Table of Contents Introductory Material 1 Lab Policies and General Information ................................................................................ 3 2 Measurement and Uncertainty ............................................................................................ 4 3 Uncertainty Propagation ..................................................................................................... 9 4 Simplifying Complicated Uncertainty Problems .............................................................. 11 5 Final Calculated Results and Significant Figures ............................................................. 13 6 Comparing Experimental Results: “Accuracy” ................................................................ 15 7 Understanding “Precision”................................................................................................ 17 8 Averaged Values and Their Uncertainties ........................................................................ 19 9 Elements of a Report......................................................................................................... 23 10 Example Notebook............................................................................................................ 28 11 Example Typed Report ..................................................................................................... 30 12 Data Analysis .................................................................................................................... 31 13 Helpful Strategies in the Lab ............................................................................................ 31 14 Practice Lab ...................................................................................................................... 32 15 Graphing ........................................................................................................................... 34 Experiments 1 Uniform Linear Motion..................................................................................................... 46 2 Friction.............................................................................................................................. 49 3 The Ideal Spring................................................................................................................ 51 4 The Simple Pendulum....................................................................................................... 54 5 Collisions in One Dimension............................................................................................ 56 6 Collisions in Two Dimensions.......................................................................................... 60 7 Uniform Circular Motion.................................................................................................. 64 8 Projectile Motion .............................................................................................................. 68 9 Moment of Inertia ............................................................................................................. 72 10 The Physical Pendulum..................................................................................................... 76 11 Resonance of Air Columns ............................................................................................... 79 12 Canimals ........................................................................................................................... 84 Appendices 1 The Basics of Using Excel................................................................................................ 90 2 Inserting Symbols and Equations into Word Documents ................................................. 96 3 Graphing with Curvefitting.xls ......................................................................................... 97 4 Graphing with Linegraph.xls .......................................................................................... 100 5 The Photogate Sensor ..................................................................................................... 102 6 National Instruments' LabVIEW Interfacing System ..................................................... 103 7 Common Derivatives and Rules of Calculus .................................................................. 104

Credits This manual is the product of a collaboration of both staff and faculty, past and present, of Kwantlen University College’s Physics Department. The principal authors of the introductory material and appendices are Len Arboleda, Kim Callaghan, Mike Coombes, Jennifer DeBenedictis, Laura Flinn, Jillian Lang, and Lara Pow. Major contributions to the design of the experiments were made by Bob Chin. Significant contributions to the overall content were made by Carmen Ciubotariu, Jana Kolac, Alex Koveshnikov, Darryl Loewen, Don Mathewson, Jennifer McKay, Tom McMath, Dan Peirce, and Tak Sato, as well as other faculty and staff members both past and present. The Physics Department of Kwantlen University College also wishes to thank the many past students who have helped shape this manual with their valuable feedback and suggestions.

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1 Lab Policies and General Information A schedule is posted on the door of your lab room so you can prepare for the lab in advance. Since your report may be due at the end of the lab period, complete as much as possible before coming to the lab (e.g. your lab notebook can be completed up to the Data section). The lab reports and lab exam (which is worth the equivalent of two lab reports) count for 25% of your physics course grade. Lab notebooks/reports must be submitted in a timely manner. Your lab instructor will give you hand-in deadlines. Late reports will be penalised and generally any report not submitted before the next lab period will receive a zero mark.

1.1 Exemptions If you have taken this course before, and received a mark in the lab portion of the course of 7/10 or higher, you may be exempt from doing the labs again. You must speak to your lab instructor in person to arrange this. Your lab instructor will submit your previous lab mark.

1.2 Working with a Partner If there’s room in the lab, you may work alone or choose a partner. Feel free to change partners at any time; working with various people is a good experience. Both you and your partner must work on the entire report together: it is not acceptable to split up the work.

1.3 Late Arrival Policy Since the pre-lab discussion is usually very short, it’s important to be on time. If you’re late, you will not be allowed to join your partner if they’ve already started taking measurements. You will be required to work on your own, on a different set of equipment, and complete your own report. If no equipment is available, you will not be allowed to do the lab at that time.

1.4 Missed Lab Policy The labs are considered part of the course requirements. Any student who misses a lab will receive an F grade for the entire course! If you know in advance that you must miss a lab, discuss the situation with your lab instructor as soon as possible. In the case of illness or other unforeseen events, a make-up for the missed lab may be arranged. Speak to your lab instructor as soon as possible or, at the latest, on the day you return to school to schedule the make-up lab; do not wait until the next lab period. In the case of illness, a doctor’s note with phone number will be required. Third-party documentation will be required for unforeseen events also.

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2 Measurement and Uncertainty Measurement is at the heart of experimental science. By measuring, we investigate the universe and discover the rules that describe it. No measurement is infinitely precise. Regardless of how carefully a measurement is taken, there is always a limit to how accurate it is. Thus, to be completely specific about an item of data, you must note both your best estimate of the value of it and also its uncertainty. An uncertainty is a realistic estimate of how far “off” your value might be. Figure 1 will be used to illustrate this.

Scale A 6 7

Figure 1 – Pointers on scales of varying precision

6.00

7.0

Scale B

6.0 6.1 6.2 6.46.3 6.5 6.6 6.7 6.8 6.9

Scale C

7.00 6.1 6.2 6.46.3 6.5 6.6 6.7 6.8 6.9

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Scale A In Scale A, the pointer is located somewhere between 6 m and 7 m. We are uncertain about what the “true” value is, because we have reached the limits of the scale’s precision. One person might say the pointer’s location is 6.4 m, while another might say it is at 6.3 m or 6.5 m. Thus, a good uncertainty for this value would be ± 0.1 m, because that’s a reasonable estimate of how far “off” our value might be from the “true” value. Another way to think about it is that since the space between the divisions is relatively large, we can estimate the pointer’s position accurately to about 1/10th of the smallest division. Since 1/10th of a metre (or 0.1 m), is the limit of our accuracy, it’s also a good uncertainty for us to choose. We would write the pointer position as: 6.4 ± 0.1 m. Scale B On Scale B, there are more divisions. For this reason, we can get a more accurate reading. The best estimate of the pointer position in this case might be 6.38, where “8” is an estimated digit (another person might estimate that digit differently than you). Because the division spacing is still large, we can again use 1/10th of the smallest division as our uncertainty, or: 6.38 ± 0.01 m. Note that a person with better or poorer eyesight than average is justified in making their uncertainty smaller or larger than this value. Scale C On Scale C, our best estimate of the value might be 6.375 m, where the “5” is estimated. However, the division spacing on Scale C is so small that most people could no longer estimate their final digit accurately to 1/10th of the smallest division. In this case we might then use the more realistic uncertainty of 1/4th of the smallest division as our uncertainty, or 0.25 cm. The value would be written as: 6.3750 ± 0.0025 m.

2.1 Types of Uncertainty There are many types uncertainty, but the uncertainty you list beside a value should be large enough to account for all of them. In this section, we outline some common sources of uncertainty. At the end, we will discuss how to combine them to get the correct uncertainty for a single data point.

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2.1.1 Reading Uncertainty The source of uncertainty you will come across most often in the lab is called reading uncertainty. It is set by how accurately you can read a value off a scale. All the uncertainties discussed in the previous section, for Scales A, B and C of Figure 1, were reading uncertainties. Reading uncertainty is determined both by how precise the measuring device is, and how accurately you can read a measurement off it. That includes such factors as the pointer being shaky or thick, as well as the spacing of the scale increments and how good your eyesight is. In Scales A, B and C, the precision of the measuring device is set by how many divisions per unit length the scale has. You decided how accurately you could read measurements off the scale when you decided whether your uncertainty should be 1/10th or 1/4th of the smallest division. Ruler Measurements When you take a measurement of length with a ruler, keep in mind that you have two reading uncertainties involved. One of them comes from reading the length of the object, and the other comes from lining up the start of the object with zero on your ruler. Because of this, when measuring a length with a ruler that has small divisions (like millimetres), it is appropriate to make your uncertainty: 2 × (1/4th the smallest division) = 2 × (0.25 mm) = ± 0.5 mm

2.1.2 Physical Uncertainty Sometimes the quantity you are measuring has an intrinsic uncertainty that is larger than the reading uncertainty of the device you are using to measure it. For example, if you try to measure the length of a cotton ball, the “fuzziness” of the cotton ball is where most of your uncertainty comes from – not the precision of the ruler. Your ruler might be subdivided into millimetres, but because the cotton ball has a poorly defined edge (it’s “fuzzy”), your physical uncertainty might be ± 1 millimetre or larger. Another example would be trying to measure how far a cart rolls up a hill before it reverses direction. The fact that the cart only pauses for an instant at the place where you need to take your measurement can make your uncertainty very large. Again, your ruler might have millimetre divisions, but your physical uncertainty in reading an accurate value could be ± 1 centimetre or larger (depending on the experiment).

2.1.3 Instrument Uncertainty The instrument you are taking a measurement with can contain an in-built uncertainty that has nothing to do with your ability to read the device.

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For example, a device with a digital display may seem perfectly easy to read, but you don’t actually know what the uncertainty of that value is until you read the manufacturer’s specifications for the device. The instrument uncertainty can be much larger or much smaller than the smallest digit on the display. In Physics 1101/1120, whenever you use an instrument with a digital display, the uncertainty will be provided either in the lab manual or by your lab instructor. In most other cases, the instrument uncertainty is much smaller than the reading uncertainty and can be neglected. For example, the markings on rulers are usually printed very accurately, so you don’t need to worry that the manufacturer didn’t space the increments correctly. Accuracy and Digital Displays Please note that a device with a digital display is often not any more accurate than a device with a pointer on a scale. Digital displays are easy to read, but the device attached to them is not necessarily of any higher quality.

2.2 Random and Systematic Uncertainties All uncertainties fall into two broad categories, random and systematic. Random uncertainties are uncertainties where your estimate of the “true” value is just as likely to be too large as too small. Another way to think about it is that if many people took the same measurement, their values would tend to cluster around the “true” value in roughly equal numbers above and below it. Systematic uncertainties are uncertainties where your estimate of the “true” value is always skewed in one direction only. For example, if the wheels of your car are larger than the manufacturer assumed they would be, then the speed registered by your speedometer will always be smaller (but never larger) than your true speed. This is a systematic uncertainty. Below are listed some important considerations regarding random and systematic uncertainties.

2.2.1 When You Have a Large Source of Random Uncertainty When your measurement has a small source of random uncertainty, such a reading uncertainty, you may simply estimate the size of that uncertainty. However, sometimes you have a large source of random uncertainty. For example, wind speed might change a lot from moment to moment due to gusting, and it changes in an essentially random way. There is no variable in the experiment that you can control to make it stop doing so.

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The standard way to deal with a large random uncertainty is to take many measurements and average them. The averaged value is then used instead of an instantaneous value. Deciding what uncertainty to use on averaged values is a subtle matter, and is discussed in detail in Section 8, page 19.

2.2.2 Assumptions and Approximations When you make assumptions or approximations about your experiment, systematic uncertainties can occur. For example, if you calculate how far a metal ball travels horizontally after being shot out of a tube, but neglect the effects of friction and air resistance, then your calculated values will always be larger (never smaller) than your measured values. When you make assumptions or approximations in your calculations, be sure to make the uncertainty on your measured values large enough to account for the influence of the neglected factors.

2.2.3 Stopwatches Most stopwatches can record time accurately to 0.01 seconds. However, human reaction time is closer to 0.15 seconds. Because of this, a reasonable uncertainty for a measurement taken with a stopwatch is ~ 0.15 seconds. It is usually not necessary to double this value, to take into account the fact that you must both start and stop the stopwatch at the right moment. This is because reaction time is a systematic uncertainty; the act of you pressing the button usually happens after the true event, so both your starting and stopping times are skewed in the same direction. Thus, the total time measured is roughly correct. The uncertainty you record for a stopwatch is really chosen to take into account the fact that your reaction time can vary from one moment to the next (which is a random uncertainty.)

2.2.4 How to Determine Your Total Uncertainty Choosing the correct uncertainty on a single data point is a judgement call. You want to choose a value large enough to take into account ALL the sources of uncertainty, but small enough to accurately represent the precision of your measurement. If you have uncertainty from several different sources, then you should estimate what the uncertainty is for each individual source and add them together. For example, if you are measuring the length of a cotton ball, you might have ± 0.5 mm of reading uncertainty and ± 1.0 mm of physical uncertainty. The correct uncertainty to use would then be: ± 1.5 mm.

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3 Uncertainty Propagation When a calculation is made with measurements that have uncertainties, then the final result will also have an uncertainty, which must be calculated too.

3.1 Terminology For A ± δA: A is the value

δA is the absolute uncertainty of A

AAδ is the fractional uncertainty of A

%100×AAδ is the percent uncertainty of A

Note: Absolute uncertainties have units. Fractional and percent uncertainties do not have units (they cancel out).

Note Also: Always use absolute values for uncertainty calculations. Never round off values in calculations. Round off the final answer only.

3.2 Addition and Subtraction The absolute uncertainty of Y is the sum of the absolute uncertainties of A, B, C. If: then: CBAY −+= CBACBAY δδδδδ ++=−+= )(

3.3 Multiplication and Division The absolute uncertainty of Y is equal to Y times the sum of the fractional uncertainties.

If: CABY = then: ⎟

⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

CC

BB

AA

CAB

CABY δδδδδ

3.4 Constants Definition: A constant is a value with an uncertainty equal to zero. Treat the constant k like a normal variable, but set δk equal to zero in your final expression. Example:

If: then: AkY = ( ) ( ) AkAA

kkAkAY δδδδ =⎟

⎠⎞

⎜⎝⎛ +==

0

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3.5 Roots and Powers

If: kAY = then: ⎟⎠⎞

⎜⎝⎛==

AAkAAY kk δδδ )()(

3.6 Trigonometric Functions Important Note: When dealing with trigonometric functions, δθ must be in RADIANS! If: θsin=Y then: δθθθδδ )(cos)(sin ==Y If: θcos=Y then: δθθθδδ )(sin)(cos ==Y

If: θtan=Y then: θ

δθθδδ 2cos)(tan ==Y

3.7 Exponentials If: then: AeY = AeeY AA δδδ )()( ==

3.8 Natural Logs

If: then: )ln(AY = ( )AAAY δδδ == )ln(

3.9 Averaged Values Please see Section 8, page 19, for details on how to correctly determine the uncertainty of an averaged value. A summary of the rules is given here.

When taking an average of n measurements, n

AAAYY nave

)...( 21 +++== , the uncertainty on the

average value should be the larger of the following two quantities:

a) n

AAAY nδδδδ +++=

...21 , or

b) n

Y σδ = , where σ is the standard deviation of Aave: 1

)(1

2

−

−=

∑=

n

AAn

iavei

σ , which is easily

calculated in Excel using the function: STDEV(A1,A2,...,An)

3.10 Using Calculus for Uncertainty Calculations You are not required to use calculus to calculate your uncertainties in Physics 1101/1120. However, if you wish to do so, an outline for how to is given in Appendix 7, along with a summary of some common derivatives.

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4 Simplifying Complicated Uncertainty Problems When faced with a complicated calculation, it is sometimes difficult to see how to calculate its uncertainty. This section outlines how to approach such problems. The Problem: Imagine you are presented with working out the uncertainty δY for the following:

( )Fe

BADY += ,

D, A, B and F are all values that have uncertainties, i.e. δD, δA, δB, and δF. Note that this expression involves addition, multiplication, division, and a special function. Which uncertainty propagation rule should you apply first? How to Begin: The basic procedure is to choose small sections of the expression where only one uncertainty rule needs to be applied, and then simplify those sections. The sections are then substituted back into the original expression. Here, we simplify the expression for Y by defining two new variables, α and β. Note that only one uncertainty propagation rule will need to be applied to find the uncertainties δα and δβ.

FeLetBALet

=

+=

β

α )(

With these substitutions, we can rewrite the original expression in a simpler form:

( )βαD

eBADY F =

+=

Note that the uncertainty for this new form of the expression can also be found by applying just one uncertainty propagation rule to it – the rule for multiplication/division. The uncertainty for this simplified expression is:

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=

βδβ

αδαδ

βαδ

DDDY

Although we know that α = (A + B) and β = e F , we don’t yet know what δα and δβ are. The uncertainty calculations for α and β are done in the next two steps:

)(exp)/(

ruleonentialFetheneIfrulensubtractioadditionBAthenBAIf

FF δδββ

δδδαα

==

+=+=

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Now that we know the exact form of δα and δβ, we can substitute them back into the expression for δY. We’ll also substitute back in the original forms of α and β.

( ) ( )

( ) )()()(

)()(

formfinale

FeBABA

DDY

ee

BABA

DD

eBAD

DDDY

F

F

F

F

F

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++

+⎟⎠⎞

⎜⎝⎛ +

=

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=

δδδδ

δδδ

βδβ

αδαδ

βαδ

You will probably note some cancellations that could be made to the final expression, but mathematically, this is correct and complete. In Summary: When faced with a complicated problem in uncertainty propagation, follow the procedure outlined below:

a. Choose a small section of the problem for which you only need to perform one type of uncertainty analysis. Define a variable for that section and substitute the variable back into the original problem.

b. Keep doing this until you have simplified your original problem to the point where you

only need to apply one uncertainty propagation rule to solve it. Work out the solution to this simplified expression.

c. Find the uncertainties for all your defined variables.

d. Substitute the defined variables and their uncertainties back into the original equation

(nesting them one inside each other, as required) to get the final form for the original equation’s uncertainty.

As you become better at doing uncertainty calculations, you will probably start to notice “shortcuts” and patterns that help you work out answers more quickly. Feel free to use these! There is more than one way to get to the correct answer, so use whatever method makes the most sense to you.

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5 Final Calculated Results and Significant Figures References: An Introduction to Error Analysis, J.R. Taylor, University Science Books, Oxford

University Press, 1982, pp. 14 – 18. Data Reduction and Error Analysis for the Physical Sciences (2nd Ed.), P.R.

Bevington and D.K. Robinson, McGraw-Hill Inc., 1992, pp. 4 – 5.

5.1 Rounding Uncertainties After you perform a calculation, your calculator will often give you a long list of digits for both the value and its uncertainty. For example:

g = 9.85196 ± 0.04385 m/s2 (1) However, not all of these digits are meaningful! If the original uncertainties on your raw data were given to one digit, then where did all this extra precision come from? The answer is that the extra digits are artifacts of the calculation. They’re not “real”. For this reason, you must round off your uncertainties (and later, your values), so that you are not implying a precision for the experiment that you didn’t actually have. The following rule should be followed:

Rule #1 – Stating Uncertainties Round the calculated uncertainty to one significant figure.

Note that zeroes that only hold the place of the decimal point are not counted as significant figures. Applying Rule #1 to the result in (1), the uncertainty δg rounds to:

δg = 0.04 m/s2 (2) Common Exception: If the leading digit in a final calculated uncertainty δx is a 1, then it is better (although not always standard practice) to keep two sig. figs. in δx. For example, if your final calculation yielded δx = 0.1437, then to round this to δx = 0.1 would result in the loss of 43.7% of your information. This is quite high, so it is better to keep two sig. figs: δx = 0.14. The same argument applies if the leading digit is a 2, but not if it is any larger (work out the percent uncertainties to check this). The exception can be summarised as:

Exception to Rule #1 If the leading digit in the calculated uncertainty is a 1 or a 2, round the uncertainty to two significant figures.

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5.2 Rounding Values Once the uncertainty is rounded, the next step is to properly round the calculated value. Going back to our calculated acceleration, so far it looks like this:

g = 9.85196 ± 0.04 m/s2 (3) Now consider what the uncertainty listed above actually means: it means that the “5” in 9.85196 could actually be as small as (5 – 4) = 1 or as large as (5 + 4) = 9. Since we’re already uncertain about what that digit really is, it should be obvious that we can’t say anything meaningful about the digits that come after the “5”. Those digits should be rounded off, because they have no significance (which is where the phrase “significant figures” comes from). The following rule applies:

Rule # 2 – Stating Values Round the final calculated value to the same number of decimal places (same order of magnitude) as its correctly rounded uncertainty.

To make the above rules absolutely clear, consider the following example:

d = 1574.3961 ± 31.892 km (5) The correctly rounded result is:

d = 1570 ± 30 km (6) If you’re unsure how the result in (5) was rounded to (6), see your lab instructor. (Hint: recall that zeroes that only act as place holders for the decimal point don’t count as sig. figs!)

5.3 Scientific Notation and Significant Figures Sometimes, it is more convenient to put your result in scientific notation. In these cases, first write the value in scientific notation and then write the uncertainty to the same power of ten as the value. (This allows you to very quickly check whether your value and uncertainty are rounded to the same number of decimal places.) For example, the result in (6), written in scientific notation, is:

d = (1.57 ± 0.03) × 103 km (7)

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6 Comparing Experimental Results: “Accuracy”

6.1 What an Uncertainty Physically Represents When you write 6.38 ± 0.01 m, you are actually specifying a range of values from 6.37 m to 6.39 m (i.e. from [6.38 – 0.01] to [6.38 + 0.01]). By writing a value with an uncertainty, you are saying that you are confident that the “true” value – whatever it is – is located within the range specified.

6.36 6.37 6.38 6.39 6.40

6.38 ± 0.01 m Figure 2 – The range of values specified by the uncertainty

6.2 Comparing Two Values All experimental values have uncertainties. When you compare two numbers, you need to take their uncertainties into account. Figure 3 shows the ranges of the following values:

a) 9.62 ± 0.04 m/s2 b) 9.70 ± 0.07 m/s2 c) 9.81 m/s2 (gravitational constant, no uncertainty) d) 9.84 ± 0.05 m/s2

9.5 9.6 9.7 9.8 9.9

(c)(a)

(d) (b)

Figure 3 – Three uncertainty ranges and one constant value We say that two numbers agree within their limits of uncertainty when the ranges specified by their uncertainties touch or overlap. Thus, value (a) agrees within uncertainty with value (b). Value (c), which has no uncertainty of its own, agrees within uncertainty with value (d), because (c) is inside the range of values specified by (d)’s uncertainty.

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When the ranges do not overlap, we say that the values disagree within their limits of uncertainty. Thus, values (c) and (d) disagree with values (a) and (b) within uncertainty. Two numbers that agree within uncertainty are considered to be the same number. They might look different, but science considers them to be indistinguishable (i.e. we can’t tell them apart.) The fastest way to check if two values, A and B, agree within uncertainty is to determine whether the difference of the two numbers is smaller than or equal to the sum of their uncertainties, i.e. is ( )BABA δδ +≤− ?

6.3 Accuracy – How to Discuss It in Your Report In the Discussion section of your lab report, you will be asked to consider the accuracy of your results. This usually means that you must state whether your experimental value agreed (within uncertainty) with a known or theoretically calculated value, and then explain what that means. A typical way to phrase your answer would be: “Our experimental value for the acceleration due to gravity, 9.84 ± 0.05 m/s2, agreed with the known value, 9.81 m/s2, within the limits of uncertainty. This verifies the theory predicting the period of a pendulum for small oscillations.”

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7 Understanding “Precision” Precision is a measure of how small an uncertainty you are capable of getting. Precision is set by the design of the experiment and the equipment used. In the Physics 1101/1120 laboratory, you don’t control these things, but we do ask that you think about precision and consider how the precision of an experiment could be improved. In the Discussion section of your report, you will be asked to answer two questions:

1) Which measurement (if any) affected the experiment’s precision the most? 2) How could you improve the precision of the experiment?

1) Which measurement affected the experiment’s precision most? To answer this question, you need to inspect your calculation of the final answer’s uncertainty to determine if one measurement in particular made a large contribution to it. By “large”, we mean that the fractional or percent uncertainty is two or more times larger than any other term’s contribution. (Note: it isn’t meaningful to compare absolute uncertainties because they usually have different units; fractional and percent uncertainties have no units.) For example, say you have the following raw data:

x1 = 0.18 ± 0.01 m x2 = 1.01 ± 0.05 m t = 0.93 ± 0.01 s With this data, you performed the following calculation:

( ) 2...919.193.0

)18.001.1(2)(222

12sm

smm

txxa =

−=

−=

The uncertainty calculation proceeds as follows:

( )

( )( )( )...0215.0...0120.0...0602.0...919.1

93.001.02

)18.001.1(01.0

)18.001.1(05.0...919.1

)(2)(

)(2

2

2

12

1

12

22

12

++=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

−⎟⎠⎞

⎜⎝⎛ −

=

sm

sm

ss

mmm

mmm

tionssimplificaaftertt

xxx

xxx

txxa δδδδ

Now notice that the first term in the second set of brackets, )(

...0602.012

2

xxx−

=δ , is the largest by

a factor of almost three. Since δx2 is the uncertainty that accompanies this term, we say that δx2 affected the experiment’s precision the most. δx2 is also referred to as the Most Significant Uncertainty, or MSU. If none of the terms is two or more times larger than any other term, then you don’t have an MSU. This will be relevant when you discuss how to improve your experiment’s precision.

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2) How could you improve the precision of the experiment? This is a hypothetical question. Your experiment’s precision is determined by the equipment and techniques used, so to improve its precision, you must suggest changes that could be made to the experiment. The most direct way to improve the precision of the experiment is to improve the precision of the MSU. If you don’t have an MSU, then you can suggest ways to improve the precision of any measurement. In the above example, the MSU was due to x2 = 1.01 ± 0.05 m. To understand how to improve this value’s precision, we start by converting its absolute uncertainty to a percent uncertainty:

%95.4%10001.105.005.02 =×⎟⎟

⎠

⎞⎜⎜⎝

⎛==

mmmxδ

As you can see, to decrease the percent uncertainty of x2 (and thus increase its precision), you would have to either:

a. decrease the size of the absolute uncertainty, 0.05 m, or b. increase the size of the measured value, 1.01 m.

In your Discussion, you need to suggest ways in which one or both of these things could be accomplished. Here’s how to think about it: a. Decreasing the size of the absolute uncertainty δx2 could be accomplished by using a

different instrument or technique to measure the distance, one that has a smaller absolute uncertainty.

b. Increasing the size of the measured value, x2, could be accomplished by redesigning the experiment so that data was taken over a longer distance interval.

Be sure to suggest a specific way to improve the precision of the measurement. You won’t get many marks for saying “A more precise instrument would improve our precision”, because that statement is so vague that it’s meaningless. Suggest an actual tool that would accomplish the job.

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8 Averaged Values and Their Uncertainties Random uncertainties occur when something influences your data in such a way that repeated, identical measurements do not give you the same value consistently, i.e. you can measure the same value twice and not get the same number. An example of this would be wind speed, which might change from moment to moment due to gusting. The standard way to deal with measurements influenced by a random uncertainty is to take many measurements and average them together. The averaged value is then used in calculations instead of any single measurement. If the source of uncertainty in your data is truly random, then a series of many measurements will cluster around the “true” value in a Gaussian distribution (also called a “bell” curve).

Figure 4 – Gaussian Distribution

8.1 How Many Measurements to Make? When taking measurements that will be averaged together, your first concern is to decide how many measurements to take before you calculate that average. There is no set number; you must study your data and make a judgment call. You need to take enough measurements to obtain an accurate averaged value (one that is close to the “true” value). If your source of random uncertainty is large, i.e. the fluctuations in the data are large, then you will need to take a lot of measurements to obtain a good average. If the random uncertainty is small, you will not need as many. To determine how many measurements to take, start by taking a small number of them. Four or five measurements is usually a good amount to start with. Table 1 – Initial sample data Trial Wind Speed (± 4 m/s)

1 12 2 17 3 18 4 13 5 13

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At this point, you need to examine your data and decide whether you think the values are clustered around some central value. That is to say, if you made a histogram of the data, like in Figure 4, would your data look like a Gaussian curve? The data in Table 1 do NOT seem to be clustered around a central value. 17 and 18 are not close to 12 or 13. When the values do not cluster around some central value, that means you need to take more data. Taking another two data points, we find: Table 2 – Initial sample data plus two more measurements

Trial Wind Speed (± 4 m/s) 1 12 2 17 3 18 4 13 5 13 6 12 7 13

Note that even with another two data points, the values still do not appear to cluster around a central value. This means you need to take even more data. Can I discard data that don’t fit the trend? At this point, you’re probably wondering if you couldn’t just get rid of the 17 and 18 m/s data points. After all, all the other data appear to cluster around a central value. Can you discard data that doesn’t fit the trend? The answer to this is NO! While it is acceptable to discard data that you know is wrong (e.g. incorrectly measured), it is never acceptable to discard data because it is unexpected (or inconvenient). To do so is essentially scientific fraud; you would be manipulating your data to fit a preconceived notion of what the result “should” be, rather than accurately recording what nature is doing. Many important discoveries have been made when an experimenter noticed something unexpected in their data. For this reason, only discard data you know is wrong. Never discard something because it doesn’t fit the trend you expected.

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By taking another three data points, we obtain the following: Table 3 – Initial sample data plus five more data points

Trial Wind Speed (± 4 m/s) 1 12 2 17 3 18 4 13 5 13 6 12 7 13 8 17 9 14 10 15

Average 14.4 ± ?? At this point, the data is finally starting to look like it clusters around a central value. Thus, it is appropriate to calculate an average for it, as is done in Table 3.

8.2 The Uncertainty of an Averaged Value The uncertainty on an averaged value is not necessarily going to be the same as the uncertainty on the individual values. That is to say, every entry in Table 3 has an uncertainty of ±4 m/s, but this is not necessarily the correct uncertainty to use on the average value, 14.4 m/s. Random uncertainty can be much larger (or smaller) than your other sources of uncertainty (which are contained in ±4 m/s). For this reason, there are two possibilities for what the uncertainty on an average value should be. These are outlined below.

8.2.1 Statistical Uncertainty Because Gaussian distributions are well understood, the field of statistics provides one possibility for what the uncertainty of an averaged value should be, which is:

nYthen

nAAAAYIf n

aveσδ =

+++==

)...( 21

where σ is the standard deviation of Aave, 1

)(1

2

−

−=

∑=

n

AAn

iavei

σ , which can be easily calculated in

Excel using the function: STDEV(A1,A2,...An) This expression is really only appropriate for large values of n and does not take into account the other sources of uncertainty present in your experiment, which is why another possibility exists.

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8.2.2 “Regular” Uncertainty The second possibility for the correct uncertainty on an averaged value is simply the uncertainty of the individual values. In the case of Table 3, this would be ±4 m/s. Note that if the individual values all have different uncertainties, then you would use the average of the individual uncertainties, i.e.

nAAAYthen

nAAAAYIf nn

ave)...()...( 2121 δδδδ +++

=+++

==

8.3 So... Which One Do I Use? You have two possibilities for what the uncertainty of an average value should be. These are:

a) n

Y σδ =

b) n

AAAY n )...( 21 δδδδ +++=

You should use the larger of these two values. That is to say, you should calculate both of the above quantities and pick whichever one is the bigger number as your uncertainty for your average value.

8.3.1 Time-Saving Tip! If every value in your list of data agrees within its limits of uncertainty with every other value, then n

AAA n )...( 21 δδδ +++ will always be larger thann

σ . The fastest way to check whether all your data values agree with each other within uncertainty is to check whether the biggest and smallest ones do. For example, the largest number in Table 3 is 18 ± 4 m/s. The smallest is 12 ± 4 m/s. Since these two values agree within their limits of uncertainty, that means all the values in the table agree within uncertainty. That in turn means that ±4 m/s will be larger than the calculated value of

nσ . Therefore, ±4 m/s

is the correct uncertainty to use on the average of the values in Table 3. The correct final value is (after rounding): 14 ± 4 m/s.

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9 Elements of a Report All students of Physics 1101/1120 are required to do the same work and analysis. However, different lab instructors have different standard formats for handing in that work. Your lab instructor will tell you which format he or she expects. A complete lab report may include up to four different elements. These are:

A laboratory notebook (discussed below) A typed report (discussed below) Graphs, when required (discussed in Section 15, page 34) Printouts of any calculations done by computer (discussed in Appendix 1)

Table 4 – What is required in the notebook and typed report A typed report always includes the following sections:

If only a notebook is required, it must include the following sections:

If both a notebook and a typed report are required, they must include the following sections:

Notebook: Typed Report: Identifying information

Objective Theory Apparatus / Procedure

Results Discussion

Identifying information Objective Theory Apparatus / Procedure Data Calculations Discussion

Identifying information

Objective Apparatus / Procedure

Data Calculations

Identifying information

Objective Theory Apparatus / Procedure

Results Discussion

Detailed instructions are given in Section 9.2 on what each element in the above table is and how it should be written in your lab notebook / typed report. There is also an example laboratory notebook on page 28, and an example typed report on page 30.

9.1 First: A Few Guidelines

9.1.1 The Lab Notebook - Permanence A scientist’s notebook is supposed to be a permanent record of his or her work – a record of what they actually did in the lab. For this reason, you must follow a few guidelines: Write in indelible ink only. If you make a mistake, cross it out with a single horizontal line and write the correction

nearby. The mistake must still be readable! Never use white-out. Never cross something out to the point it can’t be read anymore. Never write on scrap paper with the intention of copying it into the notebook later.

Everything should be recorded directly into the notebook, including rough work. Never tear pages out of your notebook.

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Your notebook needs to be neat enough to be followed, but not neat enough to be published. Don’t waste time trying to make your work look perfect; only the quality of the science matters.

9.1.2 The Typed Report – Presentation Use complete sentences and (to the best of your ability) correct spelling and grammar in your typed report. If tables are included, use the border tool to make rows and columns. The report should be written so it is understandable by someone who has not done the experiment, but who understands the technical and scientific concepts you are using. Your typed report must look professional, so you must use a word processing program that allows you to insert equations and mathematical symbols. You may not write equations and symbols in by hand. Microsoft Word can do this and is available in the laboratory. Appendix 2 provides an overview of how to insert equations and mathematical symbols into Word.

9.2 Elements of the Lab Report

9.2.1 Prelab material Some experiments require you to do prelab work before coming to the laboratory. The prelab work must be completed before you begin the experiment. You will not be allowed to enter the lab room until it is.

9.2.2 Identifying Information Experiment title, date, name and partner’s name should appear at the top of the report.

9.2.3 Objective If the lab manual lists an objective for the experiment, then copy this into your report. If the manual does not list an objective, then read through the experiment and, in your own words, briefly summarise what the experiment is attempting to accomplish.

9.2.4 Theory Write a brief summary of the theory in your own words. (Do not copy the theory section of the lab manual!) You should try to state what the theory predicts, and how the experiment will confirm those predictions. Try to be concise. List the equations you used in the experiment and define their variables. List only those equations you actually used. Do not include derivations, intermediate steps or diagrams. Finally, list any assumptions or approximations made. For example, if you neglect the effect of friction in your calculations, you would mention that in your Theory section. In summary, the Theory section should:

a. Explain what the theory predicts

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b. Explain how the experiment can verify those predictions c. List the equations actually used in the experiment and define all variables that appear in

those equations d. List any assumptions or approximations made in the experiment

9.2.5 Apparatus/Procedure Refer the reader to the lab manual for details on the apparatus and procedure. Give the full title of the manual and list its relevant page numbers. If the method or equipment used for the experiment was different than what is described in the lab manual, then record those changes in your Apparatus/Procedure. Note: If you think up and begin using a better technique midway through your experiment, then record a note about that change when and where it occurred, rather than in the Apparatus/Procedure section.

9.2.6 Data (notebook only) Before you begin to take data, “play” with your equipment for a while. Do a few trial runs to ensure you know how to use the equipment correctly before you start. Record all data (measurements) directly into the notebook with correct uncertainties and units. Everything must be written to the correct significant figures. Record data exactly as measured, without conversions (conversions are done with your other calculations). Single Measurements Record single measurements using the following format:

Constants Constants should be recorded the same way, except without uncertainty information. Repeated or Related Trials Repeated or related trials should be recorded in a ruled, numbered and titled table. Units and uncertainties may be put in the column heading. If the uncertainties are different for each entry in the column, then write the uncertainties alongside the value, rather than in the column heading.

Written description Symbol Measurement Estimated uncertainty

Length of wooden block, L = 12.20 ± 0.15 cm Unit

Explanation of uncertainty estimate

The ruler could be read to 0.05 cm. However, since the block is rough at both ends, I added an extra 0.10 cm to allow for this.

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Numbered table with clear description of contents

Table 1: Extension of a Spring, demonstrating Hooke’s Law.

Extension of Spring Total Mass Applied to Spring x (±0.05 cm)

If you make a mistake, cross it out with a single horizontal line and write the correction nearby. The mistake must still be readable! Never cross something out to the point that it can’t be read anymore. Never use white-out.

Any data you change must have a note explaining why it was changed.

Record comments about techniques used, as well as problems encountered and what was

done to deal with them.

If your results do not “work out”, you are expected to find the problem and solve it, retaking data if necessary. If you can’t solve the problem, ask for help. Troubleshooting is a normal part of the lab.

The mark for the notebook will also be based on effort/participation in the lab and the quality of data (i.e. taking careful measurements and obtaining “good” results.)

9.2.7 Calculations (notebook or Excel only) Note: If all your calculations are to be done in Excel, your lab instructor may allow this section to be simplified or omitted from your notebook. Show at least one sample calculation for every calculation done in the experiment. Show enough steps that your lab instructor can follow your logic. Work out your uncertainty formulas algebraically (i.e. in symbols) first, then provide a numerical sample calculation for them. Only round your final answers to the correct significant figures. Intermediate steps should be left un-rounded.

9.2.8 Results (typed report only) Record your final results, with the correct uncertainties, units and significant figures. Intermediate values are not recorded; if you aren’t sure whether a given value is important enough to be recorded in your Results section, check the Objective for hints.

m (±0.05 g)

22.50 42.50 62.50

0.84 1.59 2.43

Written description

Symbol, uncertainty and units The uncertainty for the balance is listed as 0.05 g.

The mirrored scale could be read to 0.05 cm.

Explanation of uncertainty estimates

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9.2.9 Discussion Discuss your results, not the theory or procedure. Concentrate on what actually happened, not on the ideal case or what should have happened. Explain how your results verify or don’t verify each goal in the experiment’s Objective. Back up your statements with evidence from your experimental results. Use words like “verify”, “confirm” or “agrees with” to talk about your results, rather than words like “prove” or “disprove”. In science, you can assemble a convincing case for or against a claim, but it is actually very hard to completely prove or disprove it. If there is a graph:

State whether the shape of the graph agrees within uncertainty with what the theory predicted. For example, you might say “Our graph was linear within uncertainty, because our best line passed through the uncertainty boxes of all the data points.” State whether the slope/y-intercept values were equal within uncertainty to the expected values (if there were any). State whether the theory that predicted the graph’s shape, slope and y-intercept was verified. Hint: If your graph was the expected shape (e.g. linear) within uncertainty, and had the correct slope and y-intercept values within uncertainty, then you can say that the graph verifies the theory.

Accuracy: If appropriate, state whether your experimental values agreed with the known or theoretically predicted values within uncertainty. If appropriate, state whether your results are consistent with theory (and be specific about which theory). (Note that if you had a graph, you will have already discussed the accuracy of your slope and y-intercept values.)

Precision: a. State which (if any) of your measured values affected your precision most (i.e. did you

have a Most Significant Uncertainty?) b. Suggest a specific way to improve the experiment’s precision.

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10 Example Notebook Caveat: Your lab instructor may have requirements for your laboratory notebook that differ from this example. He or she will explain those to you in the lab.

28

Example Notebook – Continued

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11 Example Typed Report Experiment 1: Density Jan. 8, 2007

Ima Jean-Yuss Vlad Fiziksislov

Objective: To measure the density of three copper blocks and (a) verify that density is independent of an object’s mass and volume, and (b) confirm that the blocks are copper by comparing their average density to the known density of copper, 8.92 g/cm3. Theory: Density is a property of the material something is made from, so the theory predicts we should get the same value for the density of each of three copper blocks, even though the blocks have different masses and volumes. To get density, we use:

VM

=ρ

where ρ is density, M is mass and V is volume. We measure the volume of the blocks by dipping them in water and measuring the apparent change in volume of the water, using:

12 VVV −=

where V is the volume of the block, V2 is the volume of the water plus the block and V1 is the volume of just the water alone. We assumed that the hook and string used to dip the blocks in water had negligible mass and volume, and would not affect the density measured for the copper blocks. Apparatus/Procedure: Refer to pages 7 – 9 of the Physics 1000 lab manual, “Introductory Physics Experiments”. Results: The density of block A was 8.9 ± 0.8 g/cm3, while block B was 8.4 ± 0.5 g/cm3, and block C was 9.4 ± 0.5 g/cm3. The average density of all the blocks was 8.9 ± 0.6 g/cm3. Discussion: The densities of the three blocks agreed with each other within uncertainty, but their masses and volumes did not. This means the density was independent of mass and volume, because it didn’t change even though the mass and volume did. The average density was 8.9 ± 0.6 g/cm3, which agrees within uncertainty with the known value of 8.92 g/cm3. This means we confirmed that the blocks were made of copper. Our most significant uncertainty was due to the volume measurement. To improve its precision, we could use a graduated cylinder with smaller divisions (smaller absolute uncertainty) or measure larger blocks using the same cylinder. Both things would reduce our fractional uncertainty and therefore make our precision better.

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12 Data Analysis By the end of the semester, you will be using a spreadsheet program called Microsoft Excel to mathematically analyse your data. Your lab instructor may choose to introduce this spreadsheet program to you now, or may do so later in the semester. Appendix 1, The Basics of Using Excel, provides an overview of how to use the program.

13 Helpful Strategies in the Lab

13.1 Saving Time We recommend that you make a habit of the following time-saving strategies: 1. Always read the experiment ahead of time to ensure you are adequately prepared for it. Think

about what sorts of measurements and calculations will be required. 2. Have everything up to the Data Section of your notebook written before you come to the lab,

so you can begin taking data immediately. 3. If your lab instructor requires a typed report to be turned in every week, have everything up

to the Results Section typed and saved to diskette ahead of time. 4. If you are doing all your calculations in Excel, prepare your spreadsheet (i.e. type in all the

equations) and save it to diskette ahead of time.

13.2 What to Do When Things Go Wrong You are expected to fix problems encountered in the lab and retake data if necessary. If you run out of time to do this, then in your Discussion, report honestly what your results imply (even if you know it’s wrong), then explain what you think caused the poor results. The important thing is to analyse your data correctly, even if you think the data are incorrect. Never try to hide or misrepresent what your results show! Science relies on the accuracy and truthfulness of its experimenters. While you may lose marks for whatever mistake caused the problems, you will not lose marks for correctly analysing incorrect data.

13.3 When Using Computers Save your work early and save it often. The computers in the laboratory have been known to freeze up! If this happens, you will lose all your work up to the last time you saved, so try to remember to save regularly. Please be aware that memory sticks and flash drives do not always work on the lab computers. It may be helpful to save the driver for your device to floppy disk and bring it with you to the lab, but it is always best to test your device for compatibility before you rely on it.

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14 Practice Lab Measurement of Reaction Time This experiment will give you practice in writing up a lab report, as well as working with averaged values and their uncertainties. Recall that an example notebook report is provided in Section 10, page 28, and an example typed report is shown in Section 11, page 30. Objective To determine a person’s reaction time. Theory For an accelerating object, the kinematics equation relating the distance travelled d to the initial velocity vo, acceleration a, and time t is given by 2

21 attvd o += . We will be measuring the

distance an object drops from rest, so vo = 0 and a = g = 9.81 ± 0 (the acceleration due to gravity; the uncertainty is so small that it is valid to assume it’s neglibible). Solving for t, we get:

gdt 2

= (1)

The average human reaction time is ~ 0.15 s. Apparatus Ruler Procedure 1) Hold the ruler so that your partner's fingers are lined up with the zero mark (but in an open

position). Your partner will catch the ruler as quickly as possible after you drop it. 2) Record the distance fallen as read from the ruler, estimating a reasonable uncertainty for this

measurement. 3) Measure and record a suitable number of trials (see Section 8, page 19, Averaged Values and

Their Uncertainties, for more details). Repeated measurements should always be recorded in a table. Do measurements for one person’s reaction time only.

4) Calculate the average distance fallen dave and its uncertainty. Don't round off this value as

it will be used in further calculations.

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5) Using Equation (1), calculate the time t it took the ruler to fall the average measured distance dave.(i.e., use dave to calculate one value of t). Don't round t off yet (we need to calculate δt before we can round t to the correct significant figures).

6) Calculate the uncertainty of t in symbols. Use the rules listed in Section 3, page 9,

Uncertainty Propagation, and the technique described in Section 4, page 11, Simplifying Complicated Uncertainty Problems.

7) Evaluate the expression by substituting in the values from your data. The uncertainty of g is

so small that it is insignificant here. Therefore, assume δg = 0. Finally, round the result and its uncertainty to the correct number of significant figures.

8) Analyse your final results and finish writing up your report. Reminders for when you write your Discussion: (Refer back to Section 9.2.9, page 27, for more detail.) Did you satisfy the objective of the experiment? Was your final result accurate? i.e. Did your measured reaction time agree within uncertainty

with the average value of ~ 0.15 s? (Note that, in this experiment, it’s nothing to be concerned about if your value doesn’t agree with average human reaction time – reaction time varies a lot from person to person.)

What measurement affected your final answer’s precision most? Describe a way to improve the precision of the experiment.

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15 Graphing Graphing is a powerful means of analysing data because it allows you to see the relationship between two variables at a glance. Graphing also provides a way to see what the data would look like if the fluctuations due to uncertainty were “smoothed out”.

15.1 Theories and their Predictions A theory makes predictions about how a system will behave. By taking measurements, we can confirm whether the system behaved the way the theory predicted it would. In Physics 1101/1120, you will mostly be dealing with linear graphs, so let’s look at a linear theory: When a body is moving in a straight line at a constant velocity v, the equation governing its total displacement d is given by:

d = v t + do where do is the initial position of the body and t is the amount of time the body has been in motion. Note that d and t are variables, and v and do stay constant. Now compare the above equation to that of a straight line on a graph:

d = v t + doy = m x + b

Note the similarities! The equation is a linear theory because if you make a graph with displacement d on the y-axis, and time t on the x-axis, you will get a straight line graph with the slope equal to v and the y-intercept equal to do. The following diagram shows the connection between the two equations.

d = v t + do y = m x + b

v and do are the constants.

d and t are the variables.

d corresponds to the y variable, and should be plotted on the y-axis.

The slope m will be equal to v.

The y-intercept b will be equal to do.

t corresponds to the x variable, and should be plotted on the x-axis.

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A theory that is linear makes three predictions: 1) It predicts that if you graph its two variables, you will get a linear graph. 2) It predicts what the value of the graph’s slope will be. 3) It predicts what the value of the graph’s y-intercept will be.

Let’s look at another example. Newton’s second law can be written as F = M a, where F is the force applied to an object, M is the mass of that object, and a is the acceleration the object experiences due to the force. Comparing to y = m x + b, we see that F = M a is a linear theory:

F = M a + 0 y = m x + b

The three predictions made by the equation F = M a are:

1) If you graph F vrs. a, you will get a linear graph. 2) If F is plotted on the y-axis and a is plotted on the x-axis, then the slope of the graph will

be equal to the mass M, and 3) the y-intercept will be equal to zero.

15.1.1 Linearising Non-Linear Equations Most equations that are non-linear can be plotted in such a way that their graph becomes linear. In these cases, you plot a function of the variable on one or both axes. Example 1: The period of a pendulum T is the time it takes for the pendulum to complete one full back-and-forth swing. If the length L of the pendulum changes, then the period T will also change. For small oscillations, the relationship is given by:

gLT 22 4π=

T and L are the variables, but if you made a graph of T vrs. L, you would not see a linear relationship. You would see a square root relationship. However, if you graph T 2 vrs. L, then the graph will be linear. The diagram below shows how to relate the equation to a straight line on a graph.

T2 = L + 0 y = m x + b

T and L are the variables. The predicted

slope m is 4π 2 4π 2T2 should be plotted on the y-axis.

g g

The predicted y-intercept b is zero.

L should be plotted on the x-axis.

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Example 2: In Coulomb’s Law, the force F between two electrical charges Q1 and Q2, separated by a distance r, is given by the formula

221

rQkQF =

where k is the Coulomb constant. If Q1 and Q2 are constant also, then F and r are the variables. To obtain a linear graph from this expression, you would have to graph F vrs. 2

1r

.

15.2 Drawing Graphs You will learn to graph by hand first. Later, your lab instructor will show you how to use a computer program to make your graphs. When done correctly, a hand-drawn graph can be just as accurate as one done by computer, but the program saves you time in the lab. You may draw graphs in pencil, since it is common to need to make corrections. There is an example graph in Section 15.5, page 41.

15.2.1 Axes and Scales Draw your axes right along the edges of your page. Larger graphs are more accurate. Choose your scales so that data fills at least half the page in both directions. You can start your axes at values other than (0,0), if it is more convenient to. Give your axes a scale that will make it easy to graph the data. For example, have each block of the graph equal a multiple of 1, 2, 4, 5, etc., but not a multiple of 3, 7, or any other value that makes it difficult to figure out where to graph decimal values.

15.2.2 Labels Label the axes clearly, with quantity name, symbol and units. Give the graph a number and an informative title related to the experiment (don’t just list what the axes are called.)

F = kQ1Q2 + 0 y = m x + b

F and r are the variables. The predicted

slope m is kQ1Q2 1 F should be plotted on

the y-axis. r2 The predicted y-intercept b is zero.

1 should be plotted r 2

on the x-axis.

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15.2.3 Drawing in the Data Points After you have located where a data point should be drawn, put a dot there. Next, draw in your uncertainty bars. These should be drawn to scale, showing how large the uncertainties on the x- and y-axes are for that one data point (see diagram.) Note that the two uncertainty bars form an uncertainty box. The “real” data point lies somewhere inside that box.

6.5 6.0

3 y-axis data: 2.2 ± 0.6

+ 0.6 x-axis data: 2 – 0.6 6.3 ± 0.4

uncertainty box – 4 + 4

15.2.4 Best Fit Line Once your data is plotted, you are ready to draw the best fit line. This line represents your best estimate of where the data would fall if you had no uncertainties. To draw the best fit line, visually determine the “stripe” that contains all your uncertainty boxes. You may find it helpful to lay two rulers or two pieces of paper on your graph to help you block off the area. In Figure 5, the regions outside the uncertainty box “stripe” have been shaded. Once you have blocked off the “stripe”, draw a line down its centre. This is the best fit line, and it should represent the overall trend of your data.

Figure 5 – The best fit line is drawn down the centre of the “stripe” containing the uncertainty boxes.

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15.2.5 Bounding Lines (Upper and Lower Slope Lines) You will be calculating your slope from the best fit line. However, to determine the estimated uncertainty on that slope, you will need an upper and lower bounding line (also called the upper and lower slope lines.) Draw one line from the top of your uncertainty box “stripe” on the right-hand side to the bottom of the “stripe” on the left-hand side. Draw a second line from the bottom of the right-hand side to the top of the left-hand side of the “stripe”, forming an “X” shape. Your graph should look like Figure 6.

Figure 6 – The bounding lines span the “stripe” containing the uncertainty boxes.

15.2.6 Finding the Slope and its Uncertainty The slope of the graph is determined from the best fit line and is called mb. To calculate the slope mb, find two points on the line (not data points) that are far apart. The slope of a straight line is defined as:

12

12

xxyym

−−

=

where (x1, y1) and (x2, y2) are the coordinates of your two points. After you have calculated the slope of the best fit line, you will need its uncertainty, which is given by:

lub mmm −= 21δ

where mu and ml are the slopes of your upper and lower bounding lines, respectively.

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15.2.7 Finding the Y-intercept and its Uncertainty The y-intercept b of a straight line is defined as the value of y when the value of x is zero. If your x-axis starts from zero, you can read your y-intercept directly off the graph. Simply extend your best fit line until it crosses the y-axis and read off bb. The uncertainty of bb is given by:

lub bbb −= 21δ

where bu and bl are the y-intercepts of your upper and lower bounding lines, respectively. If your x-axis does not start from zero, you will have to calculate bb. Choose a point on the best fit line and, using its coordinates and your slope value, calculate bb using y = m x + b. Do the same for bu and bl, in order to find δbb.

15.2.8 The Best Fit Equation Write the equation of the best fit line directly on your graph. Include numerical values for the slope and y-intercept, and include their units and uncertainties. For example, if the theory governing the graph is LT g

242 π= , then the equation you write on your

graph might look like this: T 2 = (3.8 ± 0.4 ms 2 ) L + (0.4 ± 1.2 s2)

15.3 Analysing the Graph’s ACCURACY Recall that a linear equation is a theory that makes three predictions:

1) It predicts that if you graph its two variables, you will get a linear graph. 2) It predicts what the value of the graph’s slope will be. 3) It predicts what the value of the graph’s y-intercept will be.

To analyse your graph’s accuracy, you need to determine how well your data agrees with these three predictions. Ask yourself the following questions: 1) Is my graph linear within the limits of uncertainty?

Your graph is linear within the limits of uncertainty if (and only if) the best fit line passes through every data point’s uncertainty box.

2) Does my slope agree with the predicted value (if there is one) within the limits of

uncertainty?

Compare your slope to the predicted value and see if they agree within uncertainty.

Note: In some experiments, the predicted slope (or y-intercept) may be a constant that depends on the equipment, so there won’t be a known value to compare your value to.

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3) Does my y-intercept agree with the predicted value (if there is one) within the limits of

uncertainty? Compare your y-intercept to the predicted value and see if they agree within uncertainty.

4) Does my data verify the theory?

If you answered “yes” to all the above questions, then your data has verified the theory!

15.4 Analysing the Graph’s PRECISION Precision of the y-intercept The precision of the y-intercept is set only by the uncertainty of whatever is plotted on the y-axis. Precision of the slope The uncertainty for the slope of a graph is given below:

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

+−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=12

12

12

12

12

12

xxxx

yyyym

xxyym δδδδδδ

To determine whether the x-axis quantity or the y-axis quantity affected your precision most,

simply determine which term is bigger, 12

12

xxxx

−+ δδ or

12

12

yyyy

−+ δδ .

Note: If the uncertainty bars in one direction are much larger than in the other direction, you’re safe in saying that the axis with the larger uncertainty bars affected your precision most (assuming your data fills roughly the same amount of the page in both directions.)

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15.5 Example Graph

41

15.6 Making Graphs with a Graphing Program Computers can create graphs quickly and accurately, but they can’t do your thinking for you. You must make sure the variables are plotted on the correct axes and that the results are reasonable. You will be provided with one of the following two programs in the lab: Curvefitting.xls or Linegraph.xls. An overview of how to use Curvefitting.xls is in Appendix 3. An overview of how to use Linegraph.xls is in Appendix 4.

15.7 Trouble-Shooting Your Graph Is your graph reasonable? How can you tell if it is reasonable? Below are some examples of common problems that that show up in graphs. The most common problem is that your graph is not linear within uncertainty (see Example 1 below). If your graph has that problem, try the following to fix it: a. Check you haven’t misplotted your value or drawn its uncertainty too small. b. Check any calculations you did to come up with that data point. c. If there is still a problem, go back to the experimental apparatus and re-measure the

data for any points that do not fall on the line within uncertainty. d. If there is still a problem, even after re-measuring, explain why in your report. 1)

Time (s)

Vel

ocity

(m/s

)

This graph is not linear within uncertainty. The best line does not pass through one data point’s uncertainty box. This is most likely due to misplotting or miscalculating the point. If you've already checked your plotting and calculations and found them to be correct, then re-measure the data for that point.

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2)

Time (s)

Vel

ocity

(m/s

)

This graph doesn’t make sense! The data fluctuates around the best line by much more than the length of the uncertainty bars. This may be due to underestimated uncertainties, a calculation mistake in the plotted points, or perhaps there was an uncontrolled variable (e.g. fluctuating temperature.) Try to determine the problem and, if necessary, retake your data.

3)

Time (s)

Vel

ocity

(m/s

)

This graph shows a systematic departure from a straight line. Perhaps one of the plotted quantities should have been raised to a different power, or there may be a systematic calibration error in one of the instruments. If the calibration of your instruments is okay, the problem may be that the theory is not valid for the conditions maintained in the experiment. Try to figure out what the problem is.

4)

Time(s)

Vel

ocity

(m/s

)

This graph is linear within experimental uncertainty, but a bit strange looking. The data points fluctuate around the best line by much less than the length of the uncertainty bars. In some instances, the experimental uncertainty is large and a graph such as this is valid. However, overestimating the uncertainty of a measurement can also produce such a graph. If you see a graph like this, double-check that you haven’t overestimated your uncertainties.

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5)

Time(s)

Vel

ocity

(m/s

)

This graph makes sense! The scatter of the data points around the best line is about the same size as the uncertainty bars. This graph is linear within the experimental uncertainty because the best line passes through all the uncertainty boxes. This is the sort of graph you hope to get!

15.8 Graphing Exercise - Hooke’s Law In this exercise, you will take data and graph it, then analyse your graph. Objective To verify Hooke’s Law. Theory In 1676, Robert Hooke discovered that, for a range of forces F, a simple metal spring would stretch a distance x that was directly proportional to the applied force on the spring. The following equation expresses this theory mathematically:

F = k x The constant k is called the “spring constant”, which is a number set by the shape and material the spring is made out of. If the spring obeys Hooke’s Law, then a graph of F vs. x will be linear within uncertainty. The slope of the graph will be the spring constant k and the y-intercept will be equal to zero within uncertainty. Note that due to varying “stiffness” of the springs, the y-intercept may be greater than zero for some springs, i.e. the best line will not pass through the origin of the graph within uncertainty. Apparatus Hooke’s Law apparatus (stand, spring, scale, hanger)

electronic balance slotted brass masses (20 and 50 g)

44

When mass is placed on the hanger, the spring will stretch from its rest position xo to its stretched position xf. You will be graphing the total displacement x, which is given by x = xo – xf. To make the math simple, you can set xo = 0. Pinch the clips on the back of the mirrored scale and slide the whole scale up or down until the pointer on the hanger is at zero. To eliminate parallax, the scale readings should be made only when the pointer is directly in front of its own image in the mirrored surface of the scale. Ask for help with this adjustment if necessary.

Figure 7 shows the Hooke’s Law apparatus. To prevent permanent deformation of the spring, do not apply a load of more than 300 g (Richmond springs) or 170 g (Surrey springs). Also, do not pull down on the spring with your hand! Procedure Add masses to the hanger and measure how far the spring stretches. Note that you want to graph the total force hanging on the end of the spring, and that will include the weight of the mass hanger itself! Collect between 6 and 8 data points to graphically verify Hooke’s Law. Choose your increments of mass such that they span the maximum possible range. For each mass increment, measure that mass on the electronic scale. Record all your data in your notebook. Graph your data, and write the equation of the best line directly on the graph. When you are finished, analyse your graph as you would for the Discussion section of a normal lab report. (See Section 9.2.9, page 27, for more details.) Reminders for your Discussion:

x

x

xoo

(a)

m

(b)

f

x

kx

Fg

7 Is the graph linear within uncertainty? What is your k value and its uncertainty?

(Note that there is no expected value for k in this experiment.) Is the y-intercept equal to the expected value within uncertainty? Did you verify Hooke’s Law? Which quantity affected your precision most? How could you

experiment?

45

Figure

improve the precision of the

Experiment 1

UNIFORM LINEAR MOTION If the force acting on a particle is constant, then the acceleration of the particle is also constant. One example of motion with constant force is the motion of an object on an inclined plane. You have been taught in class to focus on those forces that act on the object through the use of a free-body diagram and then to use the diagram to formulate the equation of motion for that object. This experiment will test not only your understanding of free-body diagrams but also your skill in analyzing experimental data by means of linear graphs. Objective To determine the acceleration of a glider on a frictionless air track inclined at an angle θ (a) theoretically, using a free-body diagram, and (b) experimentally, using a linear graph, and to compare the two values. Theory The velocity v as a function of time t for an object moving with a constant acceleration a is: v = vo + at (1) where vo is the initial velocity of the moving object. The average velocity of a moving object is defined as:

timeelapsed

positioninchangetxvav =

∆∆

= (2)

Apparatus student air track (Ealing) with air supply and glider LabVIEW interfacing system (Picket Fence VI—time intervals are ± 0.001 s) accessory photogate timer (Pasco ME-9204A)

picket fence object aluminum block tape measure ruler digital calipers (± 0.03 mm)

46

Figure 1 Set-up of experiment

The experiment makes use of a linear air track, a device which permits the observation of motion with almost complete absence of friction between the glider and the track. The glider is supported above a straight track by a cushion of air produced by blowing air (from an air supply) out of rows of small holes in the track. The airtrack is levelled by loosening the locking wing nut on the levelling screw (the single foot located near the end of the track where the air intake is) and twisting the levelling screw. The LabVIEW interfacing system will be used to measure time intervals of the pickets of a picket fence attached to the glider passing through a photogate using Picket Fence VI. Refer to Appendix 2 and the lab instructor's explanation for the details of the interfacing. As the picket fence moves through the photogate, the time between the leading edges of two consecutive pickets is measured. (If the picket fence object is moving to the left, then the “leading edge” of each picket is the left edge.)

d

You can enter the appropriate distance d between the leading edges of the picket fence into the Picket Fence VI. Hit the arrow key at the top to run the program. The timing will be triggered by the first interruption of the photogate beam.

47

Procedure 1) Before coming to the lab, draw a free-body diagram for the glider (assume the airtrack is frictionless) and derive an expression for the glider’s theoretical acceleration a in terms of b, h and g (see diagram below). Hint: Sin θ is obtained from the measurements of b and h; therefore, you don’t need to know the value of θ . Also derive an expression for the uncertainty in a.

b

hblock

two levelling feet

θ

single levelling foot

glider

air track

2) Calculate the theoretical acceleration a (and its uncertainty) using your derived expressions and the appropriate measurements. (Assume zero uncertainty for g.) 3) Level the airtrack and then place the block under the single foot of the airtrack to incline the airtrack. Release the glider on the airtrack and measure the time intervals of the pickets of the picket fence attached to the glider as it goes through the photogate using Picket Fence VI (found in the 1101-1120 Labs folder on the desktop). If the slope of the v-t graph appearing on the screen seems reasonable (i.e., close to your predicted value and, more importantly, the graph is linear and reproducible), record these time intervals. 4) Using the time intervals and any other appropriate measurements, calculate:

i) the average velocity for each time interval, ii) the “instantaneous times” at which these velocities occurred. For example, the

instantaneous time for the 5th velocity is the sum of the first 4 time intervals plus 1/2 of the 5th time interval.

5) Plot a velocity-time graph (using Linegraph) and find the acceleration from the graph.

48

Experiment 2 FRICTION

Objective To determine the coefficient of kinetic friction µk between leather and wood. Apparatus wooden inclined plane, pulley, wooden support block, mats leather bottomed wooden box with plastic tabs and string picket fence object accessory photogate timer (Pasco ME-9204A) with tall stand LabVIEW interfacing system (Picket Fence VI—slopes are ± 2%) Craftsman Universal Protractor Ohaus electronic balance (± 0.05 g) 2 slotted 50 g masses and mass hanger ruler

foam for the floor

Figure 1 Set-up of experiment

49

The acceleration of the box will be measured using the Picket Fence VI LabVIEW program as in Experiment 1. The picket fence is attached to the box, the box is released through the photogate and LabVIEW measures the time intervals for the pickets to go through the photogate. LabVIEW then calculates the velocities and plots a v-t graph where the slope is the acceleration of the box. In Experiment 1, you verified that the slope value given by LabVIEW is in fact the acceleration of the object. Therefore, for this experiment, you will simply use the slope values read from the screen as your acceleration values (i.e., don't record the time intervals this time). Procedure 1) Before coming to the lab, draw free-body diagrams for the leather bottomed box on the inclined plane (include the force of friction) and for the hanging mass. From your free-body diagrams and using Newton's Second Law and the relationship f = µkN, derive an expression for µk in terms of the masses, θ (the angle of incline), g and a (the block's acceleration). Also derive an expression for δµk. Have these equations checked by your lab instructor before taking any measurements. 2) Set up the apparatus as shown in Figure 1 (with a hanging mass of ≈ 150g) and take all necessary measurements to determine µk. Use the Picket Fence VI LabVIEW program to measure the acceleration (make sure you measure d and enter that value into the Picket Fence VI). Measure the acceleration a few times and record all valid trials (i.e., all the ones that give a linear graph, regardless of the slope value or how far apart from each other they are). 3) Use an Excel spreadsheet to calculate µk (using the average acceleration). Determine the uncertainty of the average acceleration (refer to Uncertainty Equations to calculate the uncertainty of an average value). 4) Now, change the angle and repeat the measurements for and calculation of µk.

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Experiment 3 THE “IDEAL” SPRING

In physics, “ideal” springs often appear in lecture and in problems, but what makes a spring “ideal”? Any spring that obeys Hooke's Law can be considered ideal. As well, an ideal spring will exhibit Simple Harmonic Motion. By considering these two ideas, the spring constant k for a given spring (a value that is dependent on the material and the dimensions of the spring itself) will be determined in this experiment.

Part A: Hooke's Law Objective To a) test if a given spring obeys Hooke's law and, if it does, to b) determine the spring constant of the spring. Theory Hooke's Law is a simple linear relationship for springs discovered by Robert Hooke in 1676. Hooke noticed that there was a range of force that could be applied to a spring over which “the magnitude of the applied force F was directly proportional to the change in length x of the spring.” Consider a spiral spring hung vertically as shown in Figure 1(a). Let xo be the position of the lower end of the unstretched spring (this is the spring's equilibrium position). An object of mass m (the load) is attached to the lower end of the spring as shown in Figure 1(b). Under the action of the weight of the load, the spring stretches a distance x = xf – xo (1) where xf is the position of the lower end of the stretched spring. On the lower end of the spring, the load applies a force Fg which is equivalent to the weight of the load, i.e.,

x

x

xoo

(a)

m

(b)

f

x

kx

Fg Figure 1

51

Fg = mg (see Figure 1(b)). Also, the restoring force kx of the spring itself (where k is the spring constant) acts in the opposite direction to Fg. Since the load hangs still (i.e., the system is in equilibrium), then by Newton's Second Law,

kxF

kxFF

g

g

=

=−=Σ

00

(2)

If the spring obeys Hooke's law, then the graph of Fg vs. x will be linear within uncertainty, the slope of the graph will be the spring constant k and the y-intercept will be equal to zero within uncertainty. Due to the varying “stiffness” of the springs, the y-intercept of the graph may be greater than zero; that is, for your spring the line may not pass through the origin within uncertainty. Apparatus Hooke's Law apparatus Ohaus electronic balance (± 0.05 g) slotted brass masses (20 g and 50 g) Figure 2 shows the Hooke’s Law apparatus. The scale can be moved by pressing the clips mounted behind it and sliding it along the support rod. To eliminate parallax, the scale readings should be made only when the pointer is directly in front of its image in the mirrored surface of the scale. Ask for help with this adjustment if necessary. Procedure To prevent permanent deformation of the spring, do not apply a load of more than about 300 g (this also means do not pull down on the spring with your hand). Collect between 6 and 8 data points to graphically verify Hooke’s Law (always choose your increments of the quantity you control so they are approximately equal and so that they span the maximum possible range). For each mass increment, measure that mass on the electronic scale. From the graph determine the spring constant of the spring.

Figure 2

52

Part B: Simple Harmonic Motion Objective To determine the spring constant k of a given spring through measurements of the simple harmonic motion of the spring under a load. Theory In simple harmonic motion (SHM), an object oscillates with constant amplitude. Under gravity, the vertical oscillations of a loaded spring will follow SHM similar to that of a simple pendulum where instead:

mk

=ω (3)

ω is the angular frequency (or velocity) of oscillation, k is the spring constant and m is the suspended mass on the spring. If we substitute ω = 2π/T (where T is the period of oscillation) into Equation (3), solve for the period T and square we obtain:

mk

T2

2 4π= (4)

Apparatus Hooke's Law Apparatus (Cenco #73960C) Ohaus electronic balance (± 0.05 g)

slotted brass masses (20 g and 50 g) stopwatch Procedure Test Equation (4) by means of a graph. Find the period of oscillation for each of 6 to 8 different masses added to the mass holder that range from 150g to 300g. Remember to not exceed 300g or you will damage the spring (always choose your increments of the quantity you control so they are approximately equal and so that they span the maximum possible range). To measure the period, time at least 10 periods then calculate the average period. Pull gently down on the mass 2–3 cm only then let go to start the oscillations—pulling too hard will put too much force on the spring and may cause permanent damage. Record your observations of what happens to the oscillations when the load on the spring is < 150 g. Determine the spring constant k from the graph.

53

Experiment 4 THE SIMPLE PENDULUM

At small angles from its equilibrium position, a pendulum will exhibit simple harmonic motion. A simple relationship exists between the period of the oscillation and the length of the pendulum. This experiment will investigate this relationship and the motion of the pendulum. Objective To investigate the motion of a simple pendulum and to derive a value of g, the acceleration due to gravity. Theory A simple pendulum consists of a small object (usually a sphere) of mass m suspended by a light string, where the upper end of the string is fixed as in Figure 1. The mass of the string is negligible in comparison to that of the bob, and its length is very much greater than the size of the object (i.e., the radius of the sphere). The time it takes a simple pendulum to make one complete swing (back and forth—from point A to point B and back to A again, as in Figure 1) is called its period T. If the pendulum swings through a small angle θ (not more than 5°), then the period T is given by:

gLT π2= (1)

where L is the length of the simple pendulum, as measured from the point of suspension (i.e., the point at which the pendulum pivots) to the centre of the object and g is the acceleration due to gravity. Squaring both sides of Equation (1) and rearranging, we get

Lg

T2

2 4π= (2)

θ

AB

m

Figure 1

54

Apparatus Simple Pendulum Apparatus (Sargent-Welch Model 829B) 25 mm steel pendulum bob photogate timer with memory (Pasco ME-9215A) (± 1%) meterstick

The simple pendulum apparatus is illustrated in Figure 2. A steel ball is suspended by a string from a horizontal clamp near the top of a vertical rod which is in turn mounted on a tripod base. For the measurement and adjustment of the pendulum length, a meterstick and a metal slider are clamped on to the vertical rod. The effective pendulum length can be made shorter without pulling up the string by using the metal slider. The slider has a slot in it through which the suspension string passes; by positioning the slider, the length of the pendulum can be “cut off” to any desired value (in effect, the pendulum swings from the bottom of the slot of the slider). The period of the pendulum is measured directly by the photogate timer. The photogate is turned upside down on its support shaft and the height and position of the photogate is adjusted so that the pendulum ball is centred in the photogate: Ensure that the ball does not hit and damage the photogate! Set the photogate timer in “PEND” mode (short for “pendulum”) at 1 ms resolution (the time will be read in seconds, regardless of the resolution setting) with the memory “on.” Once the pendulum is swinging, you can press the reset button on the photogate timer and the timer will measure the time for one period (one complete swing of the pendulum).

Procedure Verify Equation (2) by means of a graph and determine the value of g from the graph. Measure, record and average a reasonable number of measurements of the period T for 6 to 8 different lengths L. As always, pick the lengths so that are approximately equally spaced and that span the largest possible range.

Figure 2

55

Experiment 5 COLLISIONS IN ONE DIMENSION

In this experiment you will investigate collisions between two gliders on an airtrack. Since all motion will occur along a straight line, the collisions are one-dimensional. The conditions under which the collisions between two gliders will be observed are: the airtrack is horizontal and frictionless, the gliders have unequal masses, and one of the gliders is initially at rest. What you will be specifically looking for are physical quantities that remain constant during a collision.

Inelastic Collisions Objective To compare (a) the total linear momenta of two gliders before and after an inelastic collision, and (b) the total kinetic energies of two gliders before and after an inelastic collision in order to verify the conditions for an inelastic collision. Theory Consider the collision of an initially moving glider (call it glider A with mass mA) with a second glider (glider B with mass mB) which is initially at rest, as shown in Figure 1. They hit and stick together. By applying conservation of momentum, we obtain the following expression for this inelastic collision:

fo ABBAAA vmmvm rr )( += (1) where

oAvr is the initial velocity of glider A and fABvr is the common velocity of gliders A and B

(since they stick together after the collision). By determining the quantities involved in Equation (1), you can determine if linear momentum is conserved for this collision.

vAo vBo= 0

mB

mB

mA

mAmA mB

fv BA

>

Figure 1 A completely inelastic collision

56

The kinetic energy before the collision is 221

oAAo vmKE =∑ (since glider B is initially at rest)

and after the collision it is 221 )(

fABBAf vmmKE +=∑ . If the total kinetic energy is not the same before and after a collision, then the collision is inelastic. By calculating and comparing the total kinetic energy before and after this collision, you can determine if this collision is inelastic. Apparatus student air track (Ealing) with air supply

2 gliders equipped with Velcro “tape couplers” flag (approximately 5 cm long) 2 accessory photogates (PASCO ME-9204A) LabVIEW interfacing system (CollisionTimer VI—times are ± 1%) Ohaus electronic balance (± 0.05 g) ruler

Procedure For this part of the experiment, the set-up will be as in Figure 2 with mA > mB. Glider A will be launched so that it collides with glider B, sticks to it and continues on.

Figure 2

Take all necessary measurements to determine the total linear momentum of the system before and after the collision, and the total kinetic energy of the system before and after the collision. Use the CollisionTimer VI LabVIEW program to measure the times needed to calculate the velocities of the gliders. The flag attached to the glider should interrupt the photogate beam and therefore the distance traveled in the time measured is the length of the flag. To get the best results possible, place the two photogates as close to each other as possible so that the collision occurs as soon as the flag on glider A completely clears photogate#1 and so that the gliders then go through the second photogate as soon as possible. Also, the launch velocity should be reasonable: a very slow velocity can cause air resistance to interfere with the motion, while a very high velocity produces a “hard” collision which makes the gliders wobble on contact and lose energy.

57

Record and do calculations for your first trial that seems valid. If the results do not work out as expected, then make adjustments and do more trials recording clearly each trial, what adjustments were made and comments about what happened in each trial.

Elastic Collisions Objective

To compare (a) the total linear momenta of two gliders before and after a collision, and (b) the total kinetic energies of two gliders before and after a collision in order to determine whether the collision is elastic. Theory Consider the collision of an initially moving glider (call it glider A) with a second glider (glider B) which is initially at rest, as shown in Figure 3. They hit and bounce apart.

vAo

vAf

vBo

fvB

= 0

mB

mB

mA

mAmA mB<

Figure 3 An elastic collision By applying conservation of momentum, we obtain the following expression for this elastic collision

ffo BBAAAA vmvmvm rrr+= (2)

where and are the initial and final velocities of masses A and B. By determining the quantities involved in Equation (2), you can determine if linear momentum is conserved during the collision.

fo AA vv rr ,fBvr

The sum of the kinetic energy before the collision is 2

21

oAAo vmKE =∑ (since glider B is initially at rest) and because the gliders bounce apart after the collision, the final kinetic energy is 2

212

21

ff BBAAf vmvmKE +=∑ .

If the total kinetic energy is the same before and after a collision, then the collision is elastic. By calculating and then comparing the total kinetic energy before and after the collision, you can determine whether or not this collision is elastic.

58

Apparatus student air track (Ealing) with air supply LabVIEW interfacing system (CollisionTimer VI—times are ± 1%)

2 gliders equipped with “bumpers” 2 flags (approximately 5 cm long) 2 accessory photogates (Pasco ME-9204A) Ohaus electronic balance (± 0.05 g) ruler Procedure The two gliders in this collision have spring “bumpers” on them. Make sure the bumpers are not loose (if they are, use a screwdriver to tighten them). Take measurements for the case where mA < mB, where glider A will bounce back after colliding with glider B (see Figure 4).

Figure 4

The procedure for finding velocities is the same as that used in Part A. This time, however, you will determine two velocities after the collision: that of glider A as it bounces back through photogate #1, and that of glider B as it moves through photogate #2 (a second flag is, therefore, mounted on glider B). Take all necessary measurements to determine the total linear momentum of the system before and after the collision, and the total kinetic energy of the system before and after the collision.

59

Experiment 6 COLLISIONS IN TWO DIMENSIONS

In this experiment you will investigate a collision in two dimensions between two pucks on an air table. The conditions under which the collision between the two pucks will be observed are: the air table is horizontal and frictionless, and the objects are both initially moving with constant velocity at an angle towards each other. Objective To compare the total linear momenta and the total kinetic energies of two pucks before and after a collision in two dimensions. Based on these results, it will be determined whether the collision is elastic or inelastic. Theory For an isolated system the sum of the external forces as well as the sum of the internal forces is zero and momentum is conserved: ∑∑ = fo pp rr (1) where is the vector sum of the momenta of the system before the collision and ∑ opr ∑ fpr is the vector sum of the momenta of the system after the collision. Since vmp rr

= where m is mass and v is velocity, we have r

∑∑ = fo vmvm rr (2) For a system consisting of two objects A and B which collide, Equation (2) becomes

ffoo BBAABBAA vmvmvmvm rrrr+=+ (3)

Since velocity and, therefore, momentum are vector quantities, Equation (3) can be rewritten in x and y component forms.

fyfyoyoy

fxfxoxox

BBAABBAA

BBAABBAA

vmvmvmvm

vmvmvmvmrrrr

rrrr

+=+

+=+ (4)

To show that momentum is conserved for a collision, each component equation must be verified.

60

For the collision shown in Figure 1, we can define the x and y directions, measure the angles of the paths of the pucks with respect to the x direction and hence determine the momenta in the x and y directions.

A before collision

A after collision B before

collision

B after collision

Figure 1 A two-dimensional collision

This is most easily done by defining the x direction as in Figure 2 and extending the paths of the pucks before and after the collision so that the angles of these paths can be measured relative to the x direction. Then the x and y components of the momentum for each puck before and after the collision can be calculated. Finally, the total momenta before and after the collision can be compared to each other to determine whether or not momentum was conserved for that collision.

Figure 2 Resolving velocities into components

Path of puck B

Biv

x Afv

θBi

θBfθ Af

θ Ai

Bfv

Av i

Path of puck A

61

Collisions can be classified as either inelastic or elastic. For collisions in which the total kinetic energy of the system is the same before and after a collision (i.e., remains constant), the collision is said to be elastic. Otherwise, it is said to be inelastic. Therefore, by calculating the sum of the kinetic energy KE before the collision,

2212

212

21

oo BBAAoo vmvmvmKE +== ∑∑ , and the sum of the kinetic energy after the collision, 2

212

212

21

ff BBAAff vmvmvmKE +== ∑∑ , and comparing the two, we can determine whether a collision is elastic or inelastic. Since kinetic energy is a scalar quantity, it is not necessary to resolve it into components to compare the total kinetic energy before and after the collision. Apparatus Ohaus electronic balance (± 0.05 g)

ruler, protractor, masking tape and paper (carbon and newsprint) SRP Air Table (F-249) with air supply, pucks, spark timer (measured times have negligible (zero) uncertainty) Procedure

Warning! Hold the pucks by the plastic parts only to prevent the possibility of electric shock.

1) Set up the air table and place a piece of carbon paper on it (black side up) and on top of that, a piece of newsprint paper. Tape down the corners of the paper. Level the air table. If the table is not level, there will be an external force acting on the pucks and the conservation laws will not apply. Place a puck on the air table near the centre, turn on the air supply and adjust the feet of the table until the puck stays motionless or hovers around its position. 2) Measure the mass of each puck (or use the labelled masses) and place them on the air table. Set the period of the spark timer to a convenient value (50 ms works well for a moderately fast collision). 3) Turn on the air supply and practice launching the pucks towards each other (for example, you can launch a puck from one corner of the air table while your partner launches the other puck from the opposite corner) with various angles and speeds until you are confident that you can produce a collision near the centre of the air table (where frictional effects of the pucks with the sides of the track, etc. are minimised). Then turn on the spark timer and, while stepping on the foot pedal for the entire collision but no longer, launch the pucks and produce a collision. The resulting sparks will leave a trail of dots on the underside of the paper. Remove the paper from the air table, but make sure you record which paths belong to mA and mB before doing so!

62

Note: The launch velocity should be reasonable. A very slow velocity can cause air resistance to interfere with the motion, while a very high velocity produces a “hard” collision which makes the pucks wobble on contact.

4) The velocities before and after the collision can now be determined. Remember that the velocity of a puck, once it is set in motion, remains constant when no external force acts on it. Therefore, the velocity of puck A just before it collides with puck B is equal to its average velocity which is found by timing its motion over a known distance. The time t is found by counting the number of dots over a distance d (where the dots are evenly spaced) and multiplying this number by the elapsed time between the sparks that created the dots (the time setting on the front panel of the spark timer). Measure the distance d with a ruler. The puck's velocity is then

determined using tdv = . Make sure you assign each velocity the correct sign relative to the way

you defined your axes. Take all necessary measurements to calculate the values of the total linear momentum of the system in the x and y directions before and after the collision, and the total kinetic energy of the system before and after the collision.

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Experiment 7 UNIFORM CIRCULAR MOTION

The simplest kind of circular motion is uniform circular motion, that is, the motion of an object that has a constant tangential speed in a circular path. This type of motion is due to a net force that acts on the object and is directed towards the centre of its circular path. In this experiment, you will verify the mathematical expression for this net force. Objective To experimentally determine the net force acting on a trolley moving in a circular motion at constant speed and compare it to the expected value of this net force. Theory An object in uniform circular motion (i.e., in motion with constant tangential speed along a circular path) is accelerating toward the centre of its circular path. The magnitude of this acceleration is given by

rva

2

= (1)

where v is the object's constant tangential speed and r is the radius of its circular path. Newton's Second Law, ΣF = ma, tells us that a net unbalanced force is the cause of the object's acceleration. The magnitude of this force is given by

r

mvmaFnet

2

== (2)

where m is the mass of the object. The direction of the net force is also toward the centre of the circular path since it is this force that keeps the object moving in its circular “orbit.” The trolley moves a distance equal to the circumference of its orbit, 2πr, during T, the time of one revolution, so its tangential velocity v is:

T

rv π2= (3)

Substituting Equation (3) into (2), the net force is:

22

⎟⎠⎞

⎜⎝⎛=

TmrFnet

π (4)

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Apparatus rotational apparatus with accessories (Pasco ME-9341) wooden support stand photogate timer with memory (Pasco ME-9215A) (± 1%) Ohaus electronic balance (± 0.05 g) bubble level, cardboard flag, ruler, tape

Figure 1 Side View of Apparatus The base is placed on a support stand so that the suspended mass will hang freely below the stand. One end of the string is attached to the trolley and the other end of the string is dropped over the pulley in the centre of the track and through the hole in the track (and in the spindle) so that the string hangs below the apparatus. The trolley is placed on the track so that it rolls freely along the track. A slip knot should be used to hold the hanger to the string. Around 30–40 g is suitable for total trolley mass, while around 20–30 g is suitable for the total hanging mass. The pin on the mounting arm of the motor assembly is inserted into one of the bushings (holes) near the side of the base. The AC adapter plugs into the motor's speed control unit. To make the motor rotate the platter, the motor drive shaft must be held against the edge of the platter with a rubber band placed around the arm of the motor drive assembly and around one of the thumbscrews on the base of the rotational apparatus. The photogate clamp is inserted into the other bushing near the side of the base. The photogate clamp holds the photogate. A piece of cardboard taped on top of the main platter serves as a flag to trigger the photogate. Set the timer to Pulse mode with memory “on” and at 1 msec resolution (check the bottom of the timer if the switch is not on the top). Note: the photogate

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timer reads in seconds, regardless of the resolution setting. Position the photogate so that the cardboard flag on top of the platter will interrupt the infrared beam at each revolution of the platter. To measure the period of revolution T, press the reset button on the photogate timer to clear the memory and get it ready for the next measurement. Finally, the base can be levelled by adjusting the three levelling feet. Levelling the base ensures that, during the experiment, the circular path traversed by the trolley is horizontal and the suspended string does not touch the inside of the spindle resulting in friction between the string and the spindle. Also, a levelled base prevents uneven wear on the bearings.

Figure 2 Overhead View of apparatus

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Procedure 1) Before coming to the lab, show that the net force acting on the trolley is equal to the weight of the hanging mass by using free-body diagrams (assume zero friction) and by applying Newton's Second Law. On the free-body diagram for the trolley, be sure to indicate the direction toward the centre of rotation. 2) Take all necessary measurements to determine the net force acting on the trolley according to Equation (4). The most challenging part is to measure the radius of rotation r (as it is impossible to take a good measurement of the radius while the trolley is rotating). Devise a method to precisely measure the radius of rotation, but please never write on the equipment! Record in detail how you measured r along with your data.

Important:

Before you turn on the motor, read and make sure you understand the

following procedure; otherwise, you might cause the trolley and its added mass to “fly out” and become dangerous projectiles!

i) Turn the speed adjust knob fully counter clockwise to its minimum

position before the motor is plugged in. ii) Increase the motor speed very gradually while supporting the hanging

mass with your other hand (so that the hanging mass will not be pulled up abruptly when the static friction between the trolley and track is overcome). Adjust the speed so that mass is pulled upward slowly.

iii) Finely adjust the motor speed until the string can be let go of and the mass

remains freely suspended. At this point, the period of revolution can be measured.

3) Calculate the expected value of the net force.

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Experiment 8 PROJECTILE MOTION

Laboratory problems are often simplified when considered from an energy perspective rather than from an application of kinematics and Newton's Laws. In this experiment, energy methods (i.e., the conservation of mechanical energy and the work-energy theorem) will be applied to the motion of a steel ball rolling down through, and then projected from, a tube to the floor. Objective To predict (through energy considerations) and measure the range of a steel ball after it is projected from a tube horizontally and hits the floor. Theory Consider a steel ball placed at the release (top) end of a launch tube which is adjusted so that its exit end is at zero degrees with the horizontal (Figure 1). We can determine the total mechanical (potential plus kinetic) energy of the ball at its initial and final positions.

h2

h1

release end

launch tube

exit end

elevation screw

plumb bob

mounting board

floor Figure 1 Launching a projectile horizontally

Before the ball is released from an initial height of h1 above the floor, its gravitational potential energy, PE1 = mgh1, where m is the mass of the ball and g is the acceleration due to gravity. Its kinetic energy at h1, is equal to zero, i.e., KE1 = 0 since the ball is released from rest. As the ball rolls down the tube (and its height above the floor decreases), its gravitational potential energy decreases. At the same time, its kinetic energy increases, i.e., gravitational

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potential energy has been converted into kinetic energy. Moreover, as the ball rolls down, it is consistently in contact with the tube surface and thus “feels” a friction force, so some gravitational potential energy is also lost as work done by friction (i.e., energy was used to overcome friction). The exit end of the tube is at a height h2 above the floor, so the gravitational potential energy of the ball at the exit is given by PE2 = mgh2. The ball rolls (we assume without slipping) from h1 down the tube to height h2 where it is still rolling so the ball's kinetic energy at h2 is a sum of the kinetic energy of translation of its centre of mass, KEtr, and the kinetic energy of rotation about its centre of mass, KErot

2222

2

107

51

21 mvmvmvKE

KEKEKE rottr

=+=

+=

where v is the linear speed of the centre of mass of the ball as it leaves the tube.

The KErot term is derived as follows: 2

21 ωIKErot = (a)

The moment of inertia I for a uniform sphere of radius r about an axis through its

centre of mass is 2

52 mrI = (b)

The condition for rolling without slipping is v = ωr

or ω = vr (c)

Substituting (b) and (c) into (a) for I and ω,

22

2

51

52

21 mv

rvmrKErot =⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

With height h2 as the final position and h1 as the initial, we apply the Work-Energy Theorem and obtain: )()( 1122 PEKEPEKEWf +−+= (1) where Wf is the work done by the friction force. Substituting into Equation (1) the expressions for the kinetic and potential energies found thus far:

)0(107

122 mghmghmvWf +−⎟

⎠⎞

⎜⎝⎛ +=

or, ) (2) (7.0 122 hhmgWmv f −−=

From Equation (2) a mathematical expression for the value of v, the horizontal velocity of the ball as it leaves the tube, can be obtained. However, we first need to determine the value of Wf.

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The value of Wf is estimated by first assuming that Wf is approximately constant for the ball rolling through the entire length of the tube. Next, the tube can be adjusted using the elevation screw such that the ball reaches h2f with zero velocity (i.e., after release from h1f, the ball just reaches the exit end of the tube at height h2f, stops, then rolls back). In this situation, all the gravitational potential energy is lost as work done by friction. Applying the Work-Energy Theorem again: )0()0( 12 fff mghmghW +−+= or, )( 12 fff hhmgW −= (3) where the two zero terms correspond to the kinetic energies at heights h2f and h1f (recall that the tube was adjusted such that the velocity of the ball was zero at both heights). Substituting Equation (3) into Equation (2) and solving for v:

[ ])()(7.0 2121 ff hhhhgv −−−= (4)

The horizontal velocity v of the ball depends only on the difference between the height measurements and the value of g. The velocity v is now used to predict the range of the ball. The kinematic equation for the horizontal position of the ball is:

2

21 tatvxx xoxo ++= (5)

We assume that the effect of air resistance on the ball is negligible; hence, no horizontal forces act on the ball while it is in flight so the ball has no horizontal acceleration (ax = 0), and its horizontal velocity stays constant (vox = v). As well, if the horizontal displacement of the ball is measured from the exit end of the tube to the point of impact on the floor, we can set xo = 0 and let x = Rth, the (theoretically determined) range. Equation (5) then becomes: Rth = vt (6) where t is the time of flight of the ball. The time t is calculated in terms of the ball's vertical displacement from the exit end of the tube to its point of impact on the floor. The kinematic equation for the ball's vertical position is:

2

21 tatvyy yoyo ++= (7)

Recall that the vertical component of the ball's motion is identical to the motion it would have if it were dropped from rest from the same height above the floor. A ball dropped from rest has an initial velocity voy = 0 and an acceleration ay = g. Figure 2 is a diagram of the ball's flight path.

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v

h2 - rh2

R

plumb bob

floor

exit end

Figure 2 The flight path of the ball

If we take the exit end to be yo = 0 and only look at the motion of the centre of the ball, then the centre of the ball will fall vertically to y = (h2 - r) where r is the radius of the ball. Substituting these values into Equation (7) and solving for the time of flight t :

g

rht )(2 2 −= (8)

Equations (4), (6) and (8) are used to predict the range of the ball. Apparatus projectile tube apparatus carpenter's bubble level tape measure digital calipers (± 0.03 mm) carbon paper, blank paper ball bearing masking tape plumb bobs The projectile tube apparatus consists of a 2 meter long aluminum tube attached to a mounting board (see Figure 1). At each end of the tube is a small screw that can be used to hang a plumb bob from to ensure good vertical measurements of the heights. Also, the centre of each screw marks the approximate centre of mass of the ball at the corresponding ends of the tube. Procedure 1) Take all necessary measurements to determine the predicted theoretical range Rth before you actually project the ball from the projectile tube. 2) Determine the experimental range Rexp by launching the ball from the projectile tube. To mark where the ball hits the floor, a piece of paper can be centred on the predicted impact point and a layer of carbon paper placed on top. Hold the papers in place with masking tape. Take several trials and record all valid trials.

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Experiment 9 MOMENT OF INERTIA

In this experiment, you will solve a problem involving the rotation of a rigid body about a fixed axis. In particular, you will apply an experimental technique to determine the moment of inertia of the rotating body. Objective To determine the moment of inertia of three different objects (a disk, a ring and a bar) experimentally and compare them to calculated theoretical values. Theory Consider the system shown in Figure 1. A main platter with a step pulley attached to it is mounted over a spindle that is slid into the bushing of a base (recall the rotational apparatus used in the Uniform Circular Motion experiment). A cord is wound around the smallest step pulley of radius r and a hanging mass mh is attached to the other end of the cord over a pulley. When the hanging mass falls, a constant tension force T arises in the cord due to the weight of the hanging mass. This force T applies a torque τ on the step pulley, causing the step pulley (and, therefore, the system) to rotate with an angular acceleration α.

Figure 1 Set-up of system

From rotational kinematics, one of the equations which describes rotational motion with constant angular acceleration α is

72

tif αωω += (1) where t is the time during which the system's angular velocity changes from ωi to ωf. The graphical solution to finding the value of α predicts that if the measured values of ωf are plotted on the y-axis, and the corresponding values of t are plotted on the x-axis, the graph will be a straight line whose y-intercept is equal to the value of ωi and whose slope yields the value of α. Applying Newton's Second Law to the linear motion of the falling mass mh, we get:

amTgm

maF

hh

yy

=−

=Σ

If we substitute in the relationship between the linear and the angular acceleration, a = rα (where r is the radius of the step pulley that the cord is wound around), and then solve for the tension T we get: )( αrgmT h −= (2) Now apply the rotational analogue of Newton's Second Law to the rotating system:

αατ

ITrI

==Σ

(3)

where r is the moment arm of the applied force T. In this experiment, the cord will be wound around the smallest step pulley, therefore, the moment arm r is the radius of the smallest step pulley (see Figure 2).

step pulleyTr

Figure 2

Substituting Equation (2) into Equation (3) and rearranging, we obtain an expression for the moment of inertia of the system (where the system consists of either the apparatus by itself or the apparatus with a load on it):

2rmrgmI hh −=α

(4)

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First, the moment of inertia of the rotating apparatus without a load is determined. Then the moment of inertia of the same rotating apparatus with a load is determined. The difference between these two moments of inertia is the moment of inertia of the load by itself. The experimental value for the moment of inertia of the load will be compared to the theoretical value computed from the mass and linear dimensions of the load. The theoretical values for the moment of inertia of each object are found with the following formulas:

a) ( )22

21

ioringring RRMI += (5)

where Mring is the mass of the ring, Ro is its outer radius, and Ri is its inner radius.

b) ( )22

121 WLMI barbar += (6)

where Mbar is the mass of the bar, L is its length, and W is its width.

c) 2

21 RMI diskdisk = (7)

where Mdisk is the mass of the disk, and R is its radius. Apparatus rotational apparatus (Pasco ME-9341) with attached 1 meter length of cord LabVIEW interfacing system (Picket Disk VI—slopes are ± 2%) Ohaus electronic balance (± 0.05 g for m<800 g, ± 0.5 g for m>800 g) 5 g mass hanger and 60 g in brass masses accessory objects: ring, bar, disk picket disk table clamp and pulley digital calipers (± 0.03 mm) bubble level ruler The cord will be wound around the smallest step pulley. The spindle is inserted into the base with the long end facing up, and then the main platter is placed over the spindle with the step pulleys facing down. A “picket disk” object is mounted on the top of the main platter. The base can be levelled by adjusting the three levelling feet. A clamp holding a pulley is mounted on the edge of the table. The cord is passed over the pulley, and the mass hanger is attached to the end of the cord (around 65g total is suitable for mh). The photogate is mounted and adjusted as in the Uniform Circular Motion lab. The photogate along with the LabVIEW program Picket Disk VI times the rotation of the system as the photogate is blocked and unblocked by the rotating picket disk and measures the angular acceleration α. As noted in the Theory section, the angular acceleration α of the system is the slope of the graph of angular velocity ωf versus time t.

74

Procedure 1) Take all necessary measurements to determine the moment of inertia of the apparatus by itself and of the apparatus with each object (i.e., ring, bar and disk) on it according to Equation (4). From these results, calculate the experimental moment of inertia of each object by itself. Use the Picket Disk VI to measure the angular acceleration α of the system. You can quickly tell from the resulting graph whether the system rotates with constant angular acceleration or not. If it does not, readjust the system and try again. Repeat the measurement of α to make sure that it is consistent and then measure and record α. To get good results, it is important to eliminate, as much as possible, all sources of friction. In particular, make sure the string is not rubbing on any parts of the apparatus or the pulley, etc. 2) From the measurements of the masses and dimensions of the ring, bar and disk, calculate the theoretical moments of inertia of each object.

75

Experiment 10 THE PHYSICAL PENDULUM

A rigid body that is suspended from a point other than its centre of mass will oscillate when displaced from its equilibrium position. Such a system is called a compound or physical pendulum. In this investigation of the physical pendulum, we will make use of several concepts such as torque, moment of inertia and simple harmonic motion. Objective To compare a) the periods T for a physical pendulum that is suspended from different points from experiment and from theory, and b) the minimum period Tmin of a physical pendulum from experiment and from theory. Theory Figure 1 is a diagram of the physical pendulum for this experiment. It consists of a rectangular bar and a cylindrical mass which is attached at the centre of the bar. The pendulum has a total mass m and is suspended through the point S which is at a distance h from C, the centre of mass of the pendulum. When the pendulum is displaced through an angle θ as in Figure 1(b), its weight mg (which is shown drawn through the centre of mass) exerts a restoring torque of magnitude mghsinθ about S. The angular acceleration α of the pendulum is related to the torque τ by

θθατ sin2

2

mghdtdII −=== (1)

where I is the moment of inertia of the pendulum about the axis through the point of suspension S. For small angular displacements (i.e., θ < 10º), we can obtain from Equation (1) that the period T of the physical pendulum is

mgh

IT π2= (2)

(See the physics text by Tipler for a detailed derivation of this.)

h

C

mg

θ

(a) (b)

Sh

C

S

Figure 1

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Using the parallel-axis theorem, the moment of inertia I about an axis through S can be expressed as I = Io + mh2 (3) where Io, the moment of inertia of the pendulum about an axis through its centre of mass, is actually a sum of the moments of inertia of the individual parts of the pendulum, i.e., Io = Ibar + Icylinder (4) Ibar and Icylinder are found using the expressions for the moments of inertia for a solid bar and a cylinder (in reality, both objects are not actually solid—they have had holes either drilled in or cut out of them—we are assuming that the missing material has a negligible effect on the moment of inertia of the pendulum). Substituting Equation (3) into (2), we get

mgh

mhIT o2

2 += π (5)

Equation (5) expresses the period T as a function of h alone since Io and m are both constants for the pendulum. By applying calculus to Equation (5), the theoretical value of the minimum period Tmin is found to be

4/1

min22 ⎟

⎠⎞

⎜⎝⎛=

mI

gT oπ (6)

Thus, according to theory, Tmin depends only on the mass and dimensions of each part of the pendulum. Apparatus physical pendulum (Cenco 75037) photogate timer (Pasco ME-9215A) (± 1%) accessory photogate timer (Pasco ME-9204A) with tall stand ruler, meterstick and tape measure Ohaus electronic balance (± 0.05 g for mass < 800 g and

± 0.5 g for mass > 800 g)

77

Procedure

When changing suspension points, lift the pendulum off the knife edge; do not drag it as this will dull the knife edge. Be careful as this pendulum is quite heavy and can be dangerous if it falls.

1) Take the physical pendulum apart and measure the masses (each part must be put on the scale separately or the scales will be damaged; attach the bolt and washer to one part of the cylinder before you measure its mass) and dimensions of the bar and the cylinder. Take the two orange pieces to comprise one cylinder for the calculations. Calculate Io for the physical pendulum. Calculate the theoretical Tmin for the pendulum. 2) Using a meterstick, measure the distances h from the centre C to each of the possible points of suspension on one half of the bar only (assuming the cylinder is in the centre of the bar, the periods will be the same if measured at the holes on the other side of the bar, therefore, we will omit the measurements of the other side of the bar). Note: measure to the point of suspension, not to the centre of the hole. 3) For each h, calculate I and the theoretical period T of the pendulum. 4) Reassemble the pendulum and for each h, measure the period T as follows: Support the pendulum on the knife edge of the bracket (if the protective rubber sleeve is over the knife edge, remove it first!) at a hole on one end of the bar of the pendulum, making sure it swings freely (i.e., it is not rubbing or touching the inner spindle). Set the photogate timer in Pend mode (short for “pendulum”) at 1 ms resolution with the memory “on.” Set up the photogate at the bottom end of the pendulum (check that the pendulum does not cut the photogate beam at a hole). Pull the pendulum to the side (keep the displacement angle small) and release. Make sure the pendulum does not hit the photogate as it will damage it. The timer will measure the period T (one complete swing of the pendulum). For each h, repeat the measurement of T a few times to make sure it is consistent, and then record a single value of T. Compare each theoretical T to the corresponding measured T. 5) Draw a graph of T (measured) vs. h. Determine the experimental Tmin from the graph and record and label it directly on the graph (include its units and uncertainty). Compare this to the theoretical Tmin.

78

Experiment 11 RESONANCE OF AIR COLUMNS

You have probably seen how a system such as a stretched string can be made to resonate under certain conditions. In this experiment, you will look at another system where resonance can also occur—a column of air. On the string you see standing waves, but in this experiment, you will hear the standing sound waves set up in the air column. Much of what you have learned about the resonance of a string can be applied to the resonance of an air column. In fact, the difference in the standing wave conditions for the two systems results only from a slight change in the “boundary” conditions of the medium used. While the string is fixed at both ends, air in this system is confined to a long tube which is closed at one end but open at the other. Measurements made with resonating air columns can lead to a determination of the speed of sound in air. Objective To verify the linear relationship between the length and the resonant frequency of an air column in a closed tube and to determine the speed of sound in air. Theory Figure 1 shows a narrow vertical tube partially filled with water. Since the water level can be raised or lowered by moving the reservoir up or down, the length L of the air column above the water surface can be adjusted. Such a tube is known as a “resonance tube.”

metric scale

resonance tube

L reservoir

Figure 1 The resonance tube

79

A vibrating tuning fork placed at the top end of the tube creates a disturbance in the pressure and density of the surrounding air. The disturbance results in a series of high and low pressure regions called compression and rarefaction, respectively, that propagate through the air in the tube, i.e., sound waves are generated. As a wave passes through the column of air, the air molecules vibrate back and forth about their equilibrium (undisturbed) positions along the line of propagation of the wave. This type of wave is called a longitudinal wave. A schematic representation of a longitudinal wave is shown in Figure 2. A sinusoidal graph is superimposed to show the analogy of this wave to a transverse wave. The descriptive quantities for the velocity, wavelength and frequency of transverse waves also apply to longitudinal waves. (Note: The tuning fork is assumed to oscillate with simple harmonic motion, therefore, the resulting disturbance will also be harmonic.)

vibrating source

compression compressionrarefaction

sound wavev

Figure 2 Longitudinal wave with Transverse wave (sine curve) superimposed

Standing longitudinal waves are set up in the tube of air in a manner similar to the production of standing transverse waves in a stretched string. The disturbance generated by the tuning fork at one end of the tube is propagated along the column as a longitudinal wave and is reflected back, like an echo, from the other end. The length of the air column is adjusted so that the reflected waves are constantly being reinforced by the waves from the vibrating tuning for, i.e., the waves add in phase and standing waves are produced. This condition of resonance can be detected by ear since the sound intensity is increased. The tube or air column in this experiment is referred to as a closed tube since it is closed at one end and open at the other. The standing wave conditions for this system are analogous to that of a string with one end fixed and one end free. The bottom end of the air column (Figure 1) is closed by the water level. Since water is more dense than air, the water surface represents a rigid wall (it's as if the air molecules are so greatly confined at this end that they cannot vibrate). Therefore, the closed end of an air column is a displacement node, just as the fixed end of a stretched string is a displacement node. (As a result, the reflected wave is 180° out of phase with the incident wave.) The top end of the tube is open to the atmosphere and the air molecules have

80

complete freedom of motion, i.e., a maximum amount of to-and-fro motion. Consequently, the open end of an air column is a displacement antinode. (A compression reaching the open end expands from a confined region into an open region while a rarefaction reflects back into the tube.) For a closed tube, there must always be a displacement node at the closed end and a displacement antinode at the open end. With this restriction and remembering that two adjacent nodes (or two adjacent antinodes) are spaced one half wavelength apart, we can draw diagrams of the resonant modes of vibration of an air column, and determine how the wavelength λ for any given mode relates to the length L of the air column. Figure 3 shows the first three such resonant modes. Remember that the frequency f of the sound waves in the air column is the same as the frequency of the particular tuning fork used.

λ /4

3(λ/4)

5(λ/4)

L

L

L

(a)

(b)

(c) Figure 3 Resonant modes in an air column

The shortest length of the air column for which resonance occurs (corresponding to the fundamental mode of vibration) “just” fits one quarter of a wavelength—the distance from the

antinode at the top end to the first node. That is, 4λ

=L (Figure 3(a)).

The next higher mode occurs when the length is increased to fit three quarters of a wavelength,

⎟⎠⎞

⎜⎝⎛=

43 λL , as in Figure 3(b). In Figure 3(c), resonance occurs again when ⎟

⎠⎞

⎜⎝⎛=

45 λL .

Thus, for an air column of length L in a closed tube, the condition or requirement for the production of standing sound waves is given by:

⎟ , for n = 1, 3, 5,… (1) ⎠⎞

⎜⎝⎛=

4λnL

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Since λvf = , Equation (1) can be rewritten to give the resonant frequencies of the air column:

⎟⎠⎞

⎜⎝⎛=

Lvnfn 4

, for n = 1, 3, 5,… (2)

In deriving Equations (1) and (2) it was assumed that the displacement antinode occurs exactly at the open end of the tube—this is not strictly true. When a compression reaches an open end, it does not reach full rarefaction (expansion) until it passes somewhat beyond the end. That is, the displacement antinode actually occurs a short distance dc beyond the plane of the open end. Therefore, this distance dc is an end correction that must be added to the length L in Equations (1) and (2) above:

⎟⎠⎞

⎜⎝⎛=+

4λndL c , for n = 1, 3, 5,… (3)

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=)(4 c

n dLvnf , for n = 1, 3, 5,… (4)

(This end correction dc is of the order of the radius of the tube.) To determine the speed of sound v, we shall use the shortest length of the resonating air column; i.e., n = 1. For n = 1, Equation (4) becomes:

)(4 cdL

vf+

= (5)

or rearranging,

cdf

vL −⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

14

(6)

Comparing Equation (6) with y = mx + b, suggests that if L is plotted on the y-axis, and 1f on the

x-axis, a straight line graph will be obtained with a slope equal to v4 and with a negative y-intercept equal to the end correction dc.

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Apparatus a set of 7 tuning forks (125, 250, 320, 500, 1000, 2000, and 4000 Hz, ± 1%) resonance tube apparatus (Sargent-Welch #3308) thermometer, 1 L beaker, rubber mallet Procedure 1) Fill the cup (in its highest position) just until the water in the tube just about reaches the top. The water level in the resonance tube is adjusted by sliding the reservoir (aluminium) cup up or down the iron stand (press on the clips first). Use the metric scale mounted beside the resonance tube for length measurements. Measure the radius of the tube. 2) Start a tuning fork vibrating by striking it with the rubber mallet. Do not strike the tuning forks against hard objects (like table tops) because the tuning forks are made of a soft aluminium alloy so they are easily damaged. The vibrating tuning fork should be held so that its prongs vibrate in a vertical plane over the top of the tube (see Figure 1). 3) For each of the tuning forks, determine the shortest air column lengths L for resonance in the air column (i.e., n = 1). At resonance, the amplitude of vibration at the open end of the tube increases to a maximum. This is heard as an increase in the sound intensity of the tuning fork. It may be necessary to bring your ear close to the open end of the tube in order to distinguish “your own sound” from the many other sounds in the room. 4) The temperature of the air inside the tube should be read frequently to detect any changes while the experiment is going on.

Optional: Use the 500 Hz or the 1000 Hz tuning fork to verify the standing wave condition in a closed tube (Equation 3).

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Experiment 12 CANIMALS

* borrowed with thanks from Malaspina College The physical size of birds and animals has a significant effect on their energy requirements and, therefore, on their necessary food intake. We will consider the energy loss due to radiation from the skin. Metal cans (“canimals”) of different sizes, filled with warm water, are used to model small warm-blooded animals. The internal temperature is recorded at uniform time intervals and a cooling curve is plotted. The daily energy requirement for each is found from the slope of the cooling curve at normal body temperature and the mass and specific heat of the canimal. Conclusions are drawn concerning food requirements relative to body mass for the different sized canimals, and the existence of a lower limit on the size of real warm-blooded animals. The physical environment in which birds and animals live also has a significant effect on their energy requirements and, therefore, on their necessary food intake. In warm dry weather, very little energy is expended in maintaining body temperature. In fact, staying cool enough may be a problem. In cold or wet weather, though, just finding enough food to counteract body heat losses is a major concern. A single size of canimal is subjected to various environmental conditions, with and without the addition of ‘fat’ and ‘fur’. The internal temperature is recorded at uniform time intervals and a cooling curve is plotted. The heat loss rate in each case is found from the slope of the cooling curve at normal body temperature and the mass and specific heat of the canimal. Conclusions are drawn from these results concerning the effect of environmental conditions and the presence of heat-loss reducing materials. Objective a) To compare heat losses in different sized and shaped bodies, b) To compare heat losses under varying environmental conditions, c) To determine the effect of fur and body fat on the heat loss in a body, and d) To determine the amount of food needed for survival of various shaped bodies

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Theory The Stefan-Boltzmann law states that the rate of radiant heat loss or gain R (in watts) between a body and its surroundings is given by

)( 44ATTkA

tQR −=

∆∆

=

where k is a constant which depends on the nature of the radiating surface of the body, A is the radiating area of the body, T is the temperature (in Kelvins) of the body, and TA is the temperature (in Kelvins) of the surroundings (the ambient temperature) This may be written as R = kA(T - TA)(T3 + T2TA + TTA

2 + TA3)

If the difference between T and TA is small compared to both T and TA, the equation reduces to approximately R = 4kATA

3(T - TA) If TA remains steady during the heat exchange, the equation may be simplified even further to R = KA(T - TA) (1) where the constant K = 4kTA

3. It can be seen from this equation that the rate of radiant heat loss or gain depends on the radiating area and the difference in temperature between the body and its surroundings. This is often referred to as Newton’s law of cooling, and applies properly to radiant energy only. If convection and conduction losses are relatively small, the equation is still valid, but the constant K will include convection and conduction factors as well as the radiation factor, k. If a warm body is placed in a cool environment, it will lose energy to the environment. The loss of energy from the body is at the expense of the thermal energy stored in the body. The rate of conduction, convection and radiation heat loss (in watts) is given by

tTmc

tQR

∆∆

=∆∆

=

where m is the mass of the body, c is its specific heat capacity, and ∆T/∆t is the cooling rate. We must take into account that the body is not made entirely of water. To correct for the fact that the canimal is not made entirely of water, this equation can be written as

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tTcmcmR waterwatercancan ∆

∆+= )( (2)

where mcan is the mass of the can,

mwater is the mass of the water, ccan is the specific heat capacity of the can (ciron = 0.452 J/g0C), cwater is the specific heat capacity of water (cwater = 4.186 J/g0C), and

∆T/∆t is the cooling rate Apparatus set of canimals with lids and bases stirrer

electronic balance (± 0.05g) thermometer (± 0.05°C) stopwatch ruler body fat (foam fitted to inside of can) 4 fur (wool sock rings put on outside of can) large bath of heated water kept at 42°C plastic pitchers 2 room temp water baths in white rectangular plastic containers 2 fans

These (11) canimals must be used to be able to answer all the questions: 4 different sized cans (bare, no fat or fur, no wind or water) #2 sized can (bare with wind) #2 sized can (with dry fur; no wind) #2 sized can (with wet fur; no wind) #2 sized can (with dry fur and wind) #2 sized can (with wet fur and wind) #2 sized can (bare in room temp water bath) #2 sized can (with fat in room temp water bath) Procedure Note: This experiment is done as a class exercise. Each pair of students takes measurements for a

certain canimal under certain conditions and calculates the cooling rate tT

∆∆

at body temperature,

38°C. The results of the whole class are used to calculate the rate of heat loss R and the other relevant values to answer questions and derive conclusions. If the class is small, you may be asked to do two canimals. 1) For the canimal(s) assigned to you:

a) Record the number of the canimal (it is marked on the outside of the bottom), the mass of the empty can mcan, , and the diameter dcan and height hcan of the canimal. Put fur or fat on if required. If wind is part of your environment, place the fan so that it is 50 cm from the canimal, blowing straight toward it. The air path must be unobstructed, both behind and in front of the fan.

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b) Record the ambient (room) temperature. c) Insert the thermometer into the lid so that its tip will be about two thirds of the way down

into the water when the lid is in place. Insert the stirrer into the lid. The straight edge of the stirrer should be adjacent to the thermometer.

d) Fill the canimal to the line (inside) with water which is about 400C. e) Place the filled canimal on the base pad and set the lid assembly on the top. f) Stir the water throughout the run (but don’t continuously hold onto the can!). Record the

temperature at 60s intervals (or less if necessary; for the faster ones, you may have to go in 30s or less intervals) while it falls from 39 to 370C. (Be prepared to repeat your run if the your isn’t linear within uncertainty.)

g) If fur was used, remove it now. Record the mass of the bare canimal plus water (plus fat,

if it was used) mcan plus water. If body fat was used, one more mass is needed: Remove the fat and add water until the level is again up to the line and record the new mass of the canimal without fat.

h) Plot a cooling curve, i.e. a graph of temperature (y-axis) vs. time (x-axis). Once you have

confirmed that the graph is linear, find the slope of the graph. This slope will be referred

to as “the cooling rate at body temperature, 38 ºC.” tT

∆∆ (Note: this slope can be different

if the experiment were performed at a significantly different temperature because the cooling curve is in fact a curve. For our small range in temperature, we assume that the slope changes slowly enough that we can take it as a constant.) Express the cooling rate in ºC/s.

Correction for body fat only:

If body fat was used, a correction to the cooling rate must be made. The fat occupied volume which would have normally been occupied by water. Thus a canimal with internal fat contains less water than one without fat, and it will cool more quickly than it would have if it were full of water. The cooling rate of a “fat canimal” must therefore be multiplied by the ratio of the mass of the water in a fat canimal to that of one without fat (a number slightly less than one)

g) step (fromfat without canimal of massfat with canimal of mass

in order to account for the internal volume occupied by the fat. This adjusted cooling rate should be used in the next step.

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2) For each of the different canimals in the class, record mcan, mcan plus water, hcan, dcan and tT

∆∆ (the

“cooling rate at body temperature, 38 ºC” (in °C/s)–make sure you use the corrected tT

∆∆ for fat

canimals). For each canimal, calculate:

a) the rate of heat loss R (using equation (2)) in J/s or watts, b) the area to mass ratio of the canimal A/m (where m is the total mass)–recall that the area

of a cylinder is A = πdh, c) the radiation constant K where T is the body temperature, 38 ºC (using equation (1)), d) the daily energy loss in kJ, e) how many peanuts (in grams) would have to eaten daily in order to maintain its body

temperature assuming that the canimal’s diet consisted solely of Planters™ dry-roasted peanuts (listed on the jar as 790 kJ per 30 g serving),

f) the fraction (%) of its body mass (in peanuts) would it have to consume each day in order

to maintain its body temperature? A table such as this may help you to organize your data and calculations:

canimal mcan mcan

plus water

hcan dcan

tT

∆∆

R A/m K daily

energy loss

mass of peanuts required per day

% of its body

mass (in peanuts) required

#1 #2 #3 #4 #2 with wind #2 with dry fur and no wind

#2 with wet fur and no wind

#2 with dry fur and wind

#2 with wet fur and wind

#2 bare in water #2 with fat in water

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3) Based on the class results for the different sized canimals, answer the following questions: Can you see any trend in the value of the radiation constant K? Should there be one? What physical aspect of the canimals does K depend on? Identify and comment on any correlation between the area to mass ratio and the cooling rate. What does this suggest about the shape that a resting animal should assume in cold weather? In hot weather? Why? Identify and comment on any correlations among the area to mass ratio, the mass, and the % body mass in peanuts required per day. These relationships are best illustrated by small graphs. Given the above and the fact that the digestive system of an animal is capable of processing only a certain amount of food in a day, what can you conclude about the existence of a lower limit to the size of real animals? Is this borne out in nature? Hint: Consider shrews and hummingbirds, which have masses of about 6 and 12 grams respectively. 4) Based on the class results for the same sized canimal under different conditions, answer the following questions: How effective is dry fur in reducing the heat loss rate in still air? In the wind? Hint: Find the ratio of the cooling rate of a furry canimal to that of a bare canimal in each environment. What does this suggest about a “wind chill factor”? Explain in detail how heat is removed by the wind. Compare the cooling rates of canimals with wet and dry fur, both under windy conditions. Use a ratio. What does this say about the severity of heat loss by accidental drenching in cold windy weather? How is the heat removed? What does it say about the effectiveness of sweating and panting as heat removal mechanisms in warm weather? How is the heat removed? Compare the cooling rates of canimals with and without fat in water. Use a ratio. What does this suggest about the necessity of the blubber layer beneath the skin of sea mammals? How does fat reduce the heat loss? Animals that hibernate seek out a sheltered enclosure. During hibernation, body temperature drops dramatically. How do each of these factors contribute to the reduction of heat loss?

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Appendix 1: The Basics of Using Excel Excel is a spreadsheet program. It performs mathematical functions and displays the results in a table format. Figure 8 shows a few important features of the program. Figure 8 – Microsoft Excel

Using Excel to Analyse Data: When the program opens, you will notice that the working area of the screen looks like a table, with columns and rows. Columns are labelled with letters (A, B, C, …) and rows are labelled with numbers (1, 2, 3, …), so that each individual cell (each box in the table) has a unique label consisting of a letter and a number, e.g. B5, or H23. If you click on a cell, its name appears in a white box located in the upper left-hand corner of the screen, on the tool bar (see Figure 8). You can type four different types of information into cells: TEXT, NUMBERS, FORMULAS and SPECIAL FUNCTIONS. TEXT is simply words that you type in. For example, you should always type you and your partner’s names, the date, and the experiment title at the top of your spreadsheet. Excel treats NUMBERS differently than it does text. You can do calculations with numbers, but Excel won’t let you do a calculation using a cell that contains text. FORMULAS are what you use to perform calculations on your numbers. You type a formula into a cell, and when you hit “Enter”, Excel will calculate and display the answer. Formulas can include both numbers (e.g. 9.81) and cell addresses (e.g. A13).

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To tell Excel that you are typing a formula (rather than text), start by typing the “ = ” symbol.

Addition: If you want to add values, use the “ + ” operator: =A5+B17+6.52

Subtraction: If you want to subtract values, use the “ – ” operator: =A5-B17-6.52

Multiplication: If you want to multiply values, the program interprets the “ * ” symbol as a ×:

=A5*B17*6.52 Division: If you want to divide values, use the “ / ” symbol:

=A5/B17/6.52 Note: the program interprets what is typed above as ( )52.617

5×BA .

Powers: If you want to raise a value to a power, use the “ ^ ” symbol. =5^(1/2)

Note: the program interprets what is typed above as 5½. SPECIAL FUNCTIONS are commands you can type into your formulas to make Excel perform more sophisticated mathematical operations. Some useful ones are given in the table below. First, however, we discuss how to input lists of values.

LISTS: Some functions, such as SUM(list), require you to input a list of values. These can be entered in one of two ways:

a) EITHER enter the values individually, separated by commas, =SUM(A5, A6, A7, A8)

b) OR, for consecutive values, enter the first and last value, separated by a colon. =SUM(A5:A8)

Table of Special Functions Commonly Used in Excel SUM(list) Calculates the sum of a list of values AVERAGE(list) Calculates the average of a list of values STDEV(list) Calculates the standard deviation of a list of values SQRT(value) Calculates the square root of a value EXP(value) Calculates evalue

LOG(value,base) Calculates the logarithm of a value to the specified base, e.g. LOG(5,EXP(1)) calculates the natural logarithm of 5 (i.e. to base e = 2.71828…)

SIN(value in radians), COS(value in radians), TAN(value in radians)

Calculates the sine, cosine and tangent of a value. Note: the program assumes the value is in radians.

RADIANS(value in degrees) Converts a value in degrees to radians PI( ) Gives the numerical value of π = 3.1415…

Note: Do not put anything between the brackets! Note: You can nest one function inside another. For example, SIN(RADIANS(30)) would calculate the sine of 30 degrees.

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Correcting Mistakes: Occasionally you will mistype something. One way to fix your mistake is to click on the cell and retype it correctly. However, there is a less time-consuming way to fix it. If you click on a cell that contains a formula, the cell will display only the answer to the equation. However, the formula bar (refer to Figure 8) does display exactly what you typed into the cell. To correct a mistake in a formula, without erasing the contents of the entire cell, do the following:

1) Click on the cell you want to correct (see Figure 9, below). 2) Next, click on the formula bar. A cursor will appear in it. 3) Correct the formula by changing what is displayed inside the formula bar. 4) When you press “Enter”, the cell’s contents will be corrected and Excel will re-evaluate

its answer. Figure 9 – Fixing a mistake in a formula

Cutting and Pasting: In Excel, you can type a formula once and then apply it to a list of data without having to retype the formula. This process is called cutting and pasting. Consider Table 5 below. Data has been entered and labelled, and in the Calculations column, a formula has been typed. If you press “Enter”, Excel will evaluate the equation and return an answer.

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Table 5 – Pressing “Enter” causes Excel to evaluate the formula and return the answer. A B C A B C 1 Data Uncertainty Calculations 1 Data Uncertainty Calculations 2 4.5 0.3 =(B2/A2)*100 2 4.5 0.3 6.666666667 3 6.9 0.3

⇒ 3 6.9 0.3

4 3.2 0.3 4 3.2 0.3 To apply the same formula to the data in rows 3 and 4, without having to retype it, first click on the cell whose formula you want to copy, i.e. C2. Once you have selected C2, click “Edit” on the menu bar and then click “Copy”. The borders of cell C2 will begin to flash. Highlight cells C3 and C4 with the mouse by clicking on C3 and holding the button down, then dragging the mouse to C4 and releasing the button. Finally, click “Edit” on the menu bar again, and then click “Paste”. Numbers will appear in cells C3 and C4. Table 6 – Highlight and copy C2, then paste the contents into cells C3, C4 and C5. A B C A B C 1 Data Uncertainty Calculations 1 Data Uncertainty Calculations 2 4.5 0.3 6.666666667 2 4.5 0.3 6.666666667 3 6.9 0.3

⇒ 3 6.9 0.3 4.347826

4 3.2 0.3 4 3.2 0.3 9.375

If you click cells C3 and C4, and inspect the formula bar, you will see that each cell contains a version of the original formula. The difference is that the cell addresses have been changed. Every place the number “2” appeared in the original formula has now been replaced with the correct row number: 3 or 4. See Table 7, below. Table 7 – The formulas in column C each have the correct row numbers. A B C 1 Data Uncertainty Calculations 2 4.5 0.3 =(B2/A2)*1003 6.9 0.3 =(B3/A3)*1004 3.2 0.3 =(B4/A4)*100 When you paste a formula into a cell below the original, Excel automatically increments the row numbers. If you paste a formula to the right of the original, the column addresses will increment (from C to D, for example). In some cases, you do not want Excel to automatically increment the cell addresses. For example, in Table 8, all the data is to be multiplied by a constant, located in cell A2. However, if the

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formula is copy-and-pasted, the program will try to increment the value of A2 also, which would be incorrect. Table 8 – If C2 is copy-and-pasted, the address “A2” will increment in error. A B C 1 Constant Data Calculations2 3.14 4.5 =A2*B2 3 6.9 4 3.2 To keep the program from automatically incrementing a cell address when you cut-and-paste a formula, use the “ $ ” symbol. For example, if you type $A$2 instead of A2, then Excel will not increment either the row or column when you copy-and-paste. Table 9 – Excel does NOT increment cell addresses that include the “$” operator. A B C A B C

1 Constant Data Calculations 1 Constant Data Calculations 2 3.14 4.5 =$A$2*B2 2 3.14 4.5 =$A$2*B2 3 6.9

⇒ 3 6.9 =$A$2*B3

4 3.2 4 3.2 =$A$2*B4 The “ $ ” symbol is called the “absolute reference” operator, because it forces the program to reference only the column and row specified. Rounding Do not attempt to round off values in Excel. (The program won’t let you.) Only round off your final calculated values when you report them in your notebook or typed report. Scientific Notation in Excel To insert a number written in scientific notation, use the letter “E”. Excel understands the letter “E” to mean ×10??, so the number 5.72 × 10-3 would be typed as: 5.72E-3 Inserting Special Symbols into Excel You do not need to include special symbols, such as δ or ±, in your Excel spreadsheets. Just label your columns clearly enough for your lab instructor to understand what you mean. For example, δA can be typed as dA and ± can be typed as (+/-). A Warning! Save your work early and save it often. The computers in the laboratory have been known to freeze up! If this happens, you will lose all your work up to the last time you saved, so try to remember to save regularly.

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Printing your work: If your lab instructor requires you to do your mathematical analysis on Excel, you will need to print your Excel spreadsheets and hand them in with your report. Because your lab instructor needs to see both what your answers are and how you arrived at them, you will need to print out two copies of your spreadsheet, one that shows all your numerical values, and one that shows all your formulas. Instructions on how to print both versions are given below. To print a copy of your spreadsheet that shows your numerical values:

1) Go to the “File” menu and select “Page Setup”. 2) Click on the “Page” tab (located along the top of the dialogue box).

a. If your spreadsheet is taller than it is wide, select the “Portrait” orientation. b. If your spreadsheet is wider than it is tall, select the “Landscape” orientation. c. Click “Fit to One Page”.

3) Now, click on the “Sheet” tab (located along the top of the dialogue box). a. Select “Gridlines”. b. Select “Row and Column Headings”. c. Click “OK”.

4) Go to the “File” menu and select “Print”. To print a copy of your spreadsheet that shows your formulas:

1) Go to the “Tools” menu and select “Options”. a. Click on the “View” tab (located along the top of the dialogue box). b. Select “Formulas”. c. Click “OK”.

2) Some of your formulas will be cut off short because the columns are not wide enough for them. Resize the columns to fit the formulas as follows:

a. In the upper left-hand corner of the work area (above the row numbers and to the left of the column headings) is a small grey box. Click this box to highlight the entire spreadsheet.

b. Place your cursor on the line between A and B and then double-click that line. This will resize all the columns so they are wide enough to display everything that is typed into them.

c. Click on any cell to de-select the spreadsheet. 3) Go to the “File” menu and select “Print”. Note that the program will still print your

gridlines and row and column headings, because those options are still selected from the first time you printed.

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Appendix 2: Inserting Symbols and Equations into Word Documents In your typed reports, you are required to insert equations and special symbols properly (i.e. you may not write them in by hand). The following explains how to do this. To insert special symbols into a sentence:

1) Click on “Insert” in the menu bar. 2) Select “Symbol”. 3) A window will appear. Make sure the font selected is Symbol. 4) Select the symbol you want. Note that you have the entire Greek alphabet available (α, δ,

π, θ), as well as mathematical symbols such as ± and ×. To insert an equation into your document:

1) Click on “Insert” in the menu bar. 2) Select “Object”. 3) A window will open that has a list that you may scroll through. Find and select

“Microsoft Equation 3.0”. 4) A box and menu will appear on your document. Type your equation into the box, using

the menu to insert items like fractions, square roots, operators, superscripts and subscripts. The following information may also be helpful to you: To insert bold, italic or underlined text:

On the tool bars (see Figure 8), look for a series of three buttons marked B, I and U. Clicking these will give you Bold, Italic and Underlined text, respectively.

To insert superscripts and subscripts:

On the tool bars, look for a pair of buttons marked x2 and x2. Clicking these will give you superscript and subscript text, respectively, which is useful for typing things like v2 and do.

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Appendix 3: Graphing with Curvefitting.xls CurveFitting.xls is a macro for Microsoft Excel that allows you to graph and fit linear data. Getting started Open the program from the icon or folder on the computer desktop. The program will ask if you want to enable macros; these are needed, so please do so. The program will open a data sheet like that seen on the left-hand side of Figure 10.

Figure 10 – The “Data” tab (left) and “Graph” tab (right) of CurveFitting.xls

Along the lower edge of the window are three tabs. The “Data” tab is what you see first, but if you click the “Graph” tab, you will see a graph like the right-hand side of Figure 10. To Graph Your Own Data: 1) Click the “Data” tab. 2) Enter your name, your graph title and your graph axes labels in the appropriate locations. If

you need a superscript, use the symbol “ ^ ”. It doesn’t look nice, but your instructor will know what you mean!

3) Enter your data in the data table. 4) When you now click on the “Graph” tab, you should see your data plotted. Reformatting the Scale of the Axes: 1) Right-click the axis that you want to change the scale of. A small dialogue box will pop up.

Select “Format Axis” from this. 2) A larger dialogue box will open. Click on the “Scale” tab (located along the top of the box).

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3) You can now see what the default maximum and minimum values for your axis are. To change one or more of these, first turn off the “Auto” function by de-selecting the checked boxes to the left of the values. Then, type in the values you want and click “OK”.

Fitting Your Data: You are now ready to draw your best fit line and your two bounding lines on your graph. 1) Click on the “Graph” tab if you have not done so already.

2) Along the top of the graph, you will see the following set of tool buttons:

3) You will typically only use the “Draw Curve”, “Reset” and “Select Moveable Point” buttons.

The program will, by default, fit your data to a linear function and allow you to draw three lines on it (one best fit line and two bounding lines).

4) To put your best fit line on your graph, click on the “Draw Curve” button. A line will appear on your graph but it will not yet have the correct slope. The equation for this line also appears on the screen; you can click-and-drag the equation out of the way.

5) You will now click-and-drag the ends of the line to make the line fit your data. Click “Select Moveable Point”. One end of your line will now be highlighted with a small box. If you move the cursor over top of that box, you will see the cursor change shape.

6) Click-and-drag the highlighted end of the line to where you want it.

Note: The program has a small idiosyncrasy. It only allows you to drag the end of the line either up and down, or left and right. You can’t drag it diagonally. The easiest way to manipulate the line is to click-and-drag it horizontally first, then release the mouse, then click the end of the line a second time and drag it vertically to fit the line to your data.

7) Once you have moved the first end of the line approximately where you want it, click the

“Select Moveable Point” button again. The second end of the line will now be highlighted with a box. You can move it around in it the same way that you did the first end. By clicking “Select Moveable Point” repeatedly, you can toggle between the two ends of the line. Keep moving the ends of the line around until you have fit the line to your data to your satisfaction.

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8) Once you are happy with your best fit line, click the “Draw Curve” button again. A second

line will appear. You can move it the same way you did the first line, by clicking on “Select Moveable Point” and shifting the ends of the line. Use this second line to make your upper-slope bounding line.

9) When you are happy with your upper-slope line, click “Draw Curve” again and you will get a third line. Use this to make your lower-slope bounding line.

Fixing Mistakes: Unfortunately, if you’re not happy with one of your earlier lines, you can’t go back and fix it. You will have to erase all three lines and start again. To erase the lines, click on the “Reset” button. To redraw your lines, simply click on “Draw Curve” again. To Save Your Graph and Data Table: Click “File” on the menu bar and then “Save As...” Change the destination to your diskette and give your file a name. To Print Your Data Table and Graph: 1) Click on the “Data” tab. 2) Click “File” on the menu bar, then “Print...”, then “OK”. 3) Click on the “Graph” tab. 4) Click “File”, then “Print...”, then “OK”. Accessing Data in an Open Excel Spreadsheet: Sometimes the data you want to graph is already typed up in an Excel spreadsheet. There is an easy way to link that data directly into CurveFitting. 1) Highlight the column of data in your Excel spreadsheet that you want to transfer into your

CurveFitting data table. Click “Edit” on the menu, then “Copy”. 2) Now highlight the first cell in the CurveFitting table where you want your data to appear.

Click “Edit” and “Paste Special”. A window will pop up; in the lower left-hand corner of it is a rectangular button marked “Paste Link”. Click this button.

3) Your data will appear in the data table. 4) The nice thing about this function is that if you change your values in the Excel spreadsheet,

the values in CurveFitting will be automatically updated! Note: In step (2), you must click “Paste Special”, not “Paste”. Clicking “Paste” will put your values into the data table, but they will not be automatically updated if you then change your values in the original Excel spreadsheet.

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Appendix 4: Graphing with Linegraph.xls Linegraph.xls is a macro for Microsoft Excel that allows you to graph and fit linear data. Getting started Open the program from the icon or folder on the computer desktop. Two windows will open: a data table and a graph. Figure 11 shows these.

Figure 11 – The data table (upper screen) and graph (lower screen) of Linegraph.xls

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To Graph Your Own Data: 1) Click the “Data” tab. 2) Enter your graph title, your graph axes labels and your name in the appropriate locations. If

you need a superscript, use the symbol “ ^ ”. It doesn’t look nice, but your instructor will know what you mean!

3) Type your data into Table 1; it will be plotted automatically on the graph. 4) Linegraph calculates the best fit to your data. In Table 2, you will find your slope m and its

uncertainty δm, as well as the y-intercept b and its uncertainty δb. Reformatting the Scale of the Axes: 1) Right-click the axis that you want to change the scale of. A small dialogue box will pop up.

Select “Format Axis” from this. 2) A larger dialogue box will open. Click on the “Scale” tab (located along the top of the box). 3) You can now see what the default maximum and minimum values for your axis are. To

change one or more of these, first turn off the “Auto” function by de-selecting the checked boxes to the left of the values. Then, type in the values you want and click “OK”.

To Save Your Graph and Data Table: Click “File” on the menu bar and then “Save As...” Change the destination to your diskette and give your file a name. To Print Your Data Table and Graph: 1) Click on the data table to make the window active. 2) Click “File” on the menu bar, then “Print...”, then “OK”. 3) Click on the graph to make the window active. 4) Click “File”, then “Print...”, then “OK”. Accessing Data in an Open Excel Spreadsheet: Sometimes the data you want to graph is already typed up in an Excel spreadsheet. There is an easy way to link that data directly into Linegraph. 1) Highlight the column of data in your Excel spreadsheet that you want to transfer into your

Linegraph data table. Click “Edit” on the menu, then “Copy”. 2) Now highlight the first cell in the Linegraph table where you want your data to appear. Click

“Edit” and “Paste Special”. A window will pop up; in the lower left-hand corner of it is a rectangular button marked “Paste Link”. Click this button.

3) Your data will appear in the data table. 4) The nice thing about this function is that if you change your values in the Excel spreadsheet,

the values in Linegraph will be automatically updated! IMPORTANT: In step (2), you must click “Paste Special”, not “Paste”. Clicking “Paste” may disrupt the connection between the data table and the graph, and Linegraph will stop working. You will then have to exit and re-start Linegraph to make it work correctly again.

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Appendix 5: The Photogate Sensor A photogate consists of a light source (a photocell) that projects a narrow infrared beam (invisible to the eye) towards a photosensitive detector (see Figure 12). The detector changes its resistance to electric current by large amounts as the intensity of light incident on it is changed (as when the beam is interrupted by a flag passing through the photogate.) Such marked changes enable the detector to turn a timer on or off.

light source

infrared beam

detector

LED: lights when the beam is blocked.

Figure 12

Accuracy: Since the light source and detector of the photogate have a finite width, the true length of a flag passing through the photogate beam may not be the same as its effective length “seen” by the photogate. If the flag passes within 1 cm of the detector and “cuts” the beam at right angles, then the uncertainty to which the detector senses the length of the flag is ±1 mm. Usually, this uncertainty can be safely disregarded, because it is a systematic uncertainty. There is also a 1% uncertainty in the time t measured. This is a result of the limitations of the electronics in the timer. Always use at least ± 1% for time measurements. (Remember, δt ≠ 0.01,

but rather ttδ = 0.01.)

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Appendix 6: National Instruments’ LabVIEW Interfacing System LabVIEW is a software program made by National Instruments that is used with a hardware interfacing as an interfacing system between the computer and the environment. This system can be used for either data acquisition or data control. In this course, we will mainly be using LabVIEW for data acquisition. The programs used for this data acquisition are called Virtual Instruments (VIs) and are written in a high level programming language supplied by LabVIEW. The VIs were either written “in-house” or borrowed from others who have written them elsewhere. For each of the VIs used, there is enough documentation given on the front panel of each VI to enable you to make the proper connections and run the VI. You are given the uncertainties of the data acquired by each VI in the apparatus section of the lab, but the only way for you to really know the limitations of the VI is to look at exactly what it is doing. A full set of manuals for LabVIEW is available to you at all times and/or your instructor can show you the “diagram” of the program very easily.

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Appendix 7: Common Derivatives and Rules of Calculus function derivative function derivative

kxf =)( , k is a constant 0)(' =xf nxxf =)( 1)(' −= nxnxf )ln()( xxf =

xxf 1)(' =

xexf =)( xexf =)('

)sin()( xxf = )cos()(' xxf = )cos()( xxf = )sin()(' xxf −= )tan()( xxf =

)(cos1)(' 2 x

xf = )arctan()( xxf =

211)('x

xf+

=

function derivative

)()( xgkxf = , k is a constant )(')(' xgkxf = )()()( xhxgxf += )(')(')(' xhxgxf +=

))(()( xhgxf = )(')(' xh

dhdgxf =

)()()( xhxgxf = )(')()()(')(' xhxgxhxgxf +=

)()()(

xhxgxf = [ ]2)(

)(')()()(')('xh

xhxgxhxgxf −=

Using Calculus for Uncertainty Calculations

If f = f (x) and dxdf both exist and are continuous, then x

dxdff δδ ≈ . (This is accurate if the

uncertainty in x, δx, is small compared to x.) If f = f (x, y, z, ...), then:

...+++≈ zdzdfy

dydfx

dxdfdf δδδ

where δx, δy and δz are the absolute uncertainties in x, y and z, and dxdf ,

dxdf and

dxdf are the

partial derivatives of f with respect to x, y and z respectively.

To calculate a partial derivative, e.g. dxdf , differentiate the entire function f with respect to x

while pretending the other variables are constants. This method works with any number of variables (where a variable is a measured value that has an uncertainty) – it just gets a little messy.

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