If you had just used your data points you would have found much more variation. Here again we find that the line of best fit provides a sort of averaging.)

So far...

4. The Slope of a Graph

The slope of a graph is a measure of the steepness of the graph. Let's look at what this steepness might indicate about a particular relationship. The graph shown on the left records a certain student's cookie sales for a school fund-raiser. This student has decided to sell a certain number of boxes each day so as to be sure to reach her goal of 150 boxes at the end of a month.

After 10 days she had sold 50 boxes.  
This corresponds to the point, (t₁, c₁) or (10 days, 50 boxes) on the graph. Ten days later, at (t₂, c₂) she had sold a total of 100 boxes. The number of boxes she sold during the interval between t₁ and t₂ is just:

Δc = c₂ - c₁ = (100 boxes - 50 boxes)
= 50 boxes.

We use the capital Greek letter delta, Δ to indicate any difference or change and we compute the difference as the final value minus the initial value.

The time interval for this sale is just
Δt = t₂ - t₁ = (20 days - 10 days)
= 10 days.

  The ratio of Δc/Δt is:
50 boxes÷10 days = 5 boxes per day.

This is just the rate at which she sold cookies. As you probably realized, it is also equal to the slope of the graph between the two points we considered.

The slope of a graph is the rate at which the dependent variable changes with a given change in the independent variable.

1. On the graph you see each point labeled as well as the calculations required for finding the slope. Quickly sketch this graph.
Beside (t₁, c₁) and (t₂, c₂) fill in the corresponding numerical values. Now do the same with the calculations. Be sure to do this in the logical order of the calculation of the slope as it was worked out above.

2. Be sure to include the units in your calculations. The numbers don't mean much without them.

3. Calculate the slope between another pair of points of your choosing.

4. Make a statement about the slope between any two points on a straight line.

5. You just found out that in calculating the slope, any two points should do. You'll find however, that the farther apart the points are, the more accurate your results will be.

6. The next graph indicates the sales of another child. Find the slope of this graph.

7. Write a sentence, including numbers, that indicates the meaning of this slope.
Also comment on the meaning of the steepness of this graph relative to the first one.

A more general definition of the slope of a line follows. For a graph of y vs. x, where m is the slope and using the common terms "rise" and "run" for the quantities y₂-y₁ and x₂-x₁ respectively.

Direct Proportions Revisited

We found earlier that if y is directly proportional to x, then y equals some constant times x, i.e., y = k x. The constant is lurking in your graph. Let's find it.

Our equation for the slope of a graph of y vs. x involves a pair of points, (x₁,y₁) and (x₂,y₂). Suppose we pick the first point as the origin, (0,0). Our slope equation then simplifies to m = (y₂-0)/(x₂-0) or more simply m = y/x, or y = m x.

8. What new meaning does this give to the slope of the graph of a direct proportion?

Direct Proportions Example

Let's look at another situation that illustrates two related variables. Watch the tractor movie.

1. In the Tractor movie there are also two things changing, the time and and the distance traveled. Hopefully you can see where we're going here.

What do you think we'll find that is constant?

2. Let's take some time and distance data for our tractor.
Use the location of the back of the tractor as the "location" of the tractor. I've estimated that it's at about the 9.2-cm mark in frame zero. Let's let this position be the starting point of the tractor's motion. To do this we'll just subtract 9.2 cm from each scale reading as we go. Having done this, our starting point will be 0.0 cm. This 0.0 point is already in the data table (below).

Now move forward one frame at a time until the "Frame 15" sign appears. Since this movie runs at a rate of 15 frames per second, the tractor has now traveled for 1.00 second. (I've guessed at the precision of .01 s.)
Enter 1.00 in the time column and read the tractor's position from the meter stick. Subtract 9.2 cm from this reading to get the distance it's traveled during this first second.

4. Save your results in GA as "tractor". Print out your graph.

5. Draw your best attempt at a line of best fit.

6. Determine the slope. Remember, use your line not the data points.

7. Now have GA create a line of best fit: Click in the graph, then click the Automatic Curve Fit button . Choose "mx+b" from the stock functions. Compare the slope, m, that GA gets, with yours. If you are far off you need to stop and reassess your technique. (More on the y-intercept later.)

8. Back to direct proportions. What can we say about the relationship between the distance traveled and the time?

9. On the basis of what two facts about the graph can you make that claim?

So far...

5. Equations from Linear Graphs

1) What was the high temperature for Lexington, KY on the 6th Friday in 1995?

2) How many boxes of cookies did the first student sell in 10 days?

You may not have been able to answer either question from memory, but suppose I told you that you were going to have a quiz tomorrow where I would ask you two similar questions, just using different times. Which question would be easier to prepare for? Obviously the second question would be easier. There is a simple mathematical relationship; she just sells 5 boxes per day.

As an equation, this relationship might be stated as:

What we've found is that an equation can replace an entire graph! You might think that this might only be true for straight line graphs, but it's not. Let's see how this is done, starting with a couple of graphs we're already familiar with.

1. Get out your spring2 graph. We'll look at the stretch graph first. It's linear and goes through the origin. What did we call such a relationship?

We want to derive an equation that describes this line. That is, we want to be able to insert any value for the added weight and have the equation compute the corresponding stretch.
We'll start with our definition of the slope. (We'll use y's and x's at first for generality)

This equation is valid for any pair of points, (x₁, y₁) and (x₂, y₂). Any two points should be valid, but to simplify we'll pick the origin, (0, 0) for (x₁, y₁). After substituting, simplifying, and solving for y, the equation becomes:

General form of a direct proportion: y = m·x

In our case, y and x represent the stretch and the added weight, respectively.

Stretch = m · added weight

2. Find the slope of your line and re-write the equation using the actual value for your slope. Be sure to include units in your slope.

3. Test your equation by,

• picking a weight value and see if the equation calculates the matching stretch, and
• picking a stretch value and see if the equation calculates the matching weight.

General form of a linear relationship: y = m·x + b.

For our spring, x again represents the added weight, but this time y represents the actual length of the spring including its initial length. So, for our spring,

Length = m · added weight
+ initial length

4. Find the slope of your line and the y-intercept. Re-write the equation.

5. Again, test your equation by,

• picking a weight value and see if the equation calculates the matching length.
• picking a length value and see if the equation calculates the matching weight.

Direct Proportion	Linear Relationship
y = m·x	y = m·x + b

Of course the first equation is just a special case of the second in the same way that the direct proportion is a special case of the linear relationship. Also, note that there are other equations of straight lines. We'll just use y = mx + b. It's called "slope-intercept form."

Example of slope-intercept equation

1. To the far left you see a standard thermometer with Fahrenheit on the left and Celsius on the right. Beside it is a movie which is a close-up of this thermometer when its bulb is immersed in ice-water.

2. Read and record in the table 10 temperature pairs from the thermometer in the movie. You pick 'em. The movie starts out slow. Be patient.

Temperature (oC)

Temperature (oF)

3. Use GA to plot your Fahrenheit vs. Celsius data, including the line of best fit. Notice the values GA provides for the slope and y-intercept.

4. Print out the graph.

5. Write the equation of this line. Be sure to include units (^oC, ^oF).

6. One last thing. Quickly sketch a graph of Celsius vs. Fahrenheit.

7. Write the equation of this line.

8. Now get GA to do the last one for you. It takes about 10 seconds once you figure it out. Do it and see if GA gets the same equation that you got.

Linear Relationships Practice Questions

1) Income vs. hours worked by an hourly wage-earner

2) The total weight of a table & its contents vs. the number of books stacked on it.

3) The volume of a bath tub vs. the time required to fill it.

4) The length of your hair vs. the time since your last haircut (OK, buzz cuts notwithstanding)

5) For the two non-direct proportions above, state, in words, what the y-intercept represents.

6) Direct proportions always indicate a linear relationship.

7) ... and visa versa.

Descending Linear Graphs

Find the equation that describes the following financial arrangement. A college student buys a debit card that can be used for meals at the school cafeteria. At the beginning of the month the card's value is $300. Each day he buys three $5 meals. Your equation should relate the dollar value of the card vs. the time in days since the beginning of the month.

So far...

6. Inverse Proportions

Recall our spring experiment. We found that the stretch of the spring was directly proportional to the amount of force applied. A compressible spring works the same way. Now we want to look at a similar arrangement, except that this time we'll compress air instead of a spring.

1. In this movie, a 10-N weight is being added to a loop of wire that pushes down on the plunger of a syringe. The syringe is closed at the other end to prevent air from exiting. As a result the air which is initially at atmospheric pressure is compressed into a smaller volume and its pressure increases.

We want to determine the relationship between the volume occupied by the air and its total pressure. The total pressure is the sum of the pressure applied by the plunger and the initial atmospheric pressure of the air inside the syringe.

P_total = P_applied + P_atmospheric

The pressure applied by the plunger is equal to the force applied by the hanging weight, 5.0 N (Newtons), 10.0 N, etc., divided by the cross-sectional area of the plunger. The plunger has a diameter of 1.50 cm. From this you can compute its cross-sectional area.

This calculation will give us pressure values ranging from about .1 to 20 N/cm². To this pressure we must add the initial atmospheric pressure which is about 10 N/cm². This sum gives us the total pressure of the air in the syringe.

The volume occupied by the gas can be read directly off the syringe in units of cm³.

Here's an animated summary of our experiment.

2. Collect force and volume data from this movie. Take readings each time a force value appears on the screen. (A force of 0.0 N appears on the first frame of the movie.) Read the volume using the lower of the two black rings on the plunger. This ring is just below the top white line in the first frame. The white lines in this frame are at 7, 8, and 9 cm³. The initial volume appears to be about 8.9 cm³. Enter 0.0 and 8.9 in the table for your first force and volume readings.

To continue collecting data, run the movie until you see a yellow force box appear at the top. (You can step backwards using or the left arrow key if you run past it.) Record the force reading and read and record the volume reading. Repeat until you reach the end of the movie.

3. We'll use GA to plot volume vs. pressure. To convert our force readings to pressure we'll have to divide each force by the area of the syringe and add the 10 N/cm² of atmospheric pressure.

Let GA do this work for you using its spreadsheet capabilities. Do the following in GA.

1. Paste the force values into the first column. Double-click on the column heading ("x") and rename it "Force", giving it units of "N."
2. Do the same process in the second column with the volume. The units should be cm^3.
   A graph of Volume vs. Force is automatically plotted. We want Volume vs. Pressure.
3. Select DATA/NEW COLUMN/CALCULATED. Name the new column "Pressure" and give its units as N/cm^2.
   Click in "New Column Formula" and then using the calculator on the screen click the keys to produce
   10 + (
   In the "Columns" pull down requester, select "Force." "Force" will be added after the (
   You should see
   10 + ("Force"
   Using the calculator, enter
   /(3.14 * .75cm^2))
   This will divide the force by the cross-sectional area.
   You should now see
   10 + ("Force"/(3.14 * .75^2))
   in the box. Click OK.
   A new column will appear in the GA data table. It contains our calculated total pressures. Got it?
4. To get a plot of volume vs. pressure (instead of force), click in the abscissa label (Force) and then click on "Pressure" in the requester that pops up. You should now have your graph.

4. Is your graph linear?

(OK, now we're going to do what's called "cooking the books" in the business industry. If you don't like what the books say, put a new spin on the numbers. This skill will bring in even bigger wads of cash.)

5. We've found that the compressed gas exhibits a behavior similar to that of a spring in that the air in the syringe requires increasing amounts of force to compress it by greater amounts. However, our graph indicates that the systems are very different. The force vs. length graph of a compressed spring is just a descending straight line; our gas exhibits a more complex behavior.

Let's attempt to find a simple description of the behavior of a compressed gas. The next few steps may look sort of magical at first. Just try it and see how it works out.

6. Notice how the volume and pressure change in opposite senses. As the pressure increases, the volume decreases. That's why the graph is dropping. But it's not dropping linearly. This calls for more creative action.

Consider the descending sequence
5, 4, 3, 2, 1.
Their inverses,
.2, .25, .33, .5, 1,
ascend.

To make our pressure data change in the same way as the volume, let's try plotting inverses of the pressure. That is, we'll plot volume vs. the inverse of the pressure.

First we have to calculate the inverse (1/P) for each pressure value. But let's let GA do the work.

1. Select DATA/NEW COLUMN/CALCULATED.
2. Name the column "1/Pressure".
3. For units, use "cm^2/N."
4. Click in the "New Column Formula" box.
5. In the calculator key pad, click
   1/
6. Click on the "Columns" box and select "Pressure".
7. Click OK

By the way, when we make these changes in the data we will always work with the independent variable, the value on the x-axis.

7. Look at your new column.

8. Did this do any good? Look at the volume data and our new column of 1/P data. Find a value near the top of the volume column and another value farther down that is about half its value. Now look at the matching 1/P values.

Are they also about 2:1 in value? What this indicates is that the quantities in these two columns are directly proportional to each other. A plot should be a straight line through the origin.

9. Try plotting V vs. 1/P, as follows.

1. Click anywhere in the graph window.
2. Select WINDOW/NEW WINDOW/GRAPH (Either "New Window" will do.)
3. On the graph, click on the vertical "---" on the ordinate and chose "Volume" in the selection box. Click OK.
4. Similarly, click on the horizontal "---" on the abscissa and chose "1/Pressure" in the selection box. Click OK. Poof. It's a straight line (very) roughly through the origin.

   (If the graph doesn't start at (0,0), click in the lowest value label for each axis and change it to 0. Also, if there are no point protectors, that is, you don't see the plotted points, click anywhere inside the graph, then select Graph/Point Protectors.)

10. Get GA to draw the line of best fit. Write down the slope. (It gives you the y-intercept which is, of course, approximately zero.)

11. Write the equation of your line.

You may have thought it strange to create something like 1/Pressure, but the fact is that we found a straight line and created a useful equation. Within the limits of the precision of our apparatus, it looks as though the volume is directly proportional to 1/P.

Now we can just do some algebra. If
V = (slope) 1/P
(ignoring the small y-intercept), then
PV = slope
This is a nice, simple equation, with no 1/P's in it. You may recognize this relationship from a chemistry class. It's called Boyle's Law.

This is the general form of an inverse relationship. the product of the two variables is equal to a constant.

General form of an inverse proportion: x ·y = constant

With a direct proportion we found that doubling one variable resulted in doubling the other. How could we make a similar statement about an inverse proportion? Trying it with some numbers might help.

Suppose the constant is 360. When x = 60, y = 6. Doubling x does what to y?

For in inverse proportion, doubling one variable halves the other, tripling one, thirds the other, etc. Or in general a·x will result in y/a.
Look back at your data table and see if you can confirm this rule.

Incidentally, there are other relationships that look similar to our P vs. V graph that are not simple inverse proportions. We'll see one later. You have to draw the graph of the modified data and see the straight line before you can claim an inverse proportion.

Example of Inverse Proportion

1. Use Graphical Analysis to draw a graph of H vs. L. Be sure to convert millimeters to meters for units consistency.

2. Is the heat flow (H) directly proportional to the thickness (L)?

3. Calculate 1/thickness for each thickness. When 1/thickness doubles, does the heat flow double?

4. When 1/thickness triples, does heat flow triple?

Can you see that the heat flow is directly proportional to 1/L and thus inversely proportional to L?

Let's continue with GA and confirm this relationship, i.e., plot H vs 1/L. Use the same procedure we used above with the syringe.

5. Write the equation that relates these variables.

6. Try it. Let L = 1.5 mm. Find the corresponding H value from your equation.

7. Does the graph give the same value?

So far...

7. Quadratic Relationships

Let's reconsider the refrigerator door. The heat flow rate varied inversely with the thickness. What effect might the size of the window have? Would you expect an inverse proportion? Let's find out.

Since the window is round, the simplest variable to consider would be the diameter. In the table we have heat flow vs. diameter data for a 3-mm-thick window.

1. Use GA to plot heat flow vs. diameter. Print it out.

Your graph is clearly non-linear, but it looks much different from the heat flow vs. thickness graph. What we need to do is to "straighten it out" like we did before. But this time taking the inverse clearly won't work since both variables are increasing.

But look at how the variables are increasing. Look at the first two points. The diameter doubled, but the heat flow more than doubled. It actually quadrupled. So let's help out the diameter since it's lagging behind.

If we wanted the diameter to quadruple instead of just double, we could just square it! Sounds like cheating doesn't it? But if we plot heat flow vs. diameter squared we would not be changing the data, just the way we're plotting it. That is, if the diameter is 50 mm (.050 m), the diameter squared is really 2500 mm². The graph we get will then relate heat flow to the square of the diameter which will do nicely.

2. So give it a try. No help this time. You want a graph of heat flow vs. diameter squared. And you might as well let GA do the squaring.

3. From your graph (it will be straight, trust me), determine the equation of the line.

4. Confirm that the equation works by checking a couple of points from your original data and your graph.

I hope you've learned from these activities that graphs are an important tool to help us discover the relationships between quantities. There's still more to learn, but let's let the rest wait until we need it. For now, just be aware that you'll often be asked to use data you'll be collecting to find relationships and create equations. You should be ready!

So far...