the staff of the physics department university of...

PHYSICS 9B LAB MANUAL Fall 2012 Edition

This lab manual belongs to

_________________________________

LABORATORY EXPERIMENTS IN WAVES, OPTICS AND THERMODYNAMICS

The Staff of the Physics Department University of California, Davis, CA 95616

Physics 9 Lab Supervisor - Dr. Randy Harris

Contents

PHYSICS 9B LABORATORY MANUAL

CONTENTS

INTRODUCTION Page Purpose, Preparation, Laboratory Manual 2 Laboratory Instructor’s Role, Laboratory Reports, Coordination with Lectures, Attendance, Grading SUPPLEMENTS Page Error Analysis and Uncertainties 4 EXPERIMENTS Page Days Assigned Wave Velocity 12 Oct 1-5 Standing Waves 19 Oct 8-12 Experiments in Sound 28 Oct 15-19 Quantifying Uncertainty 38 Oct 22-26 Geometrical Optics 42 Oct 29-Nov 2 Light as a Wave 54 Nov 5-9 Thermodynamic Behaviors 63 Nov 13-19 (Tue-Fri & Mon) Low Temperature Experiments 72 Nov 26-30 A Cyclic Process 82 Dec 3-7

Updates and corrections, and late-breaking news may be found on the Physics 9 Lab website: http://www.physics.ucdavis.edu/Classes/Physics9Lab/Phy9BLab/

Supplements

2

INTRODUCTION TO THE LABORATORY

PURPOSE “Tell me, I'll forget. Show me, I may remember. But, involve me, and I'll understand”. Chinese proverb. The goal of the laboratories is to aid your understanding of physics through involvement in experiments. It is in the constant interplay between observations and hypotheses, experiment and theory, that physics has been able to come to a detailed although incomplete knowledge of the world. PREPARATION Before the first meeting of the lab, you should read this entire introduction thoroughly and look through all of the Supplements to see what is expected and what this manual contains. Before each laboratory begins, you are expected to prepare by reading the writeup in this manual. Accordingly, each lab has a prelab, comprising short questions intended to acquaint you with the most important points and with certain tasks, without which you would waste much valuable lab time. We understand that it is always possible for the unforeseen to slow you down, but don’t expect much sympathy from your laboratory instructor if you can’t complete the lab due simply to wanton lack of preparation. The material in the Supplements won’t all be needed right away. Some prelabs refer specifically to the Supplements, and so should give you a good idea of when a certain supplement will be needed, but you should be sufficiently acquainted with the Supplements to recognize when they are relevant without it being specifically mentioned. The prelabs are to be completed and turned in at the beginning of the corresponding lab meeting. (Note: There is no prelab to be turned in at the beginning of the first lab.) A short quiz on your preparation may be given at the beginning of the lab period, particularly if it becomes clear that students are not coming in sufficiently prepared. Finally, because the laboratory is a work in progress, important updates and correction arise periodically. so it is a good idea to keep an eye out for these things on the Physics 9 Lab website, http://www.physics.ucdavis.edu/Classes/Physics9Lab/. THE LABORATORY MANUAL Your suggestions for and constructive criticisms of this constantly evolving pedagogical tool are heartily encouraged, and can be relayed directly to your laboratory instructor at any time and/or put on record via the end-of-quarter lab evaluation form. LABORATORY INSTRUCTOR’S ROLE You laboratory instructor’s primary role is to provide individual help to students as they do the experiments, to guide their analysis, and to evaluate their performance. The lab instructor (sometimes referred to as a teaching assistant, or “TA”) is not a lecturer, and does not have responsibility for choosing the schedule of experiments, but is otherwise completely responsible for the conduct of the lab section. LABORATORY REPORTS You will be working in a group. It has been found that students often learn much more by thrashing things out with their peers than by simply being told something by a physics guru. The lab writeups encourage active cooperation between you and your lab partners for this reason. In most labs you will be asked to make predictions. It isn’t expected that you will always be right. In some cases it’s likely that many students will be wrong, but there

Supplements

3 is nothing wrong with this. Directly confronting a misconception is the best way to master it. You will put your predictions on record in your lab report, then you will test them and report on your newfound insight. The lab writeups also have designated “checkpoints,” where you will chat with your lab instructor, who will see where you are and provide help if needed. While your lab instructor will not in general just tell you how to proceed, this is a very good time to sound him or her out on ideas you may have. Finally, note that in many places you will be asked about possible reasons why your experimental results don’t agree with theoretical expectations. You should take these seriously. Rarely is any experiment free of such questions, and many advances in science have arisen from them. In high-level research it is often very difficult to pin these things down. You have a much easier task! Although you will work in groups, you will nevertheless complete your laboratory report individually and hand it in to your lab instructor at the end of the lab period. While the writeup in this manual is intended to be more or less complete, your laboratory instructor may pose further questions or reflections to be included in your report, and by all means add any comments or observations you think relevant to your investigations at any point. LAB/LECTURE COORDINATION Some instructors feel that the lab should precede the lecture on a given topic to make it easier for you to understand the lecture; others are convinced that the opposite order is more desirable. Even if there were agreement on this point there is still the problem that different lecturers have different schedules, ordering, and emphasis, so we try to effect the best compromise possible among these conflicting criteria in selecting experiments and scheduling them. Those most likely to precede the presentation in lecture are intended to be as self-contained as possible. Those dealing with more abstract or difficult concepts are scheduled slightly later, to give time for some lecture coverage. In any case, if you have constructive suggestions let us know, but bear with us on this complex problem. ATTENDANCE You are expected to attend the lab in which you are enrolled and be present every week. If you must miss for a valid reason (debilitating illness, for example, with documentation), you must make up the missed lab the same week in another lab section, and it is your responsibility to find out when (and if) other lab sections meet that you can attend (via the UCD Registrar’s course search tool). We expect this to occur only rarely, and when it does, your lab report must be handed in at the end of the lab period to the lab instructor of the make-up lab, who will then forward it to your own lab instructor. You need to provide your section number and TA name. Also, you should check with your own lab instructor the following week to be sure your report was received. Your lab instructor will have more detailed rules, and you are responsible for knowing them and for abiding by them. GRADING Your Physics 9 lecture instructor will use your lab grade as one component in determining your overall course grade. Your lab instructor will base your lab grade on the quality of your lab writeups, any laboratory quiz scores and your performance of the lab work itself. The lab grades will be reported as “High Pass”, “Pass”, “Low Pass” or “Fail”. A “Pass” will have a neutral effect on your course grade, causing it to be neither raised nor lowered from what it would have been from work submitted in lecture alone. “High Pass” will raise the lecture grade by one increment (C+ to B- or A- to A, for example; it will not, however, change an A in lecture to A+). Similarly, “Low Pass” will lower the lecture grade by one increment (but will not lower a D- to an F). A student who receives a “Fail” in lab, due to unsatisfactory lab work, including unsatisfactory attendance, fails the whole course.

Supplements

4

ERROR ANALYSIS Science rests on the ability to test predictions and have confidence in the results. Often the test is an experiment yielding a numerical value, but if that value is expected to be 5.00 and the experimentally measured value is 4.90, what does the result mean? It may mean success or it may mean failure, and to understand the difference we need to understand something about the quantitative limits of measurement. Error analysis is a deep and often complicated pursuit. In Physics 9 Lab we’ll learn just the basics, centering on the concept of uncertainty.

A: INSTRUMENTAL UNCERTAINTY Imagine that we wish to measure a distance x along the ground from Point A to Point B. Obviously we need a measuring instrument. Suppose it is a measuring tape with a labeled mark every meter and otherwise blank (not likely to be found at the hardware store). It is stretched out from A to B and the mark nearest Point B is labeled 137m.

Figure 1 Clearly the distance is not exactly 137m, but it is closer to 137m than to 138m or 136m. All possibilities in this range are accounted for by reporting the measurement as x = 137m±0.5m—meaning somewhere between 136.5m and 137.5m. The 0.5m is known as an absolute uncertainty. (We discuss the related percent uncertainty later.) An absolute uncertainty has the same units as the quantity itself and is denoted by a σ, e.g., σx = 0.5m. By the same logic, the measurement shown below uses a ruler marked off in divisions of tenths of a centimeter, 0.1cm, so the value might be anywhere from 0.05cm—half of a tenth of a centimeter—less than to 0.05cm greater than the value indicated; thus, the reading is 1.6cm±0.05cm.

Figure 2 This common-sense way of reporting an uncertainty due to the inherent limitations of the measuring instrument serves as a default rule—when statistical uncertainty, discussed later, can be ignored—for all individual measurements. Using an x as a generic symbol for all quantities we might measure—distance, time, mass, etc.—this rule is as follows: Instrumental Uncertainty

When an individual measurement is made of a quantity x, the absolute uncertainty !x , instrumental

is half of the smallest division of the measuring instrument.

Supplements

5 PERCENT UNCERTAINTY As an alternative to the absolute uncertainty, any uncertainty—whether instrumental or statistical—can be expressed as percent uncertainty, for which we use the symbol e. It is defined as the percentage that the absolute uncertainty represents of the actual measured value. Referring to the Point A-to-Point B example, the absolute uncertainty !

x = 0.5m represents (0.5m /137m) !100% = 0.36% of 137m, and so is equivalent to a percent

uncertainty ex

= 0.36%. Thus, 137m±0.5m and 137m±0.36% say the same thing. Clearly, if we can find percent uncertainty from absolute uncertainty, we can find absolute uncertainty from percent uncertainty: An e

x of 0.36%

means 0.0036 times a quantity, and 0.0036 times 137m is 0.5m. Again using x as a generic symbol for any measured quantity, the relationships between absolute and percent uncertainty are as follows:

Relating Absolute and Percent Uncertainties: ex=!

x

x"100% !

x= x " e

x/100% (1)

Note: Uncertainties need not themselves be very precise to convey their important information, so it is common in final reporting (though not usually before that) to round them to one significant figure, or two if the first digit is a 1.

B: STATISTICAL UNCERTAINTY In Physics 9 Lab, time constraints will often restrict us to one measurement of a quantity, but in science we generally try to avoid making statements based on a single measurement—particularly if repeated measurements vary! Suppose we repeatedly measure the distance traveled by a boulder hurled from a catapult. We obtain the distances given in Table 1 and depicted graphically in Figure 3. Clearly there is some uncertainty in the value we are attempting to measure. How do we assign that value and how do we define its uncertainty?

Figure 3 The value we assign is the mean, x , defined as the total of the individual values divided by the number of values. Denoting the individual values by xn and the number of values by N, this is:

Mean: x =!

n=1

Nx

n

N (2)

The data in the Table 1 gives a total !

n =1

N

xn of 1888.0m, and dividing by N = 20 yields

a mean of 94.4m.

Trial Distance x (meters)

1 100.8 2 89.8 3 92.5 4 92.7 5 94.6 6 95.7 7 88.8 8 95.6 9 92.7

10 98.9 11 94.1 12 97.5 13 89.4 14 94.2 15 83.2 16 98.4 17 101.7 18 96.4 19 97.6 20 93.4

Table 1

Supplements

6 To define an uncertainty we might be tempted to use the largest deviations from the mean: 101.7m - 94.4m = 7.3m and 83.2m - 94.4m = -11.2m. However, were we to conduct many additional trials, we would probably measure even larger deviations, if only rarely, which would seem to suggest that we become less certain of the answer as we conduct more measurements. Seeking a more sensible conclusion, scientists have agreed upon a logical definition of statistical uncertainty based on the fact that quantities have a decided tendency to fluctuate in a particular way, known as a normal distribution, in which many measured values are near the mean, they fluctuate symmetrically on either side of the mean, and the further a value is from the mean the less likely it is to be measured. Were we to carry out a great many trials with our catapult, we would expect Figure 3 to approach the smooth normal distribution shown in Figure 4. The characteristic shape of this distribution is also known by the names “bell-shaped curve” and “gaussian.” It can be shown that if values fluctuate according to a normal distribution, while there is always a chance that a value very far from the mean might be measured, the large majority—68%—will fall within a “distance” from the mean known as the standard deviation, defined as follows:

Standard Deviation: sx=

!n=1

N (xn" x )2

N "1 (3)

In this quantity, each deviation from the mean, x

n! x , is squared, then squares of deviations are summed, then

the total is divided by N-1. Adding squares of deviations ensures that every value that deviates from the mean, whether above or below, counts as a positive magnitude of deviation, and dividing by N then gives something like an average. (It isn’t worthwhile to go into why it is N-1 rather than just N, except to say that N is usually large, in which case it makes little difference.) Taking a final square root ensures that the standard deviation has the same units as x itself (i.e., not x2). Figure 5 gives some idea of how big the standard deviation is in a normal distribution: 68.3% of the values fall within one standard deviation of the mean, 95.4% within two, etc.

Figure 5 Now, standard deviation is still not the uncertainty we wish to define. If we carry out only the 20 trials of Table 1, giving the jagged Figure 3, we might not feel completely confident that its mean of 94.4m is correct. However, if we were to carry out 200 trials, with fluctuations of about the same extent on either side of the mean—the same standard deviation—but more like the smooth distribution of Figure 4 and with essentially the same mean, we should feel much more confident that that mean is correct. Some rather involved statistical

Figure 4

Supplements

7 arguments show that—ignoring instrumental uncertainty—the uncertainty in the mean becomes progressively smaller with an increasing number of trials according to the following rule. Statistical Uncertainty

When a measurement is repeated N times, the absolute uncertainty !x , statistical

in the mean value is given by: !x , statistical

= sx/ N

Again, as N increases, the uncertainty in the resulting mean decreases. The data in Table 1 contains 20 values, which is usually judged to be about the minimum number giving statistically significant results. The sum of squares of deviations from the mean !

n =1

N

(xn" 94.4m)

2 is 371.0m2. Dividing by N-1= 19 and taking a square root gives a standard deviation sx of 4.419m. Finally, dividing by 20 yields a statistical uncertainty of 0.988m, or, rounding to tenths of a meter, 1.0m. Thus, we would say that the distance traveled by the boulder is 94.4m±1.0m. Note that, using equations (1) above, the percent uncertainty is (1.0m /94.4m) !100% " 1.1% . In most cases in Physics 9 Lab, the computer will calculate the standard deviation automatically, so to find the uncertainty we need only divide by N .

C: UNCERTAINTY—PUTTING BOTH TOGETHER Note that each of the two previous uncertainties—instrumental and statistical—was defined with the other ignored. We now address the obvious question: What happens if repeated measurements are made and each one has instrumental uncertainty? Omitting the justifications, the logical way to do this is as follows: Uncertainty—In General !

x= (!

x, istrumental)2+ (!

x, statistical)2 (4)

How is this logical? Suppose we repeated the Point A-to-Point B measurement many times. It’s quite possible that, with the measuring tape marked off in divisions no smaller than a meter, we would obtain the same 137m reading every time. If this were the case, all values would necessarily equal the mean value x = 137m ; the standard deviation sx would be zero, because all the deviations x

n!137m would be zero; and !

x,statistical would

thus be zero. In such a case we would expect σx to be simply σx, instrumental, all uncertainty being due to the crudeness of the measuring instrument, and equation (4) agrees. Consider on the other hand the catapult example. Suppose that each measurement was made—as the data in Table 1 suggests—with an instrument marked in divisions of tenths of a meter, 0.1m. The instrumental uncertainty would be 0.05m. On the face of it, this seems rather negligible compared to the statistical uncertainty of 1.0m, and equation (4) agrees, yielding (0.05m)

2+ (1.0m)

2=1.00125m !1.0m . The uncertainty is due almost entirely to the statistical fluctuations.

In cases where neither instrumental nor statistical uncertainty is negligible, equation (4) gives an overall uncertainty somewhere between the larger of the two and the sum of the two, and this also makes sense. The overall uncertainty surely shouldn’t be smaller than either one individually, and, based on the improbability that both uncertainties would err in the “same direction,” it should not be as large as the sum. (We’ll talk more about uncertainties being “in the same direction” when we discuss combining uncertainties). SUMMARY—MOST IMPORTANT TO REMEMBER IN PHYSICS 9 LAB! • When recording only one measured value of a quantity, as with a scale or a meterstick or a timer, we will usually assume that if the measurement were repeated the result would never vary. Thus, ignore the possibility of statistical fluctuation and use the rule for instrumental uncertainty!

x , instrumental given under heading A above.

• When deliberately obtaining many values of a quantity, unless specifically instructed to consider instrumental uncertainty, use the rule for statistical uncertainty !

x , statistical given under heading B above.

• When deliberately obtaining many values of a quantity and also instructed about the inherent instrumental uncertainty, use the general rule for uncertainty given in equation (4).

Supplements

8

D: COMBINING UNCERTAINTIES FROM DIFFERENT MEASUREMENTS Almost never in Physics 9 Lab will our analysis end with measurements of only one of the fundamental quantities most commonly measured: distance, time, and mass. Rather, we combine these quantities, as when we divide a distance by a time to obtain a speed. But how do we define the uncertainty in our result when there are uncertainties in each of the two contributing measurements and the result is given by a calculation involving the two? Usually, we do this in quadrature, meaning adding squares (of uncertainties) then taking a square root. Still, there are special rules, depending on whether the calculation is addition, multiplication, etc. ADDITION AND SUBTRACTION Suppose one experiment measures the distance from Point A to Point B and another the distance, in the same direction, from Point B to Point C.

The results are: Distance from Point A to Point B: 137m±0.5m

Distance from Point B to Point C: 5.8m±0.1m The total distance is 143m, but what value do we assign to the uncertainty in the total distance? Before we answer this, let’s find the two percent uncertainties. They are, respectively, (0.5m /137m) !100% " 0.4% and (0.1m /5.8m) !100% " 1.7% . The A-B measurement has an absolute uncertainty five times larger than that of the B-C measurement (0.5m vs. 0.1m), while in percent uncertainty the B-C measurement is larger (1.7% vs. 0.4%) by about a factor of 4! The question is: Which is more important—the absolute, or the percent? Suppose we guess that it is the percent uncertainty, so we say that the total distance has a percent uncertainty of at least 1.7%. This corresponds to an absolute uncertainty in the total 143m of 0.017 !143m " 2m . But the total should not be off by this much! The first measurement is within 0.5m and the second 0.1m, so it is logical that at worst the total would be off by no more than about 0.6m. This is a good counterexample to the use of percent uncertainty when a sum is calculated. The correct way is through the absolute, and the rule, which applies also to subtraction, is as follows:

When adding or subtracting quantities x1 and x2, uncertainties are combined by adding in quadrature their individual absolute uncertainties.

!combined

= !x1

2

+!x2

2 Adding things in quadrature always yields a result between the larger of the two and the sum of the two. For the Point A-to-Point C example we’ve been considering it is (0.5m)

2+ (0.1m)

2= 0.51m ! 0.5m . It might

seem that it makes more sense to simply add the two uncertainties, but this would be assuming that both “mistakes” are in the “same direction,” either too large or too small, when it is entirely possible that one might err on the short side while the other errs on the long side. In the final analysis, if the two measurements are independent, i.e., not necessarily in the “same direction,” the odds are that the result is most likely in the range following from the quadrature rule given above. For example, if we had equal absolute uncertainties, the overall uncertainty would be 2 times (not 2 times) either. On the other hand, if one of the two absolute uncertainties is significantly larger than the other, as in our Point A-to-Point C example, the result essentially equals the larger. We will often have occasion to use this fact.

Supplements

9 Note that subtracting quantities with comparatively large absolute uncertainties can lead to results almost worthless. Suppose the two quantities are times when a projectile passes two markers: t1 = 8.0s±0.05s and t2 = 8.1s±0.05s. To find a time interval we subtract the initial time from the final, yielding 0.1s, but what is the uncertainty? The percent uncertainties are both about 0.6%, and would seem to suggest a rather small uncertainty in the result, but we don’t combine percent uncertainties when subtracting! Adding the absolute uncertainties in quadrature gives (0.05s)

2+ (0.05s)

2= 2 ! 0.05s " 0.07s . This is a percent uncertainty of

(0.07s/ 0.1s) ! 100% = 70% of the time interval 0.1s itself! We might have crudely argued that the earlier time could be as early as 7.95s and the later as late as 8.15s, giving an interval of 0.2s, or that the earlier could be as late as 8.05s and the final as early as 8.05s, giving an interval of zero! The correct uncertainty isn’t that big, but an uncertainty of seventy percent is still huge! MULTIPLICATION AND DIVISION Suppose we wish to calculate a speed by dividing a distance x of 4.00m±0.005m by a time t of 2.5s±0.05s. Clearly there is some uncertainty in the resulting speed, 4.00m/2.5s = 1.6m/s, but how do we find it? It is not through the absolute uncertainties, because we cannot add (in quadrature or otherwise) things with different units. The rule when multiplying and dividing is as follows:

When multiplying or dividing quantities x and y, uncertainties are combined by adding in quadrature their individual percent uncertainties.

ecombined = ex2+ ey

2

Note: Multiplying or dividing a measured quantity by a numerical value whose uncertainty is assumed or defined to be zero is nevertheless multiplication, and so involves the percent uncertainty. The percent uncertainty is unchanged, which means that the absolute uncertainty in the final result does change. For instance, multiplying a mass of 0.100kg±0.0005kg, or 0.100kg±0.5%, by 3 would preserve the 0.5% percent uncertainty, which would mean an absolute uncertainty in the 0.300kg total mass of 0.300kg ! 0.005 = 0.0015kg , three times the absolute uncertainty in the individual mass. While we might argue that tripling the mass is simply adding it three times, and so should require quadrature addition of the absolute uncertainties ( (0.0005kg)

2+ (0.0005kg)

2+ (0.0005kg)

2= 0.0005kg ! 3 " 0.0009kg ), we would in effect be adding

measurements that are definitely not independent (to err on the large side for one is to err on the large side for all) and quadrature addition does not apply in such cases—the absolute uncertainties simply add. QUANTITIES RAISED TO A POWER Suppose we wish to find the area of a square whose side length x we have measured to be 1.25cm±0.005cm. The area x2 is (1.25cm)2 = 1.56cm2, but what of the uncertainty? It might be argued that squaring is multiplication, so we should find the two (equal) percent uncertainties, (0.005cm /1.25cm) !100% = 0.4% , then add these in quadrature, giving 0.4% ! 2 " 0.6% . However, squaring is another example of two uncertainties that are definitely not independent. They both always err in the “same direction”; if one length errs on the large side, then both sides do, since the same value is being used. This being the case, the percent uncertainty is larger: two (not 2 ) times that of the individual measurement. Omitting the proof, the general result when raising a quantity to

the power b is as follows:

When raising quantity x to power b, the percent uncertainty is multiplied by the factor b. exb = b ex

Supplements

10

E: EXAMPLES Example 1. Suppose we wish to calculate a momentum, p, which is a mass times a speed, p = m v. The mass m of the object is measured to be 0.247kg on a scale whose smallest division is one gram. The speed is found from two measurements of distance from an origin, x1 = 32.4cm and x2 = 91.8cm, both via a measuring tape marked off in millimeters, and a measurement of the time interval, !t = 2.08s , correct to within 0.005s. From the data gathered, the momentum is calculated as follows:

p = mv = m!x

!t= m

x2" x

1

!t

Find the momentum and its absolute uncertainty. Solution The momentum is

p = 0.247kg91.8cm ! 32.4cm

2.08s= 7.05kg

cm

s

For the uncertainty, it is easiest to start from the “inside” and work outward, so we start with the subtraction. Subtraction requires that we combine absolute uncertainties in quadrature. Each of the x values has an absolute uncertainty of half a millimeter, i.e., half the smallest division:!

x1= !

x2= 0.05cm . Thus, the absolute

uncertainty in x2 - x1 is:

!"x= (0.05cm)

2+ (0.05cm)

2= 0.071cm

We will be dividing Δx by Δt, requiring use of percent uncertainties, so we need to convert σΔx to a percent uncertainty. The actual value of x2 - x1 is 59.4cm, so

e!x

=0.071cm

59.4cm"100% = 0.12%

Now, the remainder of the calculation involves only multiplication and division. We could combine in quadrature the percent uncertainties in Δt and Δx, then combine in quadrature this percent uncertainty with that for m, but it is equivalent (and faster!) to just combine in quadrature all three percent uncertainties at once. But first we need the remaining percent uncertainties. For the mass, the absolute uncertainty, half the smallest division, is 0.0005kg, and for the time interval it is given to be 0.005s.

em =0.0005kg

0.247kg!100% = 0.20% e

! t=0.005s

2.08s"100% = 0.24%

Now, combining in quadrature the percent uncertainties for m, Δx, and Δt,

ep = (0.12%)2+ (0.20%)

2+ (0.24%)

2= 0.31%

Finally, from the percent uncertainty we can find the absolute uncertainty in p.

! p = 7.05kgcm

s" 0.0031 = 0.02kg

cm

s

Thus, we report the momentum as: p = 7.05 ± 0.02kgcm

s.

Supplements

11 Example 2. A projectile lands a certain distance from the origin, given by

r = x2

+ y2

where x and y are the coordinates of the landing point measured along two perpendicular coordinate axes. The coordinates are measured to be: x = 8.6cm ± 0.05cm and y = 2.7cm ± 0.1cm Find r and its absolute uncertainty. Solution Plugging in to find r,

r = (8.6cm)2+ (2.7cm)

2= 9.0cm

For the uncertainty, each factor is squared, so we must double the percent uncertainty. The percent uncertainties for x and y alone are:

ex=0.05cm

8.6cm!100% = 0.58% and ey =

0.1cm

2.7cm!100% = 3.7%

So the percent uncertainties in their squares are e

x2 = 2 ! 0.58% = 1.2% and e

y2= 2 ! 3.7% = 7.4%

Now, x2 and y2 are added, so we need to combine in quadrature their absolute uncertainties.

!x2 = (8.6cm)

2" 0.012 = 0.89cm

2 and !y 2= (2.7cm)

2" 0.074 = 0.54cm

2

Thus, the absolute uncertainty in x2 + y2 is

!x 2 + y2

= (0.89cm2)2+ (0.54cm

2)2= 1.0cm

2 Now, to take a square root—to raise to the one-half power—we multiply the percent uncertainty by one-half. The value of x2 + y2 is (8.6cm)2+(2.7cm)2 ≈ 81cm2, so its percent uncertainty is

ex2 + y2

=1.0cm

2

81cm2!100% = 1.2%

Accordingly, the percent uncertainty in r is e

r= 1

2 !1.2% = 0.6% Finally, the absolute uncertainty in r is !

r= 9.0cm " 0.006 = 0.05cm

Note that the answer is essentially the same as the absolute uncertainty in x. While the example serves to show how the rules are carried out, it also shows that in many circumstances, with a bit of thought, we can guess the result pretty well. The x value is nearly three times the y value, so it predominates in calculating r, and correspondingly the overall uncertainty is dominated by its uncertainty.

Wave Velocity

12

WAVE VELOCITY

Name _____________________________________ Date ______________ Lab Instructor & Section _____________________ Names of Lab Partners _____________________________________ _____________________________________ _____________________________________

Circle the name of the person to whose report any joint printouts are attached. References University Physics Section 15.4. Apparatus Thin and thick white nylon cords, metal chain, heavy pulley with clamp, hook assembly, weight set,

tape measure, timer, triple-beam balance, computer. Printouts Linear Fits

Wave Velocity

13 Introduction It might seem odd to spend much time on waves, but there’s a good reason: Everything you see or

hear is a wave phenomenon, in which energy and information travel, often great distances, while intervening matter merely jiggles back and forth right where it is. There are many kinds of waves. Light is an wave of oscillating electromagnetic fields, which have no mass. Waves carried by oscillations of something with mass—such as air carrying sound waves, or a stretched string carrying waves on a piano or guitar—are known as mechanical waves. These require a medium, which communicates the wave from one point in space to another while itself going nowhere on average.

In today’s lab the medium is a stretched string. One of the first things we learn in a study of mechanical waves is that the wave’s speed depends on properties of the medium. Using the laws of mechanics to show exactly how the speed should depend on these properties, though standard fare in the textbook and lecture hall, is rather sophisticated stuff. By careful experimentation, however, we can make a pretty good guess (one that hopefully will agree with the theory). Factors upon which the speed of waves on a stretched string might plausibly depend are:

T the tension in the string, in newtons A the amplitude (size) of the wave, in centimeters µ the string’s mass density (mass per length), in kg/m

By varying only one factor at a time—the time-honored “scientific method”—you can say whether the wave speed depends on that factor at all, and if it does, you can determine fairly well if it depends on that factor to the third power, the negative one-half power, etc. Let’s give it a try!

TESTING ONE-BY-ONE

Procedure 1. Your “data run” consists of plucking the string at one end and timing how long it takes the pulse you create to travel some distance along it. You should have two nylon strings plus a “string” of metal chain. First, stretch each (or all three at once, if you wish) from the hook at one end of the lab table to the pulley at the other, then adjust the pulley location so that the shortest of the three goes over the pulley with enough left over so that a weight on the end will hang freely, hitting neither the pulley nor the table. Now, choose one of the three strings, stretch it from hook to pulley, and hang a weight of between 400g and 1.2kg. Then with a tape measure, measure the distance from hook to pulley. This portion of the string (i.e., excluding the portion hanging beyond the pulley) we refer to as the “oscillating portion”, and we’ll use the symbol l. (For the string’s total length we’ll later use an L.) Record l below.

l:_____________________m

2. You’ll pluck the string by grasping it about three inches from the hook, pulling it to the side 2 or 3cm, which we’ll call A1 (A for amplitude), and letting go. The pulse moves too fast to time accurately even a whole round-trip, so pluck the string, starting the timer simultaneously, watch carefully for the pulse’s return from its reflection at the pulley end, and continue counting returns as long as the pulse is distinguishable, hopefully at least five returns. (Eventually it may end up just swinging like a hammock—an entirely different effect we’re not studying.) If you count, say, n returns, dividing the total distance 2nl by the total time in seconds gives the speed. To make

Wave Velocity

14 sure your procedure gives consistent results, carry out your data run three times. Calculate the speed for each of your three trials, then average them. Show your work below.

3. Summon your lab instructor. Check that your procedure is sound and your speed reasonable.

4. To begin comparing the various factors, fill in row I of Table 1. In the T column enter the tension in newtons (using 9.81m/s2 for g), and in the A column enter either 2cm or 3cm, whichever was closer. We’ll address µ later. Fill in your three individual speeds and the average. As you fill in later rows, keep an eye out for suspicious inconsistencies among the three trials.

vtrial 1(m/s) vtrial 2(m/s) vtrial 3(m/s) Table 1 T (N) A (cm) µ (kg/m) vavg (m/s)

I (T1)

(A1) (µ1)

II (T2)

(A1) (µ1)

III (T3)

(A1) (µ1)

IV

(A2) (µ1)

V

(A1) (µ2)

VI

(A1) (µ3)

Wave Velocity

15

5. Now try a different tension, T2. Use a hanging weight with a substantially different value between 400g and 1.2kg, then again carry out three trials measuring the speed and calculating the average. Then change the tension again (T3) and repeat. Show your work below and record the results for the two new tensions by filling in all the columns (except µ) in rows II and III of Table 1.

6. Next vary the pulse’s amplitude. To test dependence on amplitude alone, use a tension you already used. It may be most convenient to use T3, but T2 or T1 would also do. To carry out these data runs, make the amplitude significantly larger, pulling the string perhaps 5cm sideways (A2) before letting go. Again obtain speeds for three trials and calculate the average, then fill in all the columns (except µ) in row IV of Table 1. (Changing amplitude is rather messy. so we won’t bother with any more than just the earlier small amplitude and the larger one here.)

7. Now mass density µ. Make sure to note which string you started with, then weigh each string, subtracting for the nylon strings 11g (roughly the total mass of the two end connectors), and record the results, M, in Table 2. Then measure the strings’ total lengths L and calculate µ for each. Fill in Table 2 and the µ column in Table 1.

Table 2 Mass M (kg) Total length L (m) µ = M/L (kg/m) Nylon, thin

Nylon, thick

Metal

Wave Velocity

16

8. Put one of the new strings in place and make the tension one of those you already used in row I, II, or III of Table 1. Using the same small amplitude you started with, A1, carry out the usual procedure, showing your work below, then fill in row V of Table 1. Finally, change µ (but only µ!) once more and fill in row VI.

9. Discuss your findings. Which of the factors, T, A, and µ affect wave speed? A good criterion for claiming that a factor doesn’t affect the speed is if the change from one value of that factor to another is no larger than the fluctuations from trial to trial when nothing is changed. If a factor is indeed important, indicate whether the speed is directly, or inversely, related to that factor.

10. Now here’s a guess—nothing more! Wave speed depends on T, A, and µ as follows:

v = k TaAbµc (1)

where k is a proportionality constant and a, b, and c are numerical exponents yet unknown. If this is true, then taking logs of both sides gives:

ln(v) = a ln(T) + b ln(A) + c ln(µ) + ln(k) (2)

If we look at the dependence of v on T alone, a plot of ln(v) versus ln(T) should be a straight line with slope a. Similarly, a plot of ln(v) versus ln(A) should have slope b. In other words, the slope of the line ln(v) versus ln(X)—where X might represent T, A, or µ—is the numerical exponent giving the functional dependence of v on X. This is an indispensable tool in the scientist’s toolbox! Let’s pursue it. First, fill in Table 3 (just logs of all Table 1 values).

Wave Velocity

17

Table 3 ln(T) ln(A) ln(µ) ln(vavg)

I

II

III

IV

V

VI

11. Open the file “Finding Exponents”. Though somewhat like Table 3, you have two columns (under green text) for your rows I-III ln(T) and ln(v) data. Don’t enter ln(A) and ln(µ) data in these columns, for here you’re looking at how v varies with T alone—you’ve kept A and µ constant in these rows for this very reason. You have two more columns (blue text) in which to enter your ln(A) and ln(v) data. Here you’re interested in determining separately how v varies with A alone, so you’ll use data from row IV of Table 3 and from one of rows I-III. Which row, and why?

The last two columns (red text) are for your ln(µ) and ln(v) data. This is in rows V and VI of Table 3, and one of rows I-III. Which row, and why?

Enter your Table 3 data in the appropriate places, then click in turn on each plot and choose “Linear” with the button. (If you accidentally enter data into too many rows, you

can delete a row with the button in the ln(v) vs ln(T), ln(A) and ln(µ) window.)

12. Make a printout of your plots.

13. Your plots may be nearly straight lines (of course, with only two data points, the one for A can’t help being straight). What would it mean—theoretically, i.e., given perfect experimental procedure—if they weren’t straight? If it helps, consider again equations (1) and (2).

Wave Velocity

18

14. From your plots obtain values for the exponents in equation (1) and record them below, rounded to the nearest half-integer.

a:______________ b:______________ c:______________

15. Summon your lab instructor and discuss with him/her how you obtained a, b, and c.

16. You may already have decided that one or more of the exponents should really be zero, that v doesn’t really depend on that factor. For each such factor, assume that its exponent is indeed zero, then rewrite equation (1) with numerical values for a, b, and c, leaving k as it is.

17. What units would k have to have for your equation to add up dimensionally, i.e., for the combination of factors you have on the right hand side to yield m/s, the units on the left hand side? Show your work below. (Note: A tension, in newtons, is kg·m/s2, so if your formula had, for instance, tension0.5, the units would be (kg !m / s2 )1/ 2 = kg1 / 2m1/ 2

/ s .)

18. Can you guess a value for k? And does your data agree?

Standing Waves

19

PRELAB: STANDING WAVES (TURN IN YOUR ANSWERS TO YOUR LAB INSTRUCTOR AS YOU ENTER THE LAB ROOM)

Question 1: In the Standing Waves lab we study standing waves in two different media? Which media? Question 2a: For a string stretched between two fixed ends, sketch the “fundamental” standing wave, the wave of longest wavelength that fits the fixed-ends boundary conditions. Question 2b: If now someone were to hold their finger lightly against string at a point one-third of the way from one end, preventing the string from oscillating there, but still allowing it to oscillate on both sides, sketch the longest-wavelength standing wave that could now exist on the string. Question 2c: Assuming that nothing else has changed, such as the string tension, would the speeds of the two standing waves in questions 2a and 2b be the same, or different? Briefly explain. Question 2d. How would the frequencies of the two standing waves in questions 2a and 2b be related?

Standing Waves

20

STANDING WAVES


Circle the name of the person to whose report any joint printouts are attached. References University Physics Sections 15.6-15.8 and 16.4. Apparatus Yellow “mason line” 3.0m-long string with small loops tied at each end, pulley with clamp, hook

assembly, weight set, tape measure, UCD Physics function generator connected to Pasco mechanical vibrator. Pasco box with sound sensor in Port A, 3/4-inchx30-inch aluminum rod with “mallet” (50g weight). One triple beam balance for each table.

Printouts Frequency Spectrum

Standing Waves

21 Introduction Last time we dealt with wave pulses. But a medium

oscillating at a regular time period T will exhibit periodic waves, which will also have a regular period in space, known as wavelength λ. These two are related through the wave speed v. If a wave moves one whole wavelength λ in one period T, then its speed v, distance over time, is just λ/T. Often we prefer to use the reciprocal of the period, the frequency f, rather than the period itself, so this relationship becomes

v = λ f (1)

Today we study waves that not only are periodic, but also form standing waves. These can form whenever two waves of the same frequency move in opposite directions through of a medium. Standing waves live up to their name. Traveling waves move one way or the other, and every point of the medium necessarily oscillates. But a standing wave is characterized by nodes, places where the medium never moves. Halfway between nodes are places where there is maximum oscillation of the medium, called antinodes. In Figure 2, the heavy curve represents a standing wave an one instant in time, while the others represent it as the antinodes “wobble” up and down periodically with time. Note that it is one-half a wavelength from one node to the next, and the distance between antinodes is similarly λ/2.

In today’s lab we produce standing waves by reflection. If a traveling wave reflects off the end of a medium, the reflected and incoming wave, moving in opposite directions, can form a standing wave. But standing waves must also meet certain conditions. Boundary conditions arise at the edges of the medium where the wave reflects. Not just any old wavelength will meet these conditions, but only certain discrete ones. We say that standing waves are “quantized”, meaning that only certain ones are allowed. But what are these conditions? We only need be concerned with two:

(1) If the medium is held fixed at an end or at any other point along it, there must be a node at that point.

(2) If an end of the medium is not held fixed, but is free, there must be an antinode at that end.

The first condition is sensible; if a standing wave is to reside on a medium prevented from moving at certain locations, those locations had better be nodes, where the medium wouldn’t tend to move anyway. The explanation for the second condition we leave to your textbook.

Let’s look at a few examples. If the medium is a string of length L fixed at both ends (but nowhere else), any of the standing waves in Figure 3 would fit the conditions (i.e., nodes at both ends). An integral number of “bumps”, which are λ/2 wide, must fit between the ends. These words become an equation that simply must be met:

n!

2= L n = 1, 2, 3, ... (fixed both ends) (2)

It never hurts to reiterate: Only certain wavelengths fit the conditions. Next consider a medium of length L but free at both ends, fixed

nowhere. With the conditions now being that there be antinodes at each end, any of the standing waves in Figure 4 would work. But note that again an integral number of half-wavelengths, from one antinode to the next, must fit in along the length L, so equation (2) would still apply (though n would be the number of nodes).

Figure 1

Figure 2

Figure 3

Figure 4

Standing Waves

22 Now suppose that a medium is held fixed at its left end, but

its right is free. It must have a node on the left and an antinode on the right. Because of the regular spacing of nodes and antinodes, this again leads to a “quantization condition”. Any of the standing waves shown in Figure 5 would fit the conditions; it’s just a matter of figuring out how to express them together. Look them over and you’ll see that in each case the length L is an odd number of quarter wavelengths (it being λ/4 between adjacent node and antinode). Thus

n!

4= L n = 1, 3, 5, ... (fixed one end only) (3)

If in addition to the boundary conditions at its ends, a medium is also somehow prevented from oscillating (held fixed) at some intermediate location, the result would be further restriction of the allowed standing waves, for some would require motion at a point where it is now forbidden.

ACTIVITY 1: STRETCHED STRING Introduction Probably the clearest way of generating standing waves is, as in Figure 1, a string stretched between

fixed ends and driven by an oscillator at frequency f, producing a wave that reflects back and forth at the ends. Were it not for losses to heat, we wouldn’t need the oscillator continuously on. Once established, a standing wave would persist indefinitely, its mechanical energy continually shifting between kinetic and potential. But even with a constant input of energy to replace that lost to heat, the important standing wave conditions must still be met. The oscillator can oscillate all it wants to, but if it isn’t producing a wave of one of the allowed wavelengths, it simply won’t establish a standing wave.

A formula you should already be acquainted with will come in handy in this activity. The speed of waves on a stretched string is given by v = F / µ , where F is the string tension.

Procedure 1. Weigh your string and record its mass M below. Also measure its total length Lt (we’ll use L for the portion soon to be oscillating) Then calculate its mass density µ = M/Lt in kg/m.

M:______________kg Lt:_____________m µ:_____________kg/m

2. The center pin of the Pasco mechanical vibrator should be about six inches from the hook, where one end of your string will soon be attached. Being careful not to stress this delicate pin—up, down or sideways—measure the distance L from it to the pulley. This is the length of the oscillating portion of the string. Record it below.

L:_______________m

Figure 5

Standing Waves

23

3. Given your length L in step 2, calculate numerical values for four different wavelengths that would be allowed standing waves on your string? Show how you find them and record them in Table 1 according to the number of antinodes the standing wave should have.

Number of antinodes

Wavelength (meters)

Table 1

4. Now a prediction. Suppose you were to hang a 1kg weight on the pulley end of your string (don’t do it yet!). At what frequencies (four of ‘em!) would you expect to be able to oscillate the string and produce standing waves? Show your work and enter your results in the first row of Table 2.

Predicted

frequencies

Experimental frequencies

Table 2

5. Summon your lab instructor and explain to him/her your work in steps 3 and 4.

6. Make sure that the Function Generator is OFF, then connect your string from the hook, through the fork in the vibrator’s center pin, then over the pulley, and hang a 1.0kg weight. Again, the center pin should be in contact with the string about 6 inches from the hook.

Standing Waves

24

7. Set the function generator’s “Amplitude” knob at 1/3 to 1/2 clockwise/max and its toggle switches at “Sine” and “Low”. Now carry out your experiment, producing the standing waves you considered in steps 3 and 4, and entering you results in the second row of Table 2.

8. How close were your predictions and experimental findings? If close but not exact, where could error have crept in? If not at all close, why? Recalculate below if necessary.

10. Now, to be certain of seeing what depends on what, and simply as good, thorough experimental practice, replace the 1.0kg weight with one differing by 30-50%. Enter your predictions in Table 3 then add your experimental findings. Show your calculations below.

Number of antinodes

Wavelength (meters)

Predicted frequencies

Experimental frequencies

Table 3

Standing Waves

25

11. With the new hanging weight, if the wavelengths or the frequencies were the same as before, why? And if they changed, did they increase, or decrease, and why?

12. Finally, describe what you saw when the oscillator was driving the string at a frequency not one of the allowed standing wave frequencies. What evidence (visible? audible?) was there that this system “likes” being driven at its characteristic “resonant frequencies”.

13. Turn off the function generator and LOCK the string driver.

Standing Waves

26

ACTIVITY 2: STANDING WAVES ARE EVERYWHERE Introduction Many things around us are so conducive to supporting standing waves that standing waves can be set

up with no special care. Guitarists, for instance, need not search for a string driver to get their strings to oscillate at the proper frequencies. Simply plucking them gets the job done. The reason is that plucking a string is equivalent to introducing a huge range of wavelengths and thus frequencies. (The equivalence of a jagged initial pulse to a sum of many frequencies is a fundamental idea of Fourier Analysis, possibly covered in one of your math courses.) Most of these frequencies die out right away, not being those whose wavelengths fit the standing wave conditions. Those that do fit the conditions, aided by slow loss to heat, persist for some time. Of course, as you’ve already demonstrated, for given conditions, many wavelengths are allowed, but the point is that not all are allowed. The various frequencies that are allowed, coexisting at the same time in a musical instrument, are part of that instrument’s musical quality. In this activity you’ll see multiple allowed standing waves coexisting on a musical instrument. Well, okay, it’s just an aluminum rod—not much of a musical instrument—but, as it turns out, aluminum is a rather musical metal, supporting persistent standing waves for quite a while.

Procedure 1. Make sure the sound sensor is in Port A of the Pasco box, then open the file “Everywhere”, which plots amplitude versus frequency.

2. Start recording, then balance the aluminum rod in the crook between your index finger and thumb, with the smooth end on the palm side of your hand, as in Figure 6. Minimize the area in contact with your hand, or you’ll deaden the sound you want to study. Give the rod a sharp rap on the smooth end with your mallet—just a garden variety 50g or 100g, weight. It shouldn’t sound dead, but should “sing” quite well. (Note: Our interest is longitudinal motion. Were you to strike the rod on the side near the end, rather than on the end, it would produce too much transverse motion. There will always be some, and it’s actually responsible for the lowest tone you hear, which dies out fairly quickly compared to the ones we’re studying.) Watch the spectrum after you strike the rod. In particular, after waiting perhaps 10s or so for transients to die out, notice which frequencies appear to be most and second-most prominent. When you see these clearly, stop recording, which freezes the display. For the most prominent frequency we’ll use an fo, and for the second-most we’ll use f´. Using the x-y tool, record them below.

fo (most prominent):____________Hz f´ (second most prominent):____________Hz

3. The spike at fo is the fundamental, meaning the longest-wavelength standing wave that fits the conditions in the medium. Given this definition, sketch below, from those in Figures 3, 4, and 5, the wave you think best represents your fo standing wave. (Note: We’re causing longitudinal standing waves, while Figs. 3, 4, and 5 appear transverse. Your text discusses the distinction. What’s important is where the nodes and antinodes are and how many there are, which is independent of whether the medium’s motion is transverse or longitudinal.)

Figure 6

Standing Waves

27

4. For the wavelength of the most prominent standing wave let’s us the symbol λo. Given your sketch, what numerical factor goes in the following equation: λo = ______ L.

5. Summon your lab instructor and defend your answers in steps 3 and 4.

6. Make a printout of your data.

7. You have a meterstick for measuring L and you know the frequency fo. Calculate the speed of sound in aluminum.

8. About how many times higher is your frequency f´ than fo?

9. Different allowed standing waves tend to have frequencies that are integral multiples of the fundamental frequency. Explain why your value in step 8 should be what it is. (This will take some thought: First look at other allowed wavelengths that fit the same conditions as the fundamental, and see how they are related numerically to the wavelength of the fundamental.)

10. A final challenge! Find a way to support the rod so that striking it produces a previously “missing” frequency. Describe your experiment and attach a corresponding printout.

Experiments in Sound

28

PRELAB: EXPERIMENTS IN SOUND (TURN IN YOUR ANSWERS TO YOUR LAB INSTRUCTOR AS YOU ENTER THE LAB ROOM)

Question 1: In an oscilloscope display, the horizontal axis measures what quantity? Question 2: In the Experiments in Sound lab, there is something able to reflect sound quite well to which you must pay particular care. What is it? Question 3: The figure shows two waves, the top one of 11Hz (it goes through 11 cycles in 1 second), and the other of 9Hz. At t = 0s, they are in a constructive interference condition, in that they are in phase, both at their midpoints and heading upward. By the same token they are interfering constructively at t = 1s. (a) Would you say that the interference at t = 0.5s is constructive, or destructive, and why? (b) Would you say that the interference at t = 0.25s is constructive, or destructive, and why? (c) Would you say that the interference at t = 0.75s is constructive, or destructive, and why? Assuming these are sound waves heard together, how many times per second would the overall sound get quiet?


29

EXPERIMENTS IN SOUND


Circle the name of the person to whose report any joint printouts are attached. References University Physics Sections 16.1, 16.6 and 16.7. Apparatus Pasco Box, two red-and-black pairs of hookup wires, extra black wire, voltage sensor, two speakers,

sound sensor with 6ft cord, UCD Physics function generator, meterstick. Printouts Sum of Pasco and External Generator Outputs


30 Introduction The topic of sound is very broad. Today we concentrate on just a few of its many important aspects:

amplitude and frequency (known to the layman as loudness and pitch) and quality; speed; interference; and beats. Probably the most important basic relationships to keep in mind are that we use f for the frequency of a wave, that the distance from one crest of a wave to the next is its wavelength λ, and the wave speed v is related to these two other properties by v = λ f.

Interference In some ways, the topic of interference is very simple. Two waves may reach some point in space so

that their crests arrive together. Later at this same location, their troughs arrive together, then their crests, again, etc., The result of this “in phase” relationship is constructive interference, a net wave of twice the amplitude. At some other point in space, to reach which one wave has farther to travel than the other, the crest of one may arrive at the same time as the trough of the other, then the trough of the first and the crest of the second arrive together, etc. The waves always cancel (add to zero). They are completely out of phase, and the result is destructive interference.

There is no easier way to show interference than by simply having waves spread in all directions from two sources. At points equidistant from the sources, i.e., along a perpendicular bisector to the line connecting the sources, as at Point a in Figure 3, the two waves have equal distances to travel. The so-called path difference, |L1-L2| = ΔL, is zero, so the waves must arrive in constructive interference. But even off center, there should be points of constructive interference. Suppose that to reach the off-center Point b in Figure 3, the wave from Speaker 1 has one whole wavelength farther to travel than the wave from Speaker 2, i.e., ΔL = λ. Since a sinusoidal wave looks the same if shifted over one whole wavelength, the interference must again be constructive. On the other hand, there would have to be an intervening point, Point c for instance, where ΔL is one half wavelength. Here, crests would meet troughs, and the interference would be destructive.

Beats Perhaps “beats” doesn’t sound like a name with much pizzazz, but it’s really the only one there is for

the phenomenon we study in Activity 4, one with a great host of applications. Two sound sources (or any other wave sources), if of slightly different frequency, will oscillate in phase at one point in time, then out of phase somewhat later, and back and forth. The overall effect is a sound (or wave) that is loud when they are in phase, then soft when out of phase, then loud, then soft, etc., etc., etc. If the difference in frequencies is too high, the sound goes in and out of phase too quickly to notice, but if the frequencies differ by no more than perhaps 10Hz, then the variation in and out of phase makes a noticeable and characteristic “wah-wah-wah” sound.

Figure 1

Figure 2

Figure 3


31

ACTIVITY 1: AMPLITUDE, FREQUENCY AND QUALITY

Procedure 1. In this activity you’ll “see” the air’s motion versus time plotted on the computer. Make sure that the sound sensor’s cable is in Port A of the Pasco box, then open the file “A, f, and Q”. What you see is an oscilloscope display. Historically, an oscilloscope was a bulky free-standing instrument used to display a time-varying voltage as a function of time. Nowadays we use the computer to do this.

2. Start recording. Now have someone from your group whistle near the sound sensor so as to produce a stable pattern (it’s best not to whistle so that you are actually blowing directly into the sensor). The term for the curve shown in an oscilloscope display is a trace. While continuing to whistle, stop recording, which freezes the trace. Make a rough sketch of the trace.

3. Would you say that your whistle causes a regular, periodic movement of the air? Would you say that it’s fairly sinusoidal?

4. About what is the frequency of your whistle? Note that the horizontal axis is time, and each little vertical line along it is 1ms from the next. Thus, the period of your whistle is the number of milliseconds/lines required for it to repeat. The frequency, in Hertz (cycles per second), is of course the reciprocal of the period in seconds.

5. Start recording, then while keeping a constant pitch, whistle a bit louder. What changes now?


32

6. Now, while trying to keep the loudness of your whistle steady, slowly change the pitch significantly higher or lower. What changes in the plot now?

7. Talk into the sound sensor. How does the air’s motion compare to when you were whistling? Is it periodic and regular, and if so, fairly sinusoidal?

8. Now, summon all your courage and sing as pure a tone as you can into the sound sensor so as to produce a stable pattern, then freeze it (i.e. stop recording). A simple “aaah” will do, but it should be fairly loud, and higher pitch is usually better. How does the air’s motion compare to when you were whistling? Is it periodic and regular, and if so, fairly sinusoidal?

ACTIVITY 2: SPEED OF SOUND Introduction A sinusoidal signal causes a speaker cone to oscillate sinusoidally, which in turn produces a

sinusoidal wave in front. The air is at different points in its sine wave there, and this allows a pretty good determination of the speed of sound. In all activitiesthat follow, note that sound reflects quite well from all kinds of things, even you! So, if you’re getting weird variations, it’s quite possible that you are the culprit. Careful experimental procedure pays off ! Also for the rest of the lab, whenever you’re not taking data, in courtesy to your classmates and particularly your poor lab instructor, who must bear up under the noise for several labs during the week, please keep your speakers silent.

Procedure 1. Plug the wires from one of your speakers into the Pasco Box (not the “UCD Physics Function Generator”, used in Activity 4 only) outputs, red wire into the “plus” port, , and black wire into the minus, . Then position the speaker so that there is a fair amount of uncluttered table space in front of it, lay down a meterstick with one end against it and the other straight out in front, then lay the sound sensor somewhere along the meterstick.

2. Open the file “Speed of Sound”. Make sure there’s a thin black box around “Speaker Cone”, not “Sound Sensor” (otherwise, the trace “triggers” on the wrong plot); if not, just click on “Speaker Cone”. Start recording. You’ll see two traces: one for the speaker’s sinusoidal oscillation and one for the air’s oscillation at the sound sensor’s location. Slowly move the sound sensor toward and away from the speaker to see what the traces show.


33

3. If the sound sensor were at a point where the two traces are in phase, movement of one-half wavelength would put them out of phase—right?—and another half-wavelength would put them again in phase, etc. So, while recording, move the sound sensor through several such alignments, keeping track of its location and how many half-wavelengths it moves, then, combined with the function generator’s 2,000Hz frequency, determine the speed of sound in air. Record distances and show your calculations below. (Note: For several reasons it’s best if none of your measurements is taken closer than about 25cm from the speaker.)

4. Now in the Signal Generator window click on the frequency 2000 Hz, type in 1500, hit Enter, then repeat the experiment. Again calculate v, showing your work below.

5. Would you say that the speed depends on the frequency? If not, what do you think it might depend on? (A previous lab may help guide your thoughts.)


34

ACTIVITY 3: TWO SOURCE INTERFERENCE Introduction You’ll see interference later this quarter using electromagnetic waves in not very ordinary

circumstances. But sound provides valuable experience with this important physical phenomenon using rather commonplace items: speakers driven at audible frequencies. It will be very helpful to keep Figure 3 in mind during this activity.

Procedure 1. Place two speakers about 75cm apart, on either side of the Pasco box, then plug both their red wires into the plus port, , and both black ones into the minus, . The speakers should be facing the same direction (not each other!), with a large uncluttered space in front.

2. Soon both speakers will produce a 750Hz frequency. Assuming a speed of sound of 340m/s, what would be the wavelength of the sound waves produced?

λ:___________________cm

3. Open the file “Two Speaker Interference”. Start recording, and with your ear near one speaker then the other, check to make sure they produce roughly the same loudness, which helps ensure that destructive interference is as complete as possible. If they differ greatly, tell your lab instructor. Then, with one ear pointed toward the speakers, about a meter or so away, and maybe the other ear plugged, walk slowly from left of both speakers to right of both, listening carefully for any variation in sound. Briefly describe what you hear.

4. Make sure the sound sensor is in Port A, then, while continuing to record, hold the sensor at a point about equidistant from the speakers and not too close to them, perhaps near the front/closer edge of the lab table. Move it around just a bit to make sure you’re at a point of constructive interference, where the trace is at a local maximum. (Remember: You reflect sound. Try to be as invisible as possible.)

5. With a meterstick measure the distances, L1 and L2, from the speakers to the sound sensor. Record them below and comment—do they make sense?

L1:___________________ L2:___________________


35

6. Now a prediction! If you move the sound sensor so that the sound wave from one speaker has noticeably farther to go than that from the other, i.e., that ΔL ( |L1-L2| ) is not approximately zero, what value of ΔL—an actual value in centimeters or meters!—would give constructive interference. Briefly explain your logic.

7. Summon your lab instructor and explain to him/her your answer in step 6 and how you plan to proceed in step 8.

8. Test your prediction. Show the data from your simple experiment, and discuss how, if at all, it supports your prediction. (You may find that things work better on one side than the other, particularly if your speakers don’t emit approximately equal intensity. Why?)

9. Find a point of destructive (or nearly so) interference. Measure and record L1 and L2, find ΔL, then explain why this value should indeed give destructive interference.


36

ACTIVITY 4: BEATS Introduction In Activity 3 you saw interference between two waves of the same frequency, varying between

constructive and destructive due to differing paths to the observation location. Here you’ll hear interference going back and forth between constructive and destructive at the same location due to a slight difference in frequencies between the two wave sources—known as beats.

Procedure 1. While leaving one speaker connected to the Pasco box and outputs, move the other speaker’s red and black wires to the two “Speaker” jacks on the side of the separate UCD Physics Function Generator, red into red, black into black. This signal generator we henceforth refer to as the “external generator”. Then place the speakers fairly close together, though not face-to-face—you just want to listen to them both at once.

2. Make sure a voltage sensor is plugged into Port B of the Pasco box, and insert its small red and black plugs into the “Speaker” jacks on the external generator (actually, into the wires you already plugged in there). Important Note: NEVER remove voltage sensors by pulling on the wires! Grasp them only by the plastic plugs at the ends!

3. Finally, connect a black hookup wire from the port on the Pasco box to the black “Speaker” jack (or wires plugged therein) on the external generator. This final wire ensures that the two signal generators pretty much agree on the value of zero signal.

4. Before turning on the external generator, turn both its “Amplitude” and its “Volume” knobs fully counterclockwise and set its top-panel toggle switches at “Sine” and “Mid” (not High). Turn it on and adjust its frequency knob till it reads 120. You may hear a low hum. (We aren’t actually using the “Amplitude” knob, but if we don’t turn it down, we tend to pick up campus radio!)

5. Open the file “Beats”. Start recording. You’ll see two traces, the green one, the external generator, of smaller amplitude, maybe even flat. Adjust the external generator “Volume” knob till the two sinewaves are the same amplitude. Don’t worry if they’re not both exactly centered on the horizontal axis—just make sure that their peak-to-peak heights agree.

6. Note that the Pasco box signal generator is 120Hz and will remain at that frequency throughout this activity. Now, while recording, slowly vary the external generator frequency between 116Hz and 124Hz. Occasionally stop then restart the recording, to make sure you’re seeing the whole picture. Describe what you see and what you hear, and explain how they fit together.


37

7. Return the external generator to 120Hz, then from the Window menu choose “Sum of Pasco and External Generator Outputs”. This plot algebraically adds the two signals—as does your ear! Now, while recording, slowly vary the external generator frequency between 119Hz and 121Hz. Describe what you see and how it correlates with your answer in step 6.

8. Now while again recording, slowly adjust the external generator frequency to 110Hz, then stop recording to freeze the display, then make a printout.

9. The standard theoretical analysis of beats, which you’ll find in your textbook, shows that the “beat frequency” in Hz, or loud instances (wahs) per unit time, should equal the difference between the two individual frequencies, | f1 - f2 |. This means that each “wah” in the “wah-wah-wah” should last for a time 1/ | f1 - f2 |. Does your data agree—and if so, how?

10. Turn off the external generator and remove all wires from it.

Microwave Interference

38

PRELAB: QUANTIFYING UNCERTAINTY (TURN IN YOUR ANSWERS TO YOUR LAB INSTRUCTOR AS YOU ENTER THE LAB ROOM)

Question 1: If an object near Earth’s surface falls a distance y from rest, and air resistance can be ignored, for how much time was it falling? Question 2: If an experiment measuring a distance has a statistical uncertainty (obtained from a reasonably large number of repetitions) of 1.0cm and the instrument used to measure that distance has an inherent instrumental uncertainty of 0.5cm, what is the absolute uncertainty in the distance. Question 3: A quantity F depends on the quantity v according to F = 8.0 kg!sm2 v

3 . An experiment determines that F is 125N with an absolute uncertainty of 6N. Show that the percent uncertainty in v is 1.6% and its absolute uncertainty is 0.04m/s.


39

QUANTIFYING UNCERTAINTY


Circle the name of the person to whose report any joint printouts are attached. References Physics 9 Lab Manual Supplement: Error Analysis Apparatus Reaction time apparatus: Shield and measuring stick.


40 Introduction Today’s lab is a departure from the norm, and the intent is to familiarize you with some important

ideas involved in advanced experimental science and to solidify ideas of error manipulation you have already applied in Physics 9 lab. The activities center on a simple measurable quantity: reaction time.

ACTIVITY 1: PREPARATION

Procedure 1. Before we begin our experiment, think of the words "precise" and "accurate". If the center of each bullseye below is an accepted correct value of some measurable quantity, like the actual epicenter of an earthquake, and the spots are repeated results of an experiment intended to determine that value, like locating an epicenter from remote data, compare and contrast the four data sets with respect to their accuracy and their precision. Claim some place for your group on the chalkboards and write your responses so all can see them.

1 2 3 4

2. Errors can be "random" (fluctuations all around a mean) or "systematic" (all wrong in the same direction, perhaps due to a miscalibrated measuring device). Comment on how random and systematic errors might affect precision and accuracy. Write your answers on the board.

3. When all groups have reached this point, your lab instructor will have a few things to say.

ACTIVITY 2: REACTION TIME

Procedure 1. Choose a hand of one volunteer from your group, the hand that that volunteer does not use for writing. This hand will make repeated trials of reaction time. Here is the procedure. You have a stick marked in centimeters, and a shaft in which it can be held vertically. At the shaft’s bottom are two projections to serve as nail rests. The chosen hand should have its fingernails against one rest and thumbnail against the other. Another group member on the shaft’s open side should hold the stick against the stop at the shaft’s top. The chosen hand should have its thumb and fingers at the same level as a chosen point near the bottom of the stick (e.g., the 11cm mark) that will be used in all subsequent repetitions of the experiment. The other group member should release the stick so as not to allow the volunteer to anticipate it, and the chosen hand should then grab the falling stick between fingers and thumb as quickly as possible. Practice this procedure four or five times before taking “real data”. Then carry out the experiment at least 15 times, recording the distance fallen, rounded to the nearest centimeter, for each trial.

2. On the chalkboard (you can erase earlier stuff), make a table that others can see, the heading of the left column being "Trial" (i.e., trial number) and the heading of the right being "Distance" (i.e. fallen). Fill in as many rows as you have trials.

3. Now in your group discuss how you would graph the data from several different experimental groups to most quickly convey how they compare in the typical distance fallen and the variation in that distance. When you've decided, put your graphs on the board, using data from at least three groups, including your own.



41

ACTIVITY 3: HOW PRECISE?

1. In your groups discuss how you would define a single quantity that would be a good measure of the precision of a given group's data. On the chalkboard describe/clarify your chosen definition.


ACTIVITY 4: APPLYING THE IDEAS

1. Now, as a warmup example on different physics, consider this: Suppose a circle’s radius is measured as R = 1.0m ± 0.03m . Write on the board your answers to the following questions. A: Assuming for the sake of argument that the radius could be no larger than 1.03m nor smaller than 0.97m, what are the maximum and minimum possible areas, by how much in square meters do they differ from the mean area (i.e., πR2) of 3.14m2, and by what percent do they differ from 3.14m2? B: As a measure of the uncertainty in the area, which makes the most sense to use: the absolute uncertainty in the radius? the square of the absolute uncertainty in the radius? the percent uncertainty in the radius? the square of the percent uncertainty in the radius? twice the percent uncertainty in the radius?

2. Using the mean distance of your group’s data and combining the statistical uncertainty in that mean with your inherent instrumental uncertainty of 0.5cm (half the smallest division used for the measurements) to give your distance uncertainty, determine the reaction time appropriate to your group's data and use the rules of error propagation in the Physics 9 Lab Error Analysis supplement to calculate the uncertainty in that reaction time in seconds. Show your work below.

3. Discuss what you learned in this lab; for instance, what factors affect the precision of a measurement and what might affect its accuracy. You can restrict your answers to the experiment at hand, or feel free to make more general statements. Continue on the back if you wish.

Geometrical Optics

42

PRELAB: GEOMETRICAL OPTICS (TURN IN YOUR ANSWERS TO YOUR LAB INSTRUCTOR AS YOU ENTER THE LAB ROOM)

The diagram shows an object to the left of a converging lens, Lens 1, which is to the left of another converging lens, Lens 2. Light rays diverge in all directions from two representative points on the object, its top and its bottom, and some strike Lens 1. Rays from the bottom of the object are shown in gray. Rays from the top are shown in black. The questions here refer only to the black rays. As you can see, Lens 1 converges them, and they come to an image point, producing the top of an “intermediate image”. Of course they keep going, and some strike Lens 2, which converges them to another image point, the top of the “final image”. The foci of the lenses are shown as just round spots.

Question 1: (a) Five (black) rays are shown passing through Lens 1. Label the three that are principal rays, each with a number, 1, 2, or 3, corresponding to the definitions in the Introduction to the Geometrical Optics lab. Label them fairly close to Lens 1. (b) Five rays are also shown passing through Lens 2, though not necessarily the same ones shown passing through Lens 1. Label the three principal rays for this lens. Question 2a: Suppose the focal length of Lens 1, f1, is 20cm and the Object is 30cm from the lens. Calculate how far the intermediate image should be from Lens 1. Show your work. Question 2b: The rays leaving Lens 1 don’t stop at the intermediate image; they keep going, and strike Lens 2 just as they would from a real object at the location of the intermediate image. If Lens 2 is 135cm from Lens 1, how far from Lens 2 is this object that it “sees”? Show your work. Question 2c: If the focal length of Lens 2, f2, is 30cm, how far from Lens 2 will the final image be? Show your work.

Geometrical Optics

43

GEOMETRICAL OPTICS


Circle the name of the person to whose report any joint printouts are attached. References University Physics Sections 33.3 and 34.4. Apparatus Optical bench with sliding mounts, a 10cm and 15cm converging lens, object slide, screen, gray-box

(no lens) light source cross-clamped to portable stand, small ruler. One setup per room of laser/turntable/lucite half-cylinder for reflection/refraction study.

Geometrical Optics

44 Introduction Optics is usually it is broken into two categories: Physical Optics, dealing with light’s wavelike

behaviors, such as interference and diffraction; and Geometrical Optics, dealing with its tendency to travel as straight line “rays” until reflecting, as it does from a mirror, or refracting/bending, as it does in passing through a lens. Geometrical optics is important in many areas. It’s what’s behind the operation of telescopes and microscopes—and your own eyes! In an exciting new application, understanding how to manage light rays is crucial to the pursuit of the ultimate computer monitor: the 3D display, an enticing goal being researched in a number of high-power labs.

You should definitely study the portions of your textbook mentioned in the “References” at the beginning of this lab. Here we review only the bare bones. When diverging rays of light approach a lens from a single real object point a distance do from the lens, they will after passing through the lens also have a single point in common, the image point, a distance di from the lens. The rays will either converge to that point, forming a real image on the outgoing side of the lens with a positive di, or diverge as if from a point on the incoming side of the lens, forming a virtual image with a negative di. Figure 1a shows rays diverging from a real object point a distance do from the lens and converging to a real image point a distance di from the lens. Figure 1b shows he formation of a virtual image. The formula relating the object distance and image distance is the “lens equation”:

1

do+1

di=1

f (1)

The focal length f is a characteristic of a lens, depending on the radii of curvature of its surfaces and the refractive index of its glass. Lenses has two foci F, usually equidistant from the lens, a distance f away. (F is a point in space, while f is a distance from the lens to that point.) A converging lens is curved so as to tend to converge light rays toward the lens’ axis, while a diverging lens tends to diverge rays away from the lens’ axis. (We study only converging lenses today.) Figure 1 shows converging lenses in use. Note that even if the outgoing rays are diverging, as in Figure 1b, they are doing so less drastically than they were going in—the converging lens has caused them to be less divergent. The change from a real image in Figure 1a to virtual in Figure 1b is not because the lens isn’t converging the rays in Figure 1b, but because if the object is too close (i.e., within the focus), it can’t converge them enough to actually emerge converging and form a real image.

Figure 1 shows image and object points connected by only three of the infinite number of rays they share: the principal rays. They are known by this name because, while they, like all others, bend so as to obey equation (1), they do so in ways that are particularly easy to follow.

Ray 1 starts out parallel to the axis, then refracts toward the axis, passing through the focus on the lens’ far side.

Ray 2 ends up parallel, having approached the lens from the lens’ focus.

Ray 3 approaches the lens heading directly toward the lens’ center, and passes through essentially undeviated.

Geometrical Optics

45 Since a straight line (i.e., Ray 3) connects the tops of the object and image, the ratio of image

height to object heights is the same as the ratio of their distances from the lens:

hi

ho

= !di

do

(2)

Here the minus sign is simply stuck in to give the most sensible the results: A real image, with a positive di, is inverted (Figure 1a), so we insert the minus by convention to give a negative hi.

ACTIVITY 1: CONVERGING LENSES

Introduction Without knowing a len’s radii of curvature nor the type of glass, which would allow us to calculate f, we’ll simply observe them in action and infer f. First we’ll apply a fairly crude technique known as the distant-object method. In equation (1), if an object is very far away, so that do is very much larger than f or di, the first term in the equation is very small. It follows that di = f. In other words, when the object is very distant, the image is at the lens’ focus. After the distant-object method, we’ll be a bit more methodical and precise, while beginning to see what images are really all about.

Procedure 1. You have two lenses: Lens 1 and Lens 2, nominally 10cm and 15cm focal length, respectively. Hold Lens 1 up so it can “see” an object fairly far away, such as the window or an overhead light, then hold the small screen on the other side of the lens and move it back and forth till the object is in focus. Have someone measure the image distance di from the lens to the image/screen, which should be f, and record it below in cm. Repeat for Lens 2.

f1:_________________cm f2:_________________cm

2. Now a more precise way. First note that your bench has a centimeter scale along it, yielding what we’ll refer to as x values. Place your object in a sliding mount at x = 0.0. To streamline the rest of the lab, it is probably also good to briefly place Lens 1, Lens 2, and your screen all in sliding mounts near the object and adjust their heights so that all are about the same.

3. Remove Lens 2 and the screen (we’ll use them later), then position Lens 1 at x = 24.0cm.

4. Now predictions. Based on your step 1 results and your known object-lens separation, which of Figures 1a and 1b best represents how the image would be formed? Briefly explain.

At roughly what value of x do you expect to find the image?

Should the image to be upright, or inverted, and roughly what should the ratio hi/ho to be?

Geometrical Optics

46

5. Now test your predictions! With the light source behind the object to illuminate it (you may wish to hang a piece of paper over the light to dim it), move the screen, placed in its sliding mount, along the bench till the image is in focus. Discuss how your predictions compare.

6. With a ruler measure the distance between two distinctive points on the object and the same two points on the image. Note, however, that owing to restrictions in the “field of view” later, you should choose two points on the object no farther apart than about 1cm.

ho:_____________cm hi:_____________cm

Is the image oriented (upright versus inverted) as you expect, and roughly the correct height?

7. Enter in Table 1 your image distance di, i.e., the difference between the image/screen location x you read directly off the optical bench (don’t use the rough value you guessed in step 4!) and the lens location, which right now is x = 24. Also record hi and the ratio hi/ho.

do di hi hi/ho f = 1

do+ 1

di( )!1

di/do

24.0cm

19.0cm

14.0cm

Table 1

8. Remove the screen but leave its sliding mount in place, then place your eye near the end of the optical bench opposite the object. Can you see the image, and is its orientation correct?

9. About how close to the screen’s (empty) sliding mount can you place your eye and still focus clearly on this image? Briefly explain why there should be a limit. (Note: An image is a collection of points in space from which light reaches your eye exactly as from a real object there.)

Geometrical Optics

47

10. Now a methodical analysis to find f. Return the screen to its sliding mount, then move the lens to x = 19.0cm, i.e., 5cm closer to the object than it was before. Move the screen to form a clear image, then record di, hi (while ho can’t change, hi may!), and the ratio hi/ho in Table 1. Repeat the procedure, but with the lens at x = 14.0cm.

11. According to equation (1), adding the reciprocals of do and di should always give the same thing, namely, the reciprocal of f. Let’s see! Complete the last two columns in Table 1, which relate to equations (1) and (2), respectively.

12. All in all, how well are equations (1) and (2) borne out by your data? If not exactly, in what ways is error most likely to have crept in?

13. Now move the lens to x = 9cm.

14. Which of Figures 1a and 1b best represents how the image would be formed? Briefly explain.

15. Do you think you should be able to place the screen at some value of x and focus the image?

16. Test your prediction with an experiment, then explain how your experiment supports your prediction, or, if it didn’t, what was wrong with your prediction.

17. Summon your lab instructor and discuss with him/her the physics in steps 5, 9 and 16.

Geometrical Optics

48 18. Roughly where should the image be, and should it be upright, or inverted? Show your work or

explain your answer as appropriate, referring to Figure 1 if it helps.

19. With the screen removed again put your eye near the end of the optical bench. Can you focus clearly on the image?

20. Move your eye right up to the lens. Can you focus on the image? If not, why not? And if so, keeping in mind your responses in steps 9 and 18, why?

21. Average your f values from Table 1 and record it below. We’ll refer to this as f1, the subscript meaning Lens 1.

f1:__________________cm

ACTIVITY 2A: A TWO-LENS SYSTEM

Procedure 1. Many important applications of geometrical optics (e.g., telescopes, microscopes, binoculars) employ multiple lenses, light passing through one then another. To study this, we need the focal length our second lens. Two careful measurements should give reasonable accuracy. So, replace Lens 1 on the bench by Lens 2. Position it at x = 22.0cm, form the image on the screen, record the image (lens to screen!) distance, then calculate f using equation (1).

do: 22.0cm di:___________cm f:___________cm

Move the lens farther from the object, then repeat the procedure, again obtaining f.

do:___________cm di:___________cm f:___________cm

If your values aren’t close, you should rethink your procedure. Otherwise, record the average

f2:__________________cm

Geometrical Optics

49

2. Place the screen at x = 115.0cm and Lens 2 at x = 83.0cm. Then return Lens 1 to the bench somewhere between x = 10 and x = 20 and slide it around in this region till you see a clear image on the screen. (You’ll also see a clear image with Lens 1 around x = 40. This isn’t the one we want!) If you have trouble getting an image, recheck the alignment (heights) of the various elements and perhaps reposition the object slide in its clip. Tighten the Lens 1 sliding mount. This x value is the object distance for Lens 1. Record it below.

do1:_______________cm

3. Now, which of Figures 1a and 1b do you expect best represents how the light rays move through Lens 1, and then through Lens 2? (They need not be the same.)

4. Summon your lab instructor, show him/her your f2 value, and discuss your answers in step 3.

5. A prediction! We use the term “intermediate image” for the image produced by the first lens, which becomes the object for the second lens. (This is the guiding principle in multi-lens systems!) The question here is: Where do you expect the intermediate image to be? But there are two approaches: (1) finding di1 based on knowing the first lens object distance do1 and focal length f1; and (2) finding do2 based on the second lens image distance di2 and focal length f2. Carry out both, thus giving two (perhaps equal?) values of the x where you expect it to be.

6. Now test your prediction. Remove the screen, leaving its sliding mount at 115, and hold it so as to reveal the intermediate image. Record the x value below.

xinter. image:__________cm

7. Are your various x values reasonably close?

Geometrical Optics

50

8. Finally, put the screen back in its mount at x = 115, then measure the final image height (the same two points you used in Activity 1 step 7). Record the ratio of this height to the object height (not hi itself!) below.

(experimental) hi ( final)

ho:_______________

Applying equation (2) to the lenses in sequence shows that hi(final)/ho = (-di1/do1)(-di2/do2). (Make sure you understand how the intermediate image’s height cancels.) Using the x value of step 6 for the position of the intermediate image, calculate di1 and do2 (which may differ from those in step 5). Then calculate what the ratio of final image height to object height should be.

di1:_________cm do2:_________cm

(theoretical) hi ( final)

ho=di1

do1

di 2

do2:_______________

9. How close are the theory and experiment?

10. Is the final image upright, or inverted? Explain, with simple arguments based on the qualitative behaviors in Figures 1, why it should be oriented as it is.

Geometrical Optics

51

ACTIVITY 2B: A FEW CHANGES

11. Let’s change our two-lens system a bit and see what happens. Leaving Lens 1 where it is, move Lens 2 to an x-value of 14.0cm beyond the position of the intermediate image from step 6, i.e., move it to xinter. image + 14.0cm. Record the new Lens 2 position below.

xLen 2:________________cm

12. Now, based on the distances and your known f1 and f2, which of Figures 1a and 1b do you expect best represents how the light rays move through Lens 1, and then through Lens 2?

13. The rule, again, is that the image produced by the first lens becomes the object of the second. In steps 2 and 8 you have do1 and di1, neither of which has changed, Given your new do2, find di2, and thus find where the final image should be. Show your work clearly below.

14. Do you think you should be able to place the screen at some value of x and focus the image?

15. Test your prediction with an experiment, then explain how your experiment supports your prediction, or, if it didn’t, what was wrong with your prediction.

16. With simple arguments again based on Figures 1, explain whether you expect the final image to be upright or inverted.

Geometrical Optics

52

17. Is the image indeed oriented as you expect? And how are you able to confirm it (particularly if not on a screen)?

18. Calculate, as in step 8, the ratio hi(final)/ho.

(theoretical) hi ( final)

ho=di1

do1

di 2

do2:_______________

19. A microscope produces a final image that, though it may be farther away than the actual object, is so large that fine detail can be resolved. How, if at all, does your “optical instrument” fit this description?

Geometrical Optics

53

ACTIVITY 3: AN IMPORTANT AND COOL PHENOMENON

1. Somewhere in the room is an apparatus with laser and a half-cylinder atop a turntable. This provides a vivid demonstration of reflection and refraction, but also a close look at a special phenomenon: total internal reflection. If you don’t understand it, you’ll be completely lost in many discussions of modern optics—for instance fiber-optic communication—and possibly suffer irreparable guilt for not having learned the basics in Physics 9B. To ensure that this never happens, place the half-cylinder so its straight side is centered on the turntable axis, and rotate the turntable, observing both cases shown at right, then (a) determine the plastic’s refractive index and (b) write out the following sentence, properly completed: “Total internal reflection can occur when light passes from one medium to another of [FILL IN] refractive index, and it occurs when the incident angle is [FILL IN] the critical angle.” Fully explain your observations and answers! [Important Note: In geometrical optics, angles are always defined with respect to the normal to the interface, an imaginary line perpendicular to the boundary surface between the media.]

Light as a Wave

54

PRELAB: LIGHT AS A WAVE (TURN IN YOUR ANSWERS TO YOUR LAB INSTRUCTOR AS YOU ENTER THE LAB ROOM)

Question 1: Is the “central diffraction maximum” a feature of the single, or double, slit, and what are its borders? Question 2a: Below we see the sums of two sinewaves in phase and at intervals out of phase of 1/6 λ, up to a total difference of 1/2 λ. Note that all the sums are sinewaves, but the amplitude gets progressively smaller. This is exactly as we expect in the double-slit: The intensity should get progressively smaller as the two waves go from in phase, at a constructive interference max, to completely out of phase, at a destructive interference min.

Suppose we add a third wave, each being out of phase with the next by the same amount. Though the triple-slit isn’t something used often, it shows a trend vital to understanding how a grating (many slit) works, and gratings are used everywhere! The first sum is shown. Sketch in the other three. Save time by looking only at points 1/12 of a cycle (π/6 or 30°) apart (i.e., the vertical lines) where all waves are conveniently 0, ±0.5 (sin 30°), ±0.866 (sin 60°), or ±1.

Question 2b: Compared to the 2-slit, does the wavelength shift between adjacent waves have to be as large in the 3-slit for completely destructive interference to occur, and is the in-phase maximum in the 3-slit relatively more, or less, intense than states of partial or total destructive interference?

Light as a Wave

55

LIGHT AS A WAVE


Circle the name of the person to whose report any joint printouts are attached. References University Physics Sections 35.1-35.3 and 36.1-36.5. Apparatus Laser on optical bench, variable-width single slit (brass), tape measure, slit slide in optical bench

mount, Pasco box, light sensor on translation guide, rotary motion sensor, some old CDs. Printouts Single-slit pattern traced onto page Intensity vs Distance

Light as a Wave

56 Introduction Last week we studied two-source interference using microwaves. This week we’ll see the same

phenomenon with light, and also study other important interference phenomena. SINGLE SLIT Anytime a wave passes through an aperture, it diffracts

or spreads out. Today we study the simplest case, the single slit. For details, see the relevant sections of your textbook, but perhaps the most important thing is the variation of the intensity, shown in Figure 1. Most is in a broad region known as the central diffraction maximum, bordered on both sides by minima of zero intensity. The intensity of a wave after passing through a slit of width a is given by

I

Imax

=sin(! " # l /$)! " # l /$

%

&

' (

)

2

where ! " l = asin#

where θ is the angle at which the intensity is observed. The path difference Δl´ is between waves reaching this point from the slit’s edges. The intensity I is its minimum of zero—the numerator is zero—when Δl´ is λ times any integer—except zero, where l‘Hopital’s rule shows I to be not a minimum, but a maximum. Using an m´ for the integer, this condition becomes:

Minimum: Δl´ = λ m´ or a sin θ = m´ λ m´ = 1, 2, 3, ... (1)

These diffraction minima are labeled according to m´ in Figure 1. Note that equation (1) says that decreasing a would increase the angle θ at which a given diffraction minimum would be found. In other words, the pattern’s width is inversely proportional to a.

DOUBLE SLIT While of course made up of single slits, multislit devices—

such as a diffraction grating, used much in astronomy—are important in their own right, and the first step in that direction is the double slit. For two slits whose widths are very narrow and whose center-to-center separation is d, the intensity, shown in Figure 2, is given by

I

Imax

= cos(! "l /# )( )2 where !l = d sin"

Here Δl is the path difference between the waves reaching the observation point from the two slits. The intensity will be its maximum—the cos(πΔl/λ) will be ±1, so that I will equal Imax—when Δl is λ times any integer. Using an m for the integer, this condition becomes:

Maximum: Δl = λ m or d sin θ = m λ m = 0, 1, 2, 3, ... (2)

These interference maxima are labeled according to m in Figure 2. MANY SLITS If we simply increase the number of slits, keeping their separation d the same, the interference

maxima will stay right where they are. If two waves reach a screen or detector exactly one whole wavelength out of phase, for instance, then additional waves each a certain number of whole wavelengths out of phase will surely preserve the constructive interference at that point. However, these maxima become more distinct, because more of the space between them is taken up by destructive or nearly destructive interference. You should see this clearly in today’s lab!

Figure 1

Figure 2

Light as a Wave

57 THE REAL WORLD It’s important to note a realistic difference in today’s lab. The intensity in Figure 2 is for two slits

whose widths are “very narrow”, so that the effects of diffraction aren’t noticeable. The diffraction pattern of each slit is so spread out that there is no fall-off of intensity due to it, and all the maxima are essentially equal height. But what if the slits don’t qualify as very narrow, which is quite often the case? How could we possible get nice equal-height interference maxima if the light intensity varies according to Figure 1 from each slit alone!? Well, we don’t! In such a case, the two-slit intensity pattern of Figure 2 is modulated by the single slit intensity pattern of Figure 1. Figure 3a shows such a double slit pattern, with interference maxima varying in intensity according to the limits imposed by single-slit diffraction. Note that the places where the single-slit intensity goes to zero may cause interference maxima to be absent, in this case the m = 4 interference maxima. Figure 3b shows two slits the same distance apart as in Figure 3a—the same d—giving the same spacing of interference maxima according to equation (2), but whose width a is even wider than in Figure 3a. The single-slit diffraction minima fall even closer to the center of the pattern, in this case “erasing” the m = 3 interference maxima.

THE LASER The light source in today’s lab is a helium-neon laser, whose wavelength is somewhat famous:

λHeNe = 633nm

A laser is an ideal source, for it naturally produces coherent light, i.e., of a single wavelength, unidirectional, and, most importantly, all in phase, rather than being many “pieces” of waves all out of phase with each other. Coherent light is just what is needed to show interference effects. Before proceeding any further, however, heed these warnings about lasers:

(1) Do not look directly into the laser beam! It could damage your eyes! Be content to observe the beam striking a wall or other object.

(2) Be careful when handling or working near the laser, as it is delicate. (3) Do not aim it at anyone, and even when simply swiveling it around, place

your had in front of it to ensure that it doesn’t shine in anyone else’s eyes. Don’t play with the laser!

APPARATUS Figure 4 shows today’s apparatus. The laser and a slide with different slits are mounted on an optical

bench and a light sensor is mounted at the far end of the lab table. In Activity 1 the light sensor and slit slide aren’t used. Instead, the laser is swiveled to point at the nearby wall. In Activity 2, the laser is swiveled so that its light passes through the slit slide before being analyzed by the light sensor.

Figure 4

Light as a Wave

58

ACTIVITY 1: A SINGLE VARIABLE SLIT

Procedure 1. Direct the laser beam at the nearby wall—swivel it if necessary. After turning it on, simply hold one of the brass variable-width slits (should be lying around somewhere) in front of it, then vary the slit width. Briefly describe how the pattern varies as the slit width varies.

2. Now adjust the slit width until the pattern is about as spread out as you can make it and still distinguish it fairly well. (Sometimes the slits’ edges are rough—look for a smooth region.)

3. With the slit in front of the laser, have someone from your group hold this page against the wall and trace the pattern onto the left edge. Remember that the tracing should correspond to Figure 1—dark at high intensity, less dark at low intensity. (Note: This tracing is a “joint printout”, so whichever group member’s report has it should also have the other joint printouts attached to it.)

4. Measure the distance L from the slit to the wall and record it below.

L:_______________cm

5. Record the distance Δy, measured on your tracing, from the first diffraction minimum on one side of the central diffraction maximum to the first minimum on the other side.

Δy:________________cm

6. The tangent of the angle θ from the center to the first diffraction minimum is 12 !y / L . Calculate your slit width a. (Remember: λ is 633nm.)

7. How reasonable is your result?

Figure 5

Light as a Wave

59

ACTIVITY 2: ONE, TWO, THREE, LOTS!

Procedure 1. Before taking data, make sure everything’s ready. The light sensor goes to Port A of the Pasco Box and the rotary motion sensor to Ports 1 (yellow plug) and 2 (black). (Note: Your detector and laser are on opposite ends of the lab table.) The switch on the light sensor’s top should be on 10 (not 1, or 100). And—important!—the black plastic cap on the light sensor should always be in place. It isn’t there to keep light completely out of the sensor—its 0.4mm-tall “window” allows light to enter in only a very restricted vertical range.

2. Now swivel the laser toward the slit slide mounted on the optical bench. Leave this expensive and delicate slide mounted to the bench. All you need ever do is slide it sideways within its mount and perhaps move its mounting rod vertically, to center the laser spot on slits. Turn on the laser and position the slide so the laser hits the spot marked in Figure 6 with an “a”. This is a single slit, narrow in the vertical direction, so that light spreads vertically. (Make sure you have the correct place! Note the double-slits of increasing separation along the top. If you look at the face of the slide where the one of largest separation is on the right, as in Figure 6, the single-slit marked “a” will be on the left. Check with your lab instructor if you’re uncertain.) Fine-tune the slide’s position for the brightest pattern.

3. Now, while someone moves the light sensor along its vertical travel by rotating the pulley on the rotary motion sensor (not by lifting up or pushing down on it) someone else should make sure the optical bench is oriented so the diffraction pattern falls on the center of the light sensor along its whole vertical travel. If not, adjust the optical bench’s tilt with its adjustable feet.

4. Open the file “Intensity vs Distance”. Starting with the light sensor at the bottom of its travel, begin recording, then slowly and as smoothly as you can manage, raise the light sensor by rotating the pulley on the rotary motion sensor. Stop recording when the light sensor gets to the top. (Slow movement is important! Taking at least 20 seconds to raise it would be good. If you move the sensor too fast, the computer will miss many data points and your plot will be jagged. Also, backing up usually makes for an ugly plot, so, once started, keep rotating the pulley in the same direction. The more care you take, the better will be your data!)

5. Using the scale-to-fit button, take a look at your pattern. Does it resemble Figure 1? (Note:

For several reasons not worth going into, the minima may not go to zero. Do the best you can in judging where they’re supposed to be.)

6. Measure the distance from the slide to the light sensor and record it below.

L:_______________cm

Figure 6

Light as a Wave

60

7. Now reposition the slide in its mounting so that the laser is aimed at the spot marked in Figure 6 with a “b”, fine-tuning for the brightest pattern. This is a double slit.

8. Repeat the data run as in step 4. Don’t expect perfection, but the pattern should be roughly symmetric about its center. If it’s very asymmetric, it’s probably because the optical bench is tilted, and you may want to adjust its feet then try another data run.

9. Using the scale-to-fit button, take a look at your double-slit pattern. Does it resemble one of the

double-slit pattern of Figures 2 and 3? If so, which?

10. Can you see your double-slit intensity pattern, as the introduction notes, “modulated by the single slit intensity pattern”? (Use the Data button to toggle back and forth if it helps.)

11. How does your data reflect the fact that the widths of the slits in this double slit are the same as the width of the single slit? (The data may not be able to establish that they’re exactly equal, but that they’re pretty close. What would you see if they were vastly different?)

12. Summon your lab instructor and explain to him/her your answers in steps 10 and 11.

Light as a Wave

61

13. The relatively narrow peaks you see are the double-slit interference maxima. The tangent of the angle θ from the center to the first such maximum is 12 !y / L , where Δy is the distance from the first max on one side of center to the first on the other . Calculate the slit separation d. Show your work below, clearly indicating which equation is your “starting point”.

d:______________

14. If you’ve done multiple 2-slit data runs, note the “run number” of your best. We’ll print it later with other patterns. Now again reposition the slide so the laser is aimed at the spot marked “c” in Figure 6. This simply adds a third identical slit separated from the other two by the same d.

15. Repeat the data run as in step 4. As always, repeat the run if you doubt your procedure. (Note: Here or perhaps in the next case you may begin to see some “clipping” of the highest peaks. The sensor is slightly overranging. Just pretend the peak’s tops are there.)

16. Are the interference maxima the same distance apart as in the 2 slit (refer back to your 2 slit pattern if necessary), and should they be?

17. At the points exactly in between the interference maxima, is the intensity essentially zero, as it was in the 2-slit? If not, explain what is happening here. (Hint: Remember the prelab?)

Light as a Wave

62

18. If you’ve done multiple 3-slit data runs, note the run number. Now, finally, reposition the slide so the laser strikes spot “d” in Figure 6. This gives ten equally separated slits which, if not exactly a “diffraction grating”, should get the idea across. Repeat the data run as in step 4.

19. Now print all your slit data together. Double-click on your plot, choose the “Layout” tab, then click “Create New Graph” under “Measurement Adding”. Hit “OK”, then using the

button, successively choose run numbers so your best single-slit, double-slit, 3-

slit and 10-slit runs show together (the Data button is a “toggle”: choose a run number, it appears; choose it again, it disappears). Make a printout of your four intensity patterns.

20. Based on your data, briefly describe the effect that increasing the number of sources has on an interference pattern.

ACTIVITY 3: AN APPLICATION

1. Locate a CD lying somewhere around the lab room, then swiveling the laser as necessary (be careful about your classmates’ eyes!), determine the spacing between its “slits.” Describe your assumptions and procedure, and present your data clearly. Afterward, estimating the laser beam width, estimate how many “slits” you are using and comment on its relevance.

Low Temperature Experiments

63

PRELAB: THERMODYNAMIC BEHAVIORS (TURN IN YOUR ANSWERS TO YOUR LAB INSTRUCTOR AS YOU ENTER THE LAB ROOM)

Question 1: From Physics 9A, work is force times distance, F×Δx. But if the work is done by a gas in a cylinder pushing a piston of area A, it exerts a force of pressure times area, F = PA. (Pressure is defined as force per unit area.) Assuming a constant pressure, put the work done by the gas as it expands and pushes the piston solely into terms of the (constant) pressure P and the initial and final volumes, Vi and Vf, of the gas. (Remember this result in Activity 2!) Question 2: Energy conservation is everywhere. In particular, we depend on it in the study of thermodynamic systems where the detailed interactions between particles in the system are (usually) beyond our abilities to observe. (a) Briefly, what characterizes a “thermodynamic system”? (b) Starting in Activity 2 of the Thermodynamic Behaviors lab, three energies are discussed a lot. One is the internal energy of the thermodynamic system, and the other two are ways of changing that energy, of transferring energy into or out of the system. One is mechanical work, discussed in question 1 above, and the other is heat energy, conducted either into the gas from its surroundings or out of the gas into the surroundings, by invisible microscopic collisions. If Q is the heat energy entering the gas from its surroundings (which would be negative if heat actually flowed the other way) and W is the work done by the gas (thus being energy leaving the gas), and energy is indeed conserved, then the change in the gas’ internal energy, designated ΔU, is a matter of simple arithmetic. Complete the following equation.

ΔU =


64

THERMODYNAMIC BEHAVIORS


Circle the name of the person to whose report any joint printouts are attached. References University Physics Sections 18.3, 18.5 and 19.1-19.2 Apparatus Pasco box, computer, plastic tumbler with ice-water, hotpot, 200g weight, Pasco heat engine with

light string, rotary motion sensor and 20g counterweight, low pressure sensor. Printouts Pressure vs Height


65 Introduction Today we begin a new area of study—thermodynamics. It’s one of the major branches of physics,

with many related topics, but today we focus on just one important facet: the ideal gas. In Activities 1 and 2 you’ll work with a collection of virtual atoms on the computer. There are

good reasons for this. As we study thermodynamics, never forget that a thermodynamic system is one with countless particles, and while they of course obey Newton’s laws, when there are so many of them, something “new” comes into play and assumes great importance: probabilities. We admit that we cannot know what every particle is doing, but certain behaviors of the whole system, such as average pressure, are still quite predictable simply because it is highly probable that a large number of particles should behave in certain average ways. It’s hard to grasp these things when we can’t actually see them, so computer animation comes in very handy. And with a computer we can also “see” the true origin of the work done by a thermodynamic system such as a gas. It’s just a simple energy transfer from microscopic particles to a macroscopic object, but, crude as it may seem, it’s the backbone of energy production in our modern technological world. It happens in a car engine and in both fossil-fuel and nuclear-powered electrical generating plants, where the mechanical energy is thereafter turned into electrical. (Hydroelectric plants get their mechanical energy direct from falling water, not from thermal-to-mechanical conversion.) Given its importance, it would be nice if everyone—not just university students!—had some acquaintance with the basic idea,. In Activity 3 today, you’ll see it in action, after which you’ll be prepared to spread your knowledge to all!

But first let’s look briefly at a particular ideal gas characteristic. One of the most important predictions of statistical mechanics—combining probability&statistics with mechanics—is that in a large system of particles colliding randomly, the velocities should be distributed according to a Maxwell speed distribution. A plot of the distribution, number of particles versus speed, is shown in Figure 1, and the formula is:

Number of Particles ! v2e"(m/ 2k

BT ) v2

(1) where m is the particle mass, T the temperature, kB the Boltzmann constant and v of course the

particle speed. The claim is that the speeds will ultimately end up distributed this way no matter how improbably they may have begun. Is this true?

ACTIVITY 1: WHY MAXWELL?

Procedure 1. From the Applications folder in Physics Programs, open “Atoms in Motion”. Then from the “File” menu choose “Open Simulation…” and navigate to the file “Maxwell Rules” in the Physics Programs/9B folder. Here you have 100 atoms confined in a box, 99 green ones with velocities along the z-axis of 500m/s, a speed typical of an atom in a gas, and a stationary red atom, identical except in color and velocity. The speed histogram agrees (the axes are the same as in Figure 1). This velocity distribution is, to say the least,

Figure 1

Figure 2


66 neither very random nor thus typical. Without some change it would persist forever—we’ll see what a change does! But note that this program never forces particles to behave in any average way; it applies only the laws of mechanics, collision by individual collision. Any sort of average behavior that arises is natural, simply a matter of what is most probable. Now, while it isn’t obvious, the red atom is already “selected” for possible alteration. From the Edit Atom menu choose “Edit Selected Atom(s)…” Don’t change the position (x, y or z), but do enter a new velocity. About the only restriction is that vx or vy or both should be between 20m/s (too poky) and 900m/s (unusually high). Note that the x-axis is toward you. Hit Accept and watch carefully!

2. After watching for a while—it should be clear when the “final state” is reached—describe what you saw and why it happened. Specifically address the role of the red atom: Do you think that it had to be moving exactly as it was for the system to end up as it did? To be sure, you might wish to consult other “research teams” in the vicinity.

3. From the View menu choose “Displays…” In the Y-Maximum box type 10 then hit Accept. This makes the histogram more useful in the system’s new state.

4. From the Simulation menu choose “Timing…” In the “Iteration per Display” box replace 5 with 50. This speeds things up, and while a bit jerky, you can still follow the red atom pretty well.

5. Briefly describe what you see in the histogram: About how large and how small can a particle’s speed be, and is the speed distribution steady, or fluctuating?

6. Hit the 9 key on your keyboard, which displays a list of values in the upper right. Now hit the T key till you see ACCUMULATED at the bottom of this list. This averages all data from the time you hit the key onward. The animation is still real time. Wait till at least 1.2e-09 secs before answering: How valid a prediction does the Maxwell Speed Distribution seem to be?


67

ACTIVITY 2: WORK BY A GAS?

Procedure 1. From within “Atoms in Motion”, open “Work by a Gas”. Here you have 200 atoms at 0ºC, but confined now by a piston maintaining about two atmospheres of pressure. Note that it’s been averaging for quite a while. Pressure fluctuations aren’t noticeable in air, because the number of collisions with any surface in any reasonable time interval is huge. But as numbers diminish, fluctuations become more apparent. Because computing speed limits the number of particles we can consider, we deal with the resulting sizable fluctuations by letting the computer take time averages. Record the temperature, pressure, volume, and kinetic energy. (Note: Our gas has no potential energy, so the kinetic energy KE is the total energy TE.)

T:____________ P:____________ V:____________ KE:____________

2. Hit the T key to stop the averaging, then from the Simulation menu choose “Floor Interaction…” In the “Floor Temperature Used” box, change the temperature from its previous 273 to a new value of 373 (kelvin), then hit Accept. The change takes effect slowly, so while waiting, watch the temperature readout. Once any fluctuation takes it over 390K—maybe 4 or 5 minutes—you’ve probably waited long enough, so hit the T key again till the ACCUMULATED reappears. This restarts the averaging. Wait at least till 2.5e-09 secs, then again record temperature, pressure, volume, and kinetic energy. While you’re waiting, you can move on to steps 3-6.

T:____________ P:____________ V:____________ KE:____________

3. While it’s averaging, how is the speed histogram changing, or not changing? (It may help to take note of its appearance periodically while you’re waiting.)

4. Describe in words or an equation how you can calculate work from the quantities you record above. (Remember, the pressure, fluctuations aside, is kept fairly constant by the piston.)


68

5. With your cybergas you’re able do something that confounded great minds of science for many, many years: “see” the internal energy of a system of countless little particles. Historically, determining the change in this “hidden” form of energy was assumed, on the basis of energy conservation, to be simply a matter of subtracting how much energy went out from how much went in. But you know the internal energy and are able to calculate one of only two ways energy can go in or out: work. Describe below how you can determine from your data how much energy goes in or out by the other way: the heat conducted into the gas from the “warm” floor below.

6. Summon your lab instructor and explain to him/her your answers in steps 4 and 5. Should you wish to amend either answer, use the space below.

7. Assuming you’ve waited for an accumulated time of at least 2.5e-09 secs, enter your final numerical values in the spaces provided in step 2.

8. Calculate the work done by your cybergas. (If the initial and final pressures aren’t exactly equal, what should you do?)

9. Determine the heat energy that flowed into the gas from the floor.


69

10. Calculate the ratio of heat energy that flowed into your gas to the work that it did.

11. As you’ll learn in weeks to come, for a simple gas like yours at constant pressure, this ratio should theoretically be a fairly round number (okay, maybe not an integer). Overlooking inaccuracies due to fairly large fluctuations in your small sample of cybergas, can you guess from your data what this ratio is supposed to be? Consult other “research teams” in the vicinity.

ACTIVITY 3: HEAT AS AN ENERGY SOURCE Introduction Though the thermodynamic laws you study in coming weeks place

limitations on the efficiency of the conversion, in essence, a heat engine converts disordered thermal energy into useful mechanical work. Your apparatus confines air to a cylinder with a moveable piston, and monitors pressure and piston height. Your heat engine could be used to perform the humble task shown at right, which may not look impressive, but represents the whole idea behind heat engines: You wish to do mechanical work—lifting something, for instance—and you don’t wish to expend your own energy to do it.

Procedure 1. Warning: In this activity you have a hotpot that boils water. Be careful! Pay attention to its placement, so that you don’t topple it. Moreover, don’t plug it in if the water is above the designated maximum fill line. Make sure there is water in it, then plug it in. Also, make sure you have ice-water in your plastic tumbler, and that the string attached to the piston goes around the middle pulley (not the small or large one) of the rotary motion sensor so that the pulley rotates clockwise when the piston rises—otherwise your plot will go the wrong way!

2. While holding the cylinder-piston assembly firmly in one hand, gently twist and loosen one of its hose fittings so that your lab partner can lift the piston to about the middle of the cylinder, then reconnect the hose. The piston should stay.

3. Open the file “Heat Engine”. As soon as water is boiling in the hotpot, you’re ready. To begin, place the metal can in the ice-water and let it cool down for thirty seconds or so.

4. Carry out your data run: (1) start recording, place the 200g weight on the platform (atop the piston shaft), wait a second or two, then (2) move the metal can to the hot water, then, as soon as the platform appears to have stopped moving, (3) remove the weight from the platform, wait a second or two then (4) return the can to the ice-water. When the platform again appears stationary, stop recording.

5. If your data isn’t more or less four-sided, with reasonably well-defined corners (some jaggies are inevitable), try again. When you’re satisfied, make a printout, then unplug the hotpot!


70

6. Which of the four “legs” of your data is most like the expansion you saw modeled in Activity 2? Briefly explain.

7. In Activity 2 you saw that the work a gas does in expanding against a piston is the pressure P times the change in volume. For a very small volume change, dV, this would be PdV, which in general you’d have to integrate to find a total work as the volume changes by a finite amount, i.e., the work is the integral of PdV. Your printouts show P as a function of V (if you imagine multiplying the Height by the piston area, that is). Calling upon your math knowledge, describe what on such a plot would represent the work done along a given “leg” of the multistep process.

8. We’re not so much interested in the work for each leg as we are in the total/net work done by the gas over the entire cycle. Yes it does positive work expanding, but it also does negative work—the atmosphere does work on it to compress it back to a smaller volume while it’s in the cold water. Rather than summing the work including sign for each leg separately, introducing error on each leg, there’s a much quicker way of deducing the net work. Discuss this with your lab partners then carry out your method. Show your work, briefly describing your method, and record the net work in joules below. (The piston’s area is 8.3×10-4m2.)

9. You’ve used your data to find the net work. As a rough check, about how far would such an amount of work be able to lift a 200g weight? Is your answer sensible?

10. How much work did the gas do during the expansion just after you stuck the can in the hot water? (You already found net work—this is just one leg.) Briefly explain your calculation.


71

11. Summon your lab instructor and explain to him/her your work in steps 8-10.

12. Not surprisingly, nearly all the heat energy that flows in does so during the leg mentioned in step 10. During this expansion, the ratio heat/work is, as in Activity 2, theoretically predicted to be a certain value. It depends on the kind of molecule. In Activity 2 they were monatomic. Here it’s diatomic air, and the ratio is different: 3.5. How much heat flowed in during this leg.

13. The efficiency of a heat engine is defined, sensibly, as the net work done divided by the heat that is absorbed. If all the heat absorbed were turned into useful mechanical work, the efficiency would be one, or 100%; if it absorbed heat and did no work, the efficiency would be zero. Calculate the efficiency of your heat engine. (Don’t forget, it’s the net work.)

14. Your gas ends in the same state as it started—same P, T, V, etc., so its internal energy is also he same. On balance, as much heat energy must have gone into the gas from the hot water as came out in the form of net work and heat going out into the cold water. Deduce from your answers in step 8 and 12 how much heat was “rejected” into the cold water. (We’re not being very careful here with heat exchange during the two roughly “vertical” legs, but it’s a small error.)

15. Would you say that your heat engine is very efficient? Can you speculate as to how its efficiency might be improved? (You’ll learn more about this later in the quarter.)

16. A final interesting question: Would you characterize your heat engine as an internal combustion engine, or an external combustion engine? Briefly explain.


72

PRELAB: LOW TEMPERATURE EXPERIMENTS (TURN IN YOUR ANSWERS TO YOUR LAB INSTRUCTOR AS YOU ENTER THE LAB ROOM)

Question 1: In words, what is the meaning of “heat of vaporization”? Question 2: In the Low Temperature Experiments lab, why do we use a heater, what follows directly from knowledge of its voltage and current, and why isn’t it on during the entire data run? Question 3: If a solid behaved according to classical physics (not quantum mechanics), how much heat Q in joules would be required to raise the temperature of 1mole of the solid from 77K to 295K?


73

LOW TEMPERATURE EXPERIMENTS


Circle the name of the person to whose report any joint printouts are attached. References University Physics Sections 17.6 and 18.4. Apparatus Pasco box; dewar (3 styrofoam cups) with copper-wire sling; force sensor; lead and aluminum

samples with strings for hanging, triple-beam balance, ring stand and clamp; DC power supply, heater probe, two digital multitesters; liquid nitrogen (LN)

Printouts Mass Suspended vs. Time - Activity 1 Mass Suspended vs. Time - Lead Mass Suspended vs. Time - Aluminum


74 Introduction The central player in today’s lab is liquid nitrogen (LN). LN is used routinely now in science,

medicine, and industry as a cheap, readily available source of “cold”. At atmospheric pressure the boiling point of LN is 77 K (–196°C). It is used to cool other materials (biological samples, live tissue in cryosurgery, etc.) by absorbing heat. This heat then vaporizes some of the LN. We’ll measure its crucial heat of vaporization, then use it! However, before proceeding any further, however, several warnings about LN are necessary:

1. Avoid contact. Always handle LN carefully. Its extreme cold can produce effects on your skin similar to a burn. When spilled on a surface it tends to cover it completely and intimately and therefore cool a large area. The gases issuing from LN are also extremely cold and can produce burns. Delicate tissues, such as those of your eyes, can be damaged by an exposure to these gases which is too brief to affect the skin of your hands or face. (Note: The cloudy vapor often appearing around it is condensed moisture; not the gas itself. The issuing gas is invisible.) 2. Stand clear of boiling and splashing LN and its issuing gas. Boiling and splashing always occur when charging a warm container or when inserting objects into the liquid. Always perform these operations slowly to minimize boiling and splashing. Use water to wash off any area of your body that is accidentally splashed with liquid. 3. Never allow any unprotected part of your body to touch uninsulated pipes or vessels containing liquefied atmospheric gases; the extremely cold metal may stick fast and tear flesh when you try to pull loose. Use tongs to withdraw objects immersed in liquid, and handle the tongs and the object carefully. In addition to the hazard of burns or skin sticking to cold materials, objects that are soft and pliable at room temperature usually become very hard and brittle at the temperatures of LN and are very easily broken. Be very careful with objects that you’ve been instructed to put in, and don’t put in any others—don’t play with the LN! 4. Do not expose liquid nitrogen to the air for long periods of time. Liquid nitrogen is colder than oxygen’s condensation temperature. Therefore, if liquid nitrogen is exposed to the air, oxygen from the air may condense into it. If this is allowed to continue for any length of time, the oxygen content of the liquid nitrogen may become appreciable and the liquid will require the extra precautions needed when handling liquid oxygen. Heat of Vaporization Heat of vaporization Lv is the amount of heat that must be supplied to a liquid to vaporize some

standard amount of it. Thus, the units of Lv are an energy divided by a mass, such as J/kg. One of the easiest ways to supply a known amount of heat to a liquid is to convert electrical energy to heat energy in a “resistor” immersed in the liquid. The energy input per time, the power, is then measured electrically. From basic definitions of the amp and volt we have

Electrical Power = energytime

= IV [J/s] (1)

where V is the voltage, in volts, across the resistor and I is the current, in amps, flowing through it, giving a power in joules per second, J/s. We monitor the mass of liquid vaporized by hanging the dewar-plus-liquid from a force sensor.

If no other heat entered the LN, the determination of Lv would be very simple: Merely subtract the final mass from the initial and divide the known heat input by this mass difference—J per kg.

Lv =electrical energy added

mass loss due to electrical energy (2)

Given that we read electrical power and can easily monitor the mass loss rate, dm/dt in kg/s, due to this electrical input, it is equivalent and more convenient to express this as:

Lv =(electrical power) ! time

(dm / dt)elec ! time= IV

(dm / dt)elec

(3)

Unavoidably, however, there is continuous heat conduction into the LN through the dewar walls, the resistor leads, and at the LN-air interface. So, while our electrical heater is on,

(dm /dt)heater on = (dm /dt)elec + (dm/ dt)leak (4)


75 Accordingly, equation (3) becomes

Lv =IV

(dm / dt)heater on ! (dm / dt)leak

(5)

Heat Capacity of a Solid So far as heat storage in a solid goes, the textbook “classical expectation” is as follows. Each atom

has six degrees of freedom, i.e., independent coordinates that can possess energy: a velocity component and the displacement of an electrostatic “spring” in each dimension. Its energy is thus Eper atom = 1

2 mvx2+ 1

2 kxx2+ 1

2 mvy2+ 1

2 kyy2+ 1

2 mvz2+ 1

2 kzz2 . In essence, each atom is a harmonic

oscillator oscillating in three dimensions. Now the famous Equipartition Theorem says that an atom will have 12 kBT of energy on average for each degree of freedom. Therefore, the total energy of N atoms would be E

N atoms= N 6 ! 1

2 kBT .

EN atoms

= 3NkBT = 3nRT (6)

(Note: The constant R is defined as Nav kB. And since the number of particles N is the number of moles n times Avogadro’s number, N kB = n Nav kB = n R.) If the solid does no significant work by expansion, then to raise its temperature by ΔT would require a heat of Q = n (3R) ΔT. Thus, the molar heat capacity C, in Joules per mole per degree Kelvin, should be 3R for all solids.

Q = n C ΔT where C = 3R [Classical] (7)

Today we’ll test the “classical expectation”, by measuring the heat exchanged by solids as we vary their temperature from room temperature, about 295K, to the boiling point of LN, 77K.

But quantum mechanics says that at very low temperature, springs don’t “behave classically”. They are allowed only certain quantized energies. In fact, if it is very cold, the temperature may rise, but because the jump from the spring’s lowest allowed energy to the next is so big, the material absorbs essentially no energy at all. If the energy change per unit temperature is zero, the heat capacity is zero! Yet even quantum mechanics predicts that solids should still all behave the same way, just not the classical way. Their molar heat capacities should be 3R at high temperature, and should fall to zero at low T according to the same functional dependence. Figure 1 shows lead (Pb) and aluminum (Al). (The grid covers the range of temperatures today’s lab. In fact, it breaks that area into 100 boxes, very helpful for judging percentages—keep this is mind!) The curves do have the same shape. It’s just that the one for aluminum is stretched out more, because—so the claim goes—its “springs” are stiffer, causing quantum mechanical behavior to prevail at much higher temperatures. Can we see evidence of quantum mechanical behavior?

Figure 1


76

ACTIVITY 1: HEAT OF VAPORIZATION

Procedure 1. Make sure your power supply is OFF. Then check to see that the wiring of your setup is as shown in Figure 2. One multitester, measuring V, should have wires in its VΩmA and COM ports connected, respectively, to the positive (red) and negative (black) terminals of the power supply. Set this multitester on the 200 setting in the DCV sector of its rotary dial. The other multitester, measuring I, should be set on the 10A setting on its dial. One wire from the heater probe should go to this multitester’s 10ADC port, and a wire should go from its COM port to the negative terminal of the power supply. The heater probe’s other wire shouldn’t be connected to anything yet. Soon you’ll use this free wire as a switch, inserting and removing it from the power supply positive terminal. But this wire should never be plugged into the power supply until the probe is submerged in LN! If you’re uncertain about the wiring, ask you lab instructor. Otherwise, after making sure the heater probe’s free wire is indeed not connected to anything, turn on the power supply and adjust its knob until the multitester measuring V reads 25 volts.

2. To determine the mass loss rate due to heat leakage, (dm/dt)leak, you will in step 3 start monitoring mass with the probe’s resistor element immersed in LN but not yet plugged into the power supply, and because the mass-loss rate may not be the same after running the heater as it was before, you will continue monitoring after unplugging the heater. Thus, your data run in step 3 will consist of (I) recording for 5 or 6 minutes with the heater immersed but unplugged, then (II) while continuing to monitor mass, plugging the heater’s free wire into the power supply positive terminal and letting it supply heat for 5 or 6 minutes, then (III) unplugging the heater (leaving it immersed) and continuing to monitor for 5 or 6 minutes, before finally stopping the recording. Overall, then, your data run should take 15 to 18 minutes. During this whole time, take care not to bump the table too much, create unnecessary air currents, jump up and down, or do anything else that might increase the noise the force sensor “sees”. Open the file “Mass vs. time”, then tare the force sensor.

4. Now have your lab instructor fill your dewar roughly two-thirds full, then position the heater probe in the dewar, with its resistor element near the bottom but touching neither the bottom nor sides, wait perhaps a minute for initial cooling to stabilize, then start your data run (i.e., 5 or 6 minutes without heating, 5 or 6 with, 5 or 6 without). Obviously, the data taking now won’t require your full attention, so you should start in on step 1 of Activity 2, and perhaps step 2. Still, keep an eye on the LN level; if it drops to the point of beginning to uncover the probe’s resistor element, go ahead and pull the probe wire from the power supply positive (i.e., end the part designated as II in step 2—it’ll probably still be long enough to give reasonable data). Also, don’t forget during the “middle” portion of your data, run while the heater is actually plugged in, to note the voltage V and the current I from the multitesters. Record them below.

Voltage, V (in volts, V) Current, I (in amps, A)

Figure 2


77

4. Use the scale-to-fit button, then choosing “Linear” via the button

determine the three mass-loss rates—before the heater was on, while on, and after the heater was on—and record them in the table. (Note: It’s best to avoid “elbows”, connecting one region to the next, where the slope is obviously changing considerably.)

(dm/dt)leak before (dm/dt)heater on (dm/dt)leak after

5. Make a printout of your data. As soon as you have it, label it “Activity 1”.

6. Calculate and record below (dm/dt)leak, the average of your “before” and “after” mass loss rates due to heat leakage. Attach appropriate units!

(dm/dt)leak:____________________ 7. Calculate and record below, with appropriate units, your experimental value for Lv.

8. Summon your lab instructor. Show him/her your printout and reiterate concisely what it means and how you used it to find Lv. Afterward obtain from your lab instructor the actual value of Lv. From this calculate a percent error.

9. Your value should be fairly close. Nevertheless, the procedure is not perfect and there are accordingly numerous places where error could possibly creep in. Discuss how your experimental value compares with the actual value in light of these possibilities.


78

ACTIVITY 2: HEAT CAPACITIES OF SOLIDS Introduction Now we’re ready to use what we know about liquid nitrogen to explore heat capacities of solids, and

possible deviations from “classical behavior”.

Procedure 1. First, remove any ice and dry off the outside of your dewar. Make sure your lead and aluminum samples are also dry, then weigh your samples and record their masses below.

mlead:________________kg malum:________________kg

2. The molar mass of lead is 0.207kg and of aluminum is 0.027kg. Assuming that your samples behave according to classical physics (not quantum mechanics), how much heat Q in joules would have to be removed from them to lower their temperatures from 295K to 77K? (In SI units the ideal gas constant R is 8.315J/mol-K.)

Qlead,classical: _____________J Qalum,classical: _____________J

3. To test these predictions, you’ll soon lower the room-temperature samples into the LN—one at a time, of course—and wait for them to cool to 77K, meanwhile boiling off LN. Knowing how much LN boils off and the heat required per kilogram to do so, you can determine the heat transferred into the LN from each sample. However, there is as always heat leakage to take into account. Thus, you’ll (1) initially have the dewar and both room-temperature samples hanging from the hook on the force sensor, timing for five or six minutes, then (2) gently lower the lead sample by its string into the LN till it rests on the bottom, then after the rather violent bubbling stops, (3) wait another five or six minutes, to obtain an intermediate mass-loss rate, then (4) lower the aluminum sample to the bottom, wait for it to finish cooling (usually with an exciting final flourish!), then (5) wait another five or six minutes to establish a final mass-loss rate.. We’ll use nearly the same setup as in Activity 1, but without the heater, so move it out of the way now.


79

4. Your dewar LN level is rather critical here. Each cup has a ~1 inch embossed outer border at the top. The LN should be no higher than the bottom of the bottom cup’s border (i.e., about 3 inches down from the top rim) or boiling LN might slosh out. But it shouldn’t be much lower either, or there won’t be enough LN to cool both samples. Have your lab instructor adjust the level as needed. Then, before placing the dewar in its sling, tare your force sensor.

5. Hang your samples, then carry out your data run, as laid out step 3. (Note: Minor jostling of the assembly when you lower each sample is not really a problem, as soon you’ll see, and the plot during the samples’ violent cooling isn’t terribly accurate anyway. Nor does partial exposure of the samples at the end present a problem; they’ll still be 77K.)

6. Save your file with a distinctive name on the Desktop. Your data should resemble Figure 3. Using the magnifier , select the “Lead cooling data” and make a printout, labeling it “Lead”. Afterward, zoom in on the “Aluminum cooling data”, print your plot and label it “Aluminum”.

7. Return to your lead cooling data. Given equal before and after leakage rates, these parts of the data would be parallel lines, the latter displaced downward by the extra LN boiled to cool the lead. To account for their inequality, choose Linear via the Fit button, select the data for the leakage before you lowered in the lead, then with the tool find the “height” (in kg) of this best-fit line at a point in time about halfway between the lead’s entry into the LN, where the before leakage ends, and the end of its cooling, when the after leakage begins; then select the after leakage data and find the “height” of its best-fit line at the same point in time (see Sample Data on next page). To ensure the same time, drag only the horizontal axis of the x-y crosshairs from their previous spot. (Different times would include leakage in the interim—we don’t want that!) Record the difference between the two “heights”:

ΔmLN (to cool lead):________________kg

8. Repeat step 7 for your aluminum data.

ΔmLN (to cool alum):________________kg

9. From your steps 7 and 8 data, determine how much heat energy in joules transferred to the LN from each sample as it cooled. Use the actual value, not your experimental one, for Lv.

Qlead,experimental: _____________J Qalum,experimental: _____________J

10. Summon your lab instructor, and show him/her how your values result from your data.

Figure 3


80


81

11. How do your results in step 9 compare to the predictions of step 2? Assuming for the time being that the theoretical predictions of step 2 are correct values, calculate the percent “error” in your experimental values.

12. Figure 1 says that for both lead and aluminum the actual heat required to change the temperature to 77K from 295K is less than 100% of the classical value, n (3R) ΔT. What percentage should it be? Briefly explain your procedure. (Reread the paragraph just above Figure 1 if necessary.)

% of classicallead: _____________ % of classicalalum: _____________

13. Instead of your values in step 2, what, then, should the true theoretical values be?

Qlead,theo: _____________J Qalum,theo: _____________J

14. Calculate new percent errors in your experimental values of step 9.

15. Do you think your experiment is really sensitive enough to say whether the samples are behaving classically or not? Briefly explain (Think! If the typical percent errors you’re seeing are smaller than the difference between classical and quantum-mechanical expectations, you would probably say yes. If larger, no.) Also, to minimize clutter, please delete your file from the Desktop.

A Cyclic Process

82

PRELAB: A CYCLIC PROCESS (TURN IN YOUR ANSWERS TO YOUR LAB INSTRUCTOR AS YOU ENTER THE LAB ROOM)

Question 1: How much heat flows into or out of a system in an “adiabatic” process? Briefly explain. Question 2: Theoretically, what should be the value of ! for (a) a monatomic gas, and (b) a diatomic gas “without vibrations”?

Question 3: Equation (1) in the Cyclic Process lab implies that P V!= P

iVi

! , or P =PiVi

!

V! . Use this relationship to

obtain equation (4) from equation (3).

A Cyclic Process

83

A CYCLIC PROCESS


Circle the name of the person to whose report any joint printouts are attached. References University Physics Sections 19.3-19.8, 20.2 and 20.4. Apparatus Pasco box; pressure sensor; two liter bottle with brass pressure fitting, tubing, clamp, crusher arm Printouts Pressure vs Time

A Cyclic Process

84 Introduction Today you’ll experiment with complete cycle of processes in a nearly ideal gas, by examining what

happens when you rapidly crush, then release, a gas-filled 2 liter plastic bottle. The bottle is connected to an pressure sensor, giving a plot of pressure versus time. The resulting four-step cycle, representative of a real machine that puts thermodynamics to use, you’ll then analyze as you “translate” it to a P–V diagram. Along the way, you’ll also deduce an important property of your relatively ideal gas.

A CYCLIC PROCESS

Procedure 1. Open the file “2 Liter Bottle”. Make sure that the bottle is not compressed in any way (such as the crusher arm resting on it), then record for ten or fifteen seconds. The pressure should be fairly stable in the range 135-145kPa. If it isn’t, consult your lab instructor.

2. You’ll carry out your data run as follows: By (i) starting the recording and waiting for a nice, level pressure line to appear on the screen, then (ii) quickly crushing the bottle with the crusher arm and keeping it crushed until the pressure displayed on the computer levels off (this will take a bit of exertion—and be careful about your knuckles!), then (iii) quickly releasing the crusher arm and swinging it away from the bottle, then (iv) waiting for the pressure to again level off before stopping the recording. When you’ve got this sequence down, carry out your data run.

3. Your data should consist of a flat line, then an abrupt rise to a maximum, followed by a gradual but consistent drop to another flat line, then an abrupt drop to a minimum, followed by a gradual,

consistent rise to a final flat line. If not, repeat your data run. Otherwise, use the scale-to-fit

button, then make a printout. You’ll work from your printout alone.

4. Before getting into detailed analysis, let’s make a guess! Estimate how much work in joules you (or your lab partner) did to compress the 2 liter bottle. A simple force-times-distance argument is all you need. Show your work below.

5. Once you have your printout, label on the curve with the corresponding capital letter the following important “events”:

A The steady-pressure point just before the pressure shot upward upon crushing B The maximum pressure caused by the crushing/compression C The steady-pressure point just before the pressure shot downward upon releasing D The minimum pressure caused by the releasing/expansion E The final steady pressure

A Cyclic Process

85

6. If leakage is negligible, you should find that your final pressure PE is essentially equal to the initial PA. On the basis of a well-known “law”, why should these two pressures be equal?

7. Assuming points A and E are indeed the same state, you can see that you’ve carried out a cyclic process, in four “legs”: AB, BC, CD, and DA. The standard kinds of processes we consider in thermodynamics are: Isothermal (ΔT = 0), isobaric (ΔP = 0), isochoric (ΔV = 0) and adiabatic (Q = 0). For each of your four legs, indicate which kind of process it is, and explain briefly why.

AB:

BC:

CD:

DA:

8. During which leg(s) is(are) heat flowing in or out, and how do you know?

9. At which of the points A, B, C, D is the gas at room temperature? Briefly explain how you know.

A Cyclic Process

86

10. There should be a thermometer either mounted to the wall or on a lab table. Determine the temperature of the lab room. Enter it in Kelvin in the appropriate place(s) in the table below. Note that a total volume of 2.1L for the bottle has already been entered.

Point Pressure (kPa) Volume (L) Temperature (K)

A 2.1L

B

C

D

E 2.1L

11. Now fill in the pressures at the various points from the data in your pressure vs. time plot.

12. Next, fill in the volumes. All you’ll need are PV = NkBT, ratios, and values you already know (you won’t need to find the number of particles/moles)!

13. Finally, find and enter in the table the yet undetermined temperatures. Note that something is held constant in leg BC and leg DE, so ratios are again the fastest way to go.

14. On the P vs. V axes of Figure 1, choose logical limits and increments, label the divisions, then add and label the points, A, B, C, and D, and indicate the temperature beside each. Choose your limits carefully, so your plot fills most of the area provided. In particular, you should not plot all the way down to zero pressure or volume. Note: Each student make his or her own P-V diagram.

15. Now sketch in lines or smooth curves for the various legs (adiabats aren’t straight lines, but hyperbolic curves—see your text), completing the 4-leg P-V diagram for this cyclic process. Add arrows on each leg, showing the direction of each of the four individual processes.

16. Finally add to your plot a plausible room-temperature isothermal curve, bearing in mind which of the four points it should pass through.

17. On a P-V diagram, adiabats are steeper than isotherms. How well does your diagram agree?

18. Summon your lab instructor and show him/her your P-V diagram. Explain which legs are which of the standard processes (noted in step 7), then explain why you would expect an adiabatic compression to be steeper than an isothermal one starting at the same point.

A Cyclic Process

87

Figure 1

A Cyclic Process

88 19. For an adiabatic process,

P V!= constant [adiabatic only] (1)

which means that for any two points, i and f, on the same adiabatic curve,

Pi Vi!= Pf Vf

! [adiabatic only] (2)

You have two adiabatic curves on your P-V diagram. Using your data and the above relationship, calculate ! for the air in your 2 liter bottle for each adiabat, then average the two. (Note: Taking natural logs of both sides is good idea!) Show your work below.

! average:__________________

20. What would you expect ! to be theoretically and how well does your experimental value agree?

A Cyclic Process

89

21. The work done by a gas may always be calculated as follows:

W = P(V ) dVVi

Vf

! (3)

In an adiabatic process, where Equation (1) applies, the result of this integration is:

W =Pi Vi

! "11 "

Vi

Vf

#

$

% &

'

(

! "1)

*

+

,

-

. (4)

Use this to calculate the work, in joules, done by the gas when the bottle expands on leg CD. (Note: To find work in joules, kPa must be converted to Pa and liters to cubic meters.)

22. Now calculate the work, in joules, done by the gas as the bottle was compressed on leg AB.

23. Comment on the relationship between your answer in step 22 and your guess in step 4. Should they be equal, or opposite? Are they close? Why or why not?

A Cyclic Process

90

24. Based on your data and calculations (not your guess in step 4), what is the total work in joules done by the gas in this 4-leg cyclic process?

25. Would this cyclic process better qualify as a heat engine, or as a refrigerator? Briefly explain. Also discuss how external reservoirs would have to be brought into contact with the bottle, and when, to put such a “machine” into use. (Note: It’s true that it would be a pretty anemic “machine” in practice, but we’re interested in the idea just in theory.)

the staff of the physics department university of...

Documents