Department of Physicsand Astronomy

Physics 2130Lab Manual

Spring 2012

Lab Reports and Marks

This section includes an outline for your lab reports and a sample lab report based on anexperiment called "Conservation of Energy on an Inclined Plane" (also included).

Since your marks for the lab are derived from your written reports it is important to under-stand what is expected as a report. Read the sample experimental outline "Conservationof Energy on an Inclined Plane" first, then the lab report outline, and conclude by readingthe sample report.

Lab Report Outline:

Lab reports must have the following format:

Title Page: Includes the title of experiment, the date the experiment was performed, yourname, your partner’s name(s), and lab section.

Results: List data in tables whenever possible. Each table must have a title and carryunits for each column. Show one sample calculation for each step in the analysis,carrying the units through the calculations. Consult the sample lab report to see howsample calculations should be done. Any other measurements associated with thedata (parameters, constants, etc.) should be clearly labelled within this section.

Analysis: Complete all steps and questions in the analysis section of the experimentaloutline. The analysis section contains primarily equations and numerical work thatmust be carried out in order to obtain usable results. When asked to plot a graphbe sure to label each axis including units. Each graph must have a descriptive titleplaced above the graph (note: y vs x is not a descriptive title).

Discussion: A few written paragraphs outlining the important facets and details of yourexperimental results. Examples of what questions you should be asking yourself areprovided in each lab outline, but do not feel limited by these questions. If you haveother observations or conclusions that you would like to discuss, then do so. Thissection is the most important, yet most difficult.

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Conclusion: A final sentence(s) stating the outcome(s) of the experiment written in thecontext of the objective of the experiment. Your conclusions should be supported byarguments made in your discussion.

If you glance through a scientific journal you will observe that the outline listed above is asimplified version of most journal formats. Items such as introductions and procedures havebeen provided for you within the manual. It is important to note here that presentationmatters; the sections of your report should appear in the same order listed above.

The difference between a "good" lab write-up and a poor one is often attention to detail. Asyou write your report you must assume that the person who will read it knows little aboutyour experiment. You are presenting a case upon which you will base your conclusion. Guidethe reader through the report by clearly labeling tables and graphs with descriptive titles,succinctly presenting your discussion of the experiment and results, and unambiguouslystating your conclusion(s).

Marking scheme

Each group is required to submit one lab report for each experiment unless directed oth-erwise. Some labs are designed to be done individually. Groups can be no more than 2students. When experiments are done in groups one lab report for the group is sufficient,although individual group members may submit their own report if they prefer. Whengroup reports are marked each member of the group receives the same mark.

In some cases, several groups will team up on a given apparatus to obtain their results.Each group will then share the same data, but ALL analysis must be done by each groupindividually. This includes tabulating the original data; you are not allowed to ‘share’spreadsheets.

Lab reports are typically due one week after each experiment is completed, unless instructedotherwise.

Lab reports will generally be marked out of 10 (longer labs may be marked out of 15). Yourlab reports will be marked on the following criteria:

Completeness: - Have you completed all the necessary tables, graphs, and calculations?- Are all the required sections present, i.e. title page, conclusion, etc.?

Correctness: - Do you have the proper labels and units for all graphs and tables?- Are your numerical calculations done correctly?

Presentation: - Are the sections of the report in correct order?- Are the results presented in a clear and concise manner?- Are the spelling, grammar, and punctuation correct?

Reason: - Do your conclusions and arguments seem reasonable, based on your results?

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Note that the list of comments for each section given above is not exhaustive. There aremany different aspects to a good lab report which are not given here. Refer to the LabReport Outline section above to obtain more guidance on each section.

Your final lab mark is calculated as:

Total Marks Received / Total Marks Possible

Absence from Lab sessions

If you must miss a lab section, contact your lab instructor as soon as possible to let themknow you will be absent, preferably before the lab, if possible. You will need to makearrangements to do the lab at some other time. If you are able to attend a different labsection within the same week, then let your lab instructor know. He or she can make thenecessary arrangements with the instructor of this other lab session. Under no circumstancesare you allowed to put your name on a report for a lab that you did not participate in.

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Sample Lab Outline

Objective

The objective of this experiment is to investigate the principle of conservation of energyusing an inclined plane.

Apparatus

• Air Track

• Sonar System

• Carpenter’s Level

• Meter Sticks

Background

The near frictionless surface of an air track represents a close approximation to somethingcalled an isolated system. Such a system would be expected to demonstrate the principleof conservation of mechanical energy.

By definition an isolated system does not exchange mass or energy with its surroundings.Any mechanical energy put into the system will remain there and we should be able toaccount for it as either kinetic or potential energy. If the principle of conservation ofmechanical energy is correct then the total energy in the system should remain constant.

Tipping the air track at an angle creates what physicists call an inclined plane. In theabsence of friction an object placed on an inclined plane will begin to move in the downhilldirection.

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Energy is introduced into the inclined plane system by placing the glider at the top ofthe incline. Until released all the energy in the system is in the form of potential energy.Once moving some of the potential energy is transformed into kinetic energy. As the gliderprogresses down the incline the potential energy continues to decrease while the kineticenergy increases. At any instant in time, however, the sum of the two should be constantand equal the initial amount of potential energy in the system.

Procedure

Set the air track at a slight angle to the horizontal (about 2 degrees). Place the glider atthe high end of the air track and allow it slide to the low end. Track the position of theglider during its travel using the sonar equipment. Save your data on a disk. Be sure torecord the mass of the glider and measure the angle of incline with two meter sticks andthe carpenter’s level.

Analysis

Construct a data table like the one below and make the appropriate calculations.

Time Position of Velocity of Kinetic Energy Potential Energy Total Energy(s) Glider (cm) Glider (cm/s) of Glider (J) of Glider (J) (J)

From your calculations plot on a single graph the potential, kinetic and total energies as afunction of time.

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Discussion

From your results, can you conclude that energy was conserved? If not, where might itbeen lost?

How accurately were you able to measure the angle of incline? What impact(s) might anerror in the measurement of the angle of incline have on the results of the experiment?

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Sample Lab Report

CONSERVATION OF ENERGY ON AN INCLINED PLANE

Larry BrownCurly JonesMoe Smith

Thursday, Jan. 8, 2010Lab Section #2

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Table 1. Inclined Plane Data Table

Mass of glider (g) 220 ± 0.5∆t (s) 0.0500 ± 0.00025Angle of incline (°) 2.7 ± 0.5

Time Position Velocity Potential Kinetic Total(s) (cm) (m/sec) Energy (J) Energy (J) Energy (J)1.10 198.43 0.316 0.2015 0.0110 0.2131.15 196.85 0.330 0.1999 0.0120 0.2121.20 195.20 0.428 0.1982 0.0202 0.2181.25 193.06 0.402 0.1961 0.0178 0.2141.30 191.05 0.352 0.1940 0.0136 0.2081.35 189.29 0.508 0.1922 0.0284 0.2211.40 186.75 0.450 0.1897 0.0223 0.2121.45 184.50 0.442 0.1874 0.0215 0.2091.50 182.29 0.518 0.1851 0.0295 0.2151.55 179.70 0.590 0.1825 0.0383 0.2211.60 176.75 0.568 0.1795 0.0355 0.2151.65 173.91 0.598 0.1766 0.0393 0.2161.70 170.92 0.632 0.1736 0.0439 0.2181.75 167.76 0.652 0.1704 0.0468 0.2171.80 164.50 0.644 0.1671 0.0456 0.2131.85 161.28 0.690 0.1638 0.0524 0.2161.90 157.83 0.772 0.1603 0.0656 0.2261.95 153.97 0.696 0.1564 0.0533 0.2102.00 150.49 0.812 0.1528 0.0725 0.2252.05 146.43 0.778 0.1487 0.0666 0.2152.10 142.54 0.806 0.1448 0.0715 0.2162.15 138.51 0.826 0.1407 0.0751 0.2162.20 134.38 0.826 0.1365 0.0751 0.2122.25 130.25 0.896 0.1323 0.0883 0.2212.30 125.77 0.880 0.1277 0.0852 0.2132.35 121.37 0.912 0.1233 0.0915 0.2152.40 116.81 0.932 0.1186 0.0955 0.214

Sample Calculations

Notes:

1.) The position data column represents the distance from the sonar unit to the glider.

2.) The frequency of the sonar unit was 20.0 ± 0.1 Hz, which translates into a ∆t of .0500±0.00025 seconds.

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Velocity Calculation:

Velocity is defined as: ∆Position∆time

Each entry in the velocity column was calculated as: Pn − Pn+1∆t

Sample calculation: 198.43cm− 196.85cm0.05s = 31.6 cm/s = 0.316 m/s

Potential Energy:

Potential energy is defined as U = mgh.

Given that the data can be thought of as the hypotenuse of a right-hand triangle, the heightcan be calculated as,

h = (position) sin θ∴ U = mg(position) sin θ

Sample calculation: U = (.220 kg) (9.80 m/s2) (1.9843 m) (sin (2.7°)) = 0.2015 J

Kinetic Energy:

Kinetic energy is defined as K = 1/2mv2

Sample calculation: K = (0.5) (0.220 kg) (.316 m/s)2 = 0.0110 J

Total Energy:

Total energy is the sum of the potential and kinetic energies, E = U +K.

Sample calculation: E = 0.2015 J + 0.0110 J = 0.2125 J = 0.213 J

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Figure 1: Potential, kinetic, and total energy of a single glider on an inclined plane.

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Figure 2: The velocity of the glider on the track as a function of time.

Discussion

It can be seen from Fig. 1 that energy was conserved as the glider descended the air track.There are some slight deviations from a constant energy value that appear to be due tosimilar deviations in the kinetic energy values. This is probably due to small errors in thevelocity values, most likely caused by timing errors in the motion sensor.

We checked to see if air resistance was a problem. We found that the velocity of the glider(see Fig. 2) was linear and therefore air resistance did not appear to affect our results. Thefact that the velocity was linear also tells us that acceleration on the air track is constant.

Measurement of the angle of the air track is a concern. If the angle is improperly mea-sured the potential energy calculations will be inaccurate. The result could be an apparentincrease or decrease in the total energy when in reality energy is conserved.

We measured the angle several times and used the average value as the actual angle ofincline for the air track. We found the accuracy of the measurement to be about one halfof one degree. The limiting factor in measuring the angle is the carpenter’s level. Theexperimental calculations are quite sensitive to the value used for the angle of incline. Thisis because this angle is used to calculate the potential energy. Using the same data, andvarying the angle over a range of 1.5 degrees, we were able to get a variety of results. Over

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this range of angles we could show a loss of energy or a gain of energy.

Conclusion

Energy was conserved on the inclined plane.

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LAB 1

Damped Harmonic Motion

Objective

The objective of this experiment is to determine the variables describing 2 spring-massoscillator systems and subsequently model their motion mathematically.

Background

The motion of an harmonic oscillator can be characterised by noting the position of theobject undergoing oscillation with respect to some fixed point. The most convenient pointto use as a reference is called the equilibrium position of the oscillator. This is the positionan oscillator will rest at when left unperturbed.

Consider the two oscillators shown in Fig. 1.1. The spring-mass oscillator on the left willhang at a position such that the force of gravity pulling down on the mass is counterbalanced

Figure 1.1: Two types of oscillator, a hanging spring system, and a sloped ‘well’.

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2 LAB 1. DAMPED HARMONIC MOTION

by the force of the spring pulling upward on the mass. This is its equilibrium position. Theball rolling in the trough on the right will remain at the bottom of the trough unless pushedaway from that position. If pushed away, the ball will eventually return to the bottomafter oscillating up and down along the sides of the trough. The equilibrium position for anoscillator is that point along the path of oscillation where no net force acts on the oscillator.

To characterise the motion of an oscillator we first note that this is one dimensional motion,therefore only one coordinate is required. Additionally, the position or displacement ofthe oscillator is easiest to describe with respect to its oscillation position. If we use x torepresent this displacement, then the equilibrium position would be represented as x = 0.The furthest distance the oscillator travels from this position can be referred to as xmax.This value is also referred to as the amplitude of the oscillation.

The motion of an harmonic oscillator is given by the following differential equation, wherex is a function of time (i.e. x = x(t)):

d2x

dt2+ kmx = 0 (1.1)

The solution to this equation is

x = xmax cos(ωt+ φ), (1.2)

where xmax = maximum displacement,ω =

√k/m = angular frequency,

φ = phase angle.

Note: φ = 0 if x = xmax at t = 0.

We wish to study damped harmonic motion, which has the slightly more complicated equa-tion of motion

d2x

dt2+ bm

dx

dt+ kmx = 0 (1.3)

The solution to Eq. 1.3 may be expressed as

x = xmax e−ut cos(ωt+ φ) + xeq, (1.4)

In this case xmax is the value of the ‘highest’ peak in the wave. The term e−ut describesthe damping of the oscillator. The damping coefficient u = b/(2m) is characteristic of eachparticular system. In this experiment the damping coefficient is found by graphical analysisof the position data. The xeq term is simply the equilibrium position of the oscillator. Weoriginally set this value to zero, but this general form of the equation will account for anycase where it is not.

PROCEDURE 3

If we ignore the cosine term in Eq. 1.4 and consider only the exponential term we have,

x = xmaxe−ut (1.5)

By taking the natural logarithm of both side we obtain

ln x = −ut+ ln xmax (1.6)

This equation indicates that a plot of ln x vs. t should yield a straight line of slope u.

Procedure

Part I: Air Table

Figure 1.2: Air table setup. A piece of paper is placed under the puck and dragged under-neath it as it sparks, showing the path of the puck.

1. Set up an air table as per Fig. 1.2. Note that the posts that the springs are attachedto can be loosened and moved. Before each trial, move each post roughly the samedistance in a single direction. This prevents a track from being permanently burnedinto the carbon paper.

2. Level the table and position the recording paper so that it may be easily pulledunderneath the puck. The paper should be aligned so that the long edge of thenewsprint is towards you, with one of the short edges flush with the lip of the table.This will aid in keeping the paper aligned properly. Your instructor can show youhow to position the paper correctly if you are having difficulty.

3. Using a pen or marker, draw a mark on the paper indicating which side of the paperis flush with the table; all measurements will be done from this side. Also mark theinitial position of the centre of the puck as it rests in equilibrium. This can alsobe done by turning on the spark generator and very briefly pressing the pedal; the

4 LAB 1. DAMPED HARMONIC MOTION

puck will burn the centre location onto the paper. Note that this method may makeinterpretation of the data more difficult, as there is an extra mark on the paper whichcould interfere with the oscillating data.

4. Set the spark generator to 20 Hz, and start the puck oscillating. Pull the papertowards you and step on the pedal to make the generator spark. Be sure to pull sothat the edge of the paper stays flush with the edge of the table. (NOTE: It is notnecessary to pull the paper at a constant rate). Be sure your spark trace has at least5 peaks.

Part II: Air Track

1. Set up the air track including sonar system as per Fig. 1.3. The motion sensor shouldbe set to 20 Hz; you can change or view the frequency from the setup window in thesoftware.

Figure 1.3: Air track setup. The glider is disturbed from equilibrium and oscillates alongthe track. The motion sensor records the position of the glider as a function of time.

2. Turn on the air source, and allow the glider to reach its equilibrium position. Thismay take a while; you may aid the process by slowing the glider down manually. Withthe glider motionless, click ‘Start’ in DataStudio. Record the position of the glider asthe equilibrium position. Stop the data. Note that the position may not be constant;in this case, copy down at least 10 values of the position and find the average, as wellas the uncertainty in this average.

3. Set the glider oscillating and start a new data run. Let the glider oscillate for at least10 full periods, or until it has fully damped out, whichever comes first. Once you aresatisfied with the data, export it as a text file.

Analysis

Part I: Air Table

Measure the position of each minimum and maximum, from the marked edge of the paper.Since the spark generator was set to 20 Hz, the time between each spark is 0.05 s. Use this

ANALYSIS 5

information to find the time corresponding to each peak you recorded. Assume that yourfirst point (not peak) occurs at t = 0. Record the times and corresponding positions in atable, as follows:

Time (s) Position (m) Displacement (m) ln(Disp.) Period (s) xmax (m) xtht1 d1 x1 = |d1 − xeq| ln(x1) 2(t2 − t1)

The displacement is found as shown in the table using your value for the equilibrium positionxeq. The natural log of the displacement should be straightforward. For details on how tofind the values for the final 3 columns (period, xmax, and xth), consult the rest of theanalysis below.

Plot the natural log of the magnitude of the displacement of each peak vs. time. The slopeof this graph should be the damping coefficient u (see Eq. 1.6). Use linear regression tofind the slope, with uncertainty.

From your data determine values for ω, φ, and xmax. Instructions for each one of thesevariables is below:

Angular Frequency ω: Find values for the period T using the times from your table.This time is the difference between successive time values in your table, multiplied by2 (i.e. T = 2(tn+1 − tn), as shown in the table). Repeat this calculation for as manyperiods as possible and find the mean T̄ and standard deviation of the mean σm ofall your values. Use this average T to find the angular frequency ω.

Maximum Displacement xmax: Note that in equation 1.4, a maximum occurs whencos(ωt + φ) = 1. This implies that if a peak of height x′ occurs at time t′, thenx′ = xmaxe−ut

′ . Solve this equation for xmax and use this equation in your tableto find xmax for each minimum and maximum in your table. Find the average andstandard deviation of the mean of these values.

Phase Angle φ: Measure and record the x value for your first point recorded. This isx(0), the position at t = 0. When t = 0 is plugged into equation 1.4, the resultingexpression is x(0) = xmax cos(φ). Use this expression to find φ.

Plug these values back into the theoretical damped harmonic oscillator equation 1.4 tofind the theoretical equation for x(t). Using this equation and the times for your min-ima/maxima, find theoretical values for the minima/maxima and place them in the finalcolumn of your table (xth). Plot the points determined from this equation and your originalmeasured values against time on a separate plot, as 2 separate series.

Part II: Air Track

Import your text file into a spreadsheet program. In a separate worksheet, record the timesand positions of the first 20 maxima/minima of your data. To do this, simply look throughyour original position data and pick the highest and lowest points, with their correspondingtimes. These values can simply be copied and pasted into your new worksheet.

6 LAB 1. DAMPED HARMONIC MOTION

Create a table similar to the one found in Part I, but omit the xth column. Using thesteps outlined in the Part I analysis, find values for ω, φ, and xmax. Plug these values intoequation 1.4 to find the theoretical equation that describes the wave. Using this equation,create an additional column using your original data (not just the maxima/minima, butall the points) that finds the theoretical values for x as a function of time. Plot both theexperimental theoretical results on a graph of x vs. t (as 2 series). There will be a lot ofpoints to plot; consider using a scatter plot with joined lines as opposed to marked points.Ask your instructor if you need help finding this option.

Discussion

How do your theoretical and experimental plots agree, for each experimental setup? Whichprocedure appeared to give better results? What differences do you notice between the two?Do your results convince you that a damped harmonic wave can be accurately described byequation 1.4?

Tinker with the values of the constants in your equations(s) for xth and describe whathappens when each constant is changed. If your plots do not agree, which constant seemsto be the problem? What did you have to do to this constant (or constants) to help thetheoretical and experimental results agree?

LAB 2

Vibrations on a Wire

Objective

The objective of this experiment is to examine how the propagation speed of a wave travel-ling on a wire changes with respect to the tension of the wire, and use this information tofind the linear mass density of a wire.

Apparatus

• Function Generator

• Oscilloscope

• Tool Box

• Sonometer Kit

Background

Velocity of Wave Propagation

Assuming a perfectly flexible, perfectly elastic wire, the velocity of wave propagation (v) ona stretched wire depends on two variables: the mass per unit length or linear density of thewire (µ), and the tension of the wire (T ). It can be shown that the relationship is given bythe equation:

v =√T

µ(2.1)

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8 LAB 2. VIBRATIONS ON A WIRE

Without going into the derivation of this equation, its basic form can be appreciated. Theequation is analogous to Newton’s Second Law (F = ma), providing a relationship betweena measure of force, a measure of inertia, and a quantity of motion. With this analogy inmind, it makes sense that the velocity should depend on the tension and linear density ofthe wire. That the form of the two equations is not exactly the same is to be expected. Themotion of the wire is considerably different than the motion of a simple rigid body actedon by a single force. (It could be asked whether velocity, rather than acceleration, is theright measure of motion to focus on. Since the waves on the wire do not accelerate, this isat least a reasonable assumption.)

If the analogy with Newton’s 2nd Law is accepted, and it is assumed that the wave velocitydepends only on tension and linear density, dimensional analysis shows that the form ofthe equation must be as it is. There is no other way to combined tension (with units of[M ][L][T ]−2) with linear density ([M ][L]−1) to get velocity ([L][T ]−1).

To verify this equation we will set up an experiment in which the tension of a wire isvaried and the corresponding wavelength of the wire’s fundamental resonance frequencyis measured. The velocity can then be calculated using the relationship v = λf , and theeffects of tension and linear density on velocity can be determined.

Procedure

Set up the sonometer as shown in the figure below:

Figure 2.1: Sonometer setup

1. Select one of the wires supplied and measure its diameter in thousandths of inches,using calipers. Record this number as the gauge of the wire. Also measure and recordthe mass and length of the wire, in SI units.

2. Set the end bridges roughly 60 cm apart on the sonometer and record the distancebetween them. Hang approximately 0.5 kg of mass (measure it) from the tensioninglever, and adjust the wire adjustment knob such that the tensioning lever is horizontal.A level is useful here, but note that even an angle of 5° away from the horizontal willcreate an error of only 0.4% in your results.

PROCEDURE 9

Figure 2.2: Adjusting the tension in the wire is accomplished simply by changing the notchon which the mass is resting. Each notch increases the tension by an integer multiple ofMg.

3. Position the driver coil between the bridges, approximately 5 cm from the nearestbridge, and position the detector coil at the centre of the wire (see Fig. 2.1).

4. Set the function generator to produce a sine wave of amplitude 5 V to 8 V. Set thegain of the oscilloscope channel to which the detector coil is connected to 5 mV/div.

5. Determine the tension in the wire as shown in Fig. 2.2 and record the tension in yourdata table. Wire tension in Newtons is calculated as (Mg× the notch position), asshown.

6. Beginning with a frequency of 20 Hz, slowly increase the frequency of the signal tothe driver coil in 1 Hz. When you begin to see resonance, adjust the frequency by0.1 Hz steps. Record TWICE this resonance frequency in a data table (see Note #1,below). Note that near the resonance value, the amplitude of the wave on the wirewill not vary by much, so include a reasonable estimate of the error in your values inyour results.

7. Increase the tension in the wire by moving the weight hanger to the next notch andagain find the resonant frequency. Repeat this step until you have used all 5 notches.

8. Repeat the procedure for 2 more wires, of different gauges. Be sure to properly labeland keep track of what data corresponds to which wire.

Notes:

1. The frequency observed on the wire may not be the frequency of the driver. Usuallyit is twice the driver frequency, since the driver electromagnet exerts a force on thewire twice during each cycle. It is theoretically possible for the wire to form standingwaves at the driver frequency, and at any even integer multiple of the driver frequency;although the highest multiple observed on this equipment so far has been six.

10 LAB 2. VIBRATIONS ON A WIRE

2. If the detector is placed too close to the driver, it will pick up some interference. Youcan check for this interference by observing the waveform from the detector on anoscilloscope; when they are too close, the trace will change shape. For best results,keep the detector at least 10 cm from the driver.

3. You will occasionally see higher and lower frequencies superimposed on the primarywaveform. It is possible for multiple standing waves to form. For example, the wiremay vibrate at the driver frequency and twice the driver frequency at the same time,thus causing two sets of “nodes”.

Analysis

Place your results in a table similar to the following:

Gauge Tension (N) Resonant Frequency (Hz) Velocity (m/s)g1 T1 f1 v1

......

...T5 f5 v5

g2 T1 f1 v1...

......

T5 f5 v5g3 T1 f1 v1

......

...T5 f5 v5

Use your measured distance between the bridges to find the wavelength of the wave. Usingthis value and the fundamental resonant frequency for each tension, determine the propa-gation velocity of the wave using the equation v = λf .

Construct a plot of v versus T with 3 series, one for each wire you used. Perform a power fit(add a trendline and choose ‘Power’) on each series, and display the equation on the chart.These equations should obey the form y = axn, where a and n are positive constants.

Note the value of n for each wire; they should be similar. Round n to the nearest tenthand create a new column on your table with values for Tn (be sure to include the unitsin your column header!). Create a new plot of v vs. Tn using these values, again with 3series. Perform linear regression on each line and select the option to set the intercept tozero. Present the slopes clearly, with error.

Using equation 2.1, find the linear mass density µ from your slope. Recall that for a lineargraph y = mx+ b, and note that in your graph, y = v and x = Tn. Express equation 2.1 asv = aTn, where a is a function of µ. It should be clear that m = a. Write a as a functionof µ and solve for µ. You should now have µ as a function of the slope, which can be used

DISCUSSION 11

to find the linear mass densities for each of your slopes.

Calculate the linear mass density of each wire by simply dividing the mass of each wire byits length. Include the errors in your results.

Discussion

Do the values for µ gained from the slopes of your graphs agree with the other measurementsyou took, within error? If not, discuss some possible reasons for the disagreement. Whichdo you believe is more accurate?

Acoustic guitars have strings of different thickness (gauge) for different notes (frequencies).Would it be possible to create a similar guitar using wires of all the same gauge and length?What physical quantity would you have to adjust to create the different notes? Whatpotential problems would occur using this method?

LAB 3

The Speed of Sound

Objective

The objective of this experiment is to determine the speed of sound in air by using theprinciples of resonance and standing waves.

Apparatus

• Function Generator

• Oscilloscope

• BNC Cable

• Wires, with alligator clips

• Microphone

• Resonance Tube w/ 2 ends, one containing a speaker

Background

Waves

The repetitive motion of waves breaking on a beach is probably the most familiar conceptwe have of waves. This familiar concept can be extended to represent many processes thatoccur in nature. Physical terms such as frequency (f in cycles/sec, or Hz), wavelength (λin cm or m), phase (φ in radians), and amplitude (magnitude of process represented by the

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BACKGROUND 13

Figure 3.1: Depiction of a typical transverse wave, showing the characteristics associatedwith the wave, such as amplitude and wavelength (λ).

wave) can be employed to quantify and measure properties in other cyclical processes whichbehave like the conceptually familiar water waves.

Terms such as node and antinode are also used to identify particular positions on the wave.Nodes represent points on the wave where the value of the process depicted by the waveis at a minimum. Conversely, antinodes are points depicting the maximum ( positive ornegative) value of the process depicted by the wave. For example if the wave shown aboverepresented 110 volt household AC electricity, then the amplitude would represent voltage(maximum value of about 155 volts at the antinodes). The frequency of the wave would be60 Hz and the value of the voltage at the nodes would be 0 volts.

Waves propagate from one place to another at a speed which can be defined as the frequencyof the wave multiplied by the wavelength of the wave, v = fλ.

Sound Waves

Sound waves are longitudinal waves, meaning that the direction of the ’oscillation’ is par-allel to the direction of propagation of the wave. The speed at which the wave travels(propagation velocity) is dependent on the medium through which the wave is travelling.In a simple sense the stiffer the medium the faster the speed of sound.

One indicator of ’stiffness’ in materials or media is the bulk modulus (B). When combinedwith the density of the medium (ρ), the speed of sound (v) may be calculated as:

v =√B

ρ(3.1)

14 LAB 3. THE SPEED OF SOUND

Typical values of B and ρ :

Material Bulk Modulus (Pa) Density (kg/m3)Water 0.21× 1010 1.0× 103Steel 20× 1010 7.8× 103

Aluminum 7× 1010 2.7× 103

Table 3.1: Physical properties of water, steel, and aluminum.

Propagation of sound in a gas is a slightly more involved subject. Ignoring the details, theequation for calculating the speed of sound in a gas is:

v =

√γRT

M(3.2)

where:

γ = a constant particular to individual gasesR = the gas constant (8.314 Jmol·K)T = the absolute temperature (K)M = the molecular mass of the gas (kg/mol)

Typical values of γ and M :

Gas γ M (kg/mol)Air 1.40 0.0288

Helium 1.67 0.0040CO2 1.30 0.0280

Table 3.2: Various properties of gases.

If one considers the speed of sound in air then factors such as altitude, humidity, andpressure also enter into the calculation for the speed of sound.

A sound wave consists of alternating pressure changes called compressions and rarefactions.As the wave moves it compresses the air to a pressure greater than atmospheric pressure.Between successive compressions there is a region of low pressure called a rarefaction. Thecompressions and rarefactions occur along the path the wave is travelling.

Audible sounds travelling through air will have wavelengths ranging from approximately 3to 300 cm. Wavelengths of this size are easy to study. We will employ sound resonance ina tube to determine the wavelength of the sound waves (λ), determine the frequency (f)of the sound wave with an oscilloscope frequency, consequently calculating the speed of thewave as the product of f and λ .

BACKGROUND 15

Superposition

Observations of waves in nature reveal that waves interact with each other in special ways. Ifyou throw two stones into a pond for example, and watch the wave pattern created by each,you will observe that the patterns interact with each other in passing, but the individualpatterns remain intact and continue to propagate after the interaction.

These observations are indicative of the principle of superposition. Multiple waves mayinteract with each other without losing their individual character. The interaction is gen-erally referred to as interference. For our purposes the interaction of sound waves in thisexperiment will be considered to be linear.

Superposition of sound waves moving through air can be very complicated. The air moleculesare already moving in a random motion. As a sound wave(s) travels it imposes an orderedvibrational motion on the already moving molecules. The motion of a molecule at anyinstant will be a combination of random motion and the oscillations caused by any numberof sound waves present at that instant.

Resonance

The principle of superposition leads to a phenomenon known as resonance. In this experi-ment sound resonance occurs when the wavelength of the sound "fits" the cavity in whichit is travelling. Consider a long tube closed at one end. A sound wave enters the open endand travels along the tube until it hits the closed end. Upon striking the closed end thesound wave reverses direction (reflection) and travels back down the tube toward the openend. When a wave is reflected in this manner there is a 180o phase change in the wave; thisis a very important factor for resonance.

Figure 3.2: Two examples of standing waves in a tube with one closed end and one openend. The top wave has a wavelength of L/4, where L is the length of the tube, while thebottom wave has a wavelength of 3L/4.

16 LAB 3. THE SPEED OF SOUND

Figure 3.3: Resonace tube setup

By adjusting the frequency, and therefore the wavelength of the sound, it is possible tocreate standing waves in the tube. As you can see in Fig. 3.2, if a node occurs at the closedend of the tube then the reflected wave is 180o out of phase with the incoming wave. If thewavelength is such that an antinode is located at the mouth of the tube then we have theproper conditions for a standing wave. Each wave travelling in the tube will reflect backupon itself without altering the positions of the nodes and antinodes.

The conditions for a standing wave are also the conditions for resonance. The resultinginteraction of the sound waves in the tube is the addition of their respective amplitudes(sound) thereby making a much louder sound than that of any single wave in the tube.

You can also see from Fig. 3.2 that only odd multiples of λ/4 can produce resonance in thetube and distance between successive resonances for a tube of length L will be λ/2.

Procedure

Setup

Assemble your apparatus as directed by the instructor using the set-up at the front of thelab as a guide. Your instructor will assist you with any difficulties and demonstrate how touse the oscilloscope and function generator.

The experiment will comprise three trials at different frequencies. For each trial, start withthe moveable piston at the zero mark on the scale. With the microphone located next tothe speaker slowly move the piston away from the speaker and observe the oscilloscopescreen. The wave on the oscilloscope screen is the signal from the microphone. Its heightrepresents the magnitude of the sound inside the tube. The height of the wave will rise andfall depending on the length of the tube. When the tube is at a length that corresponds toa resonance for the frequency in question the wave will have its maximum height. Recordeach length for which a maximum occurs.

ANALYSIS 17

Since this is a closed tube each resonance occurs at nλ/4, where n = 1,3,5,7,... . Thechange in the measured distances is half a wavelength. The frequency of the signal can bemeasured with the oscilloscope.

Pick three different frequencies in the range of 1000 to 3000 Hz. For each frequency alterthe length of the tube using the moveable piston. Each time the tube comes into resonancerecord the length (L) of the tube by noting the position of the piston on the scale.

Analysis

Construct a table like the following:

Frequency Length Wavelength Speedf1 L1 λ1 v1

L2 λ2 v2...

......

f2 L1 λ1 v1L2 λ2 v2...

......

To find the wavelength, note that the difference between lengths will always be λ/2. Ex-pressing this as an equation and rearranging for λ gives λ = 2∆L. Note that the value for λshould be approximately the same for each frequency. The speed is found by using v = λf .Report the speed of sound for each trial as the average of your calculations ± the standarddeviation.

The speed of sound is in fact a function of many variables, as discussed in the backgroundsection. One of these variables is temperature. Using the information from the backgroundsection of the lab (specifically, table 3.2 and equation 3.2), plot a graph of the speed of soundin air from -40oC to +40oC. To do this, construct a simple table in a spreadsheet consistingof a variety of temperatures within this range and the speed at these temperatures. Plotthis data as you normally would. You will need to convert your temperatures to Kelvin inorder to use the equation provided, but note that the x-axis of your graph must be in oC.

Using the information from the background section (specifically, equation 3.1), calculatethe speed of sound in steel and water. How many times faster does sound travel in steelthan in water? In water than in air (at 20°C)?

18 LAB 3. THE SPEED OF SOUND

Discussion

The ambient temperature in the lab is a little over 20oC. Calculate the speed of sound inair using this temperature and compare to your results. Do they agree? Which of yourfrequencies actually results in the best data? Is there any pattern in your results, such ashigher frequencies giving better results?

In practice the diameter of the tube used in the experiment also has an effect on thecalculated speed of sound. The actual wavelength of the sound is calculated as L+ 0.4d =nλ/4, where L is again the measured length of the tube and d is the diameter of the tube.If this equation was used instead of the one provided in the manual (L = nλ/4), whateffect would this have on your results? Specifically, would the results for the speed of soundincrease, decrease, or remain the same? Justify your answer.

APPENDIX A

Reading Vernier Scales

A vernier scale consists of a stationary scale (main scale) and a sliding scale (vernier scale).The divisions on the vernier scale are smaller than those on the stationary scale. N divisionsof the vernier scale equal N − 1 divisions of the main scale, (see Fig. A.1, where N = 10)

Figure A.1: Divisions on a Vernier scale.

In Fig. A.1, it may be seen that both the zero and the 10 mark on the vernier coincide withsome mark on the main scale. The first mark beyond the zero on the vernier is 1/10 of amain scale division short of coinciding with the first line to the right of the main scale zero.This difference between the lengths of the smallest divisions on the two scales represents theleast amount of movement that can be made and read accurately. This amount is called theleast count of the instrument. If the vernier scale is moved a distance 1/10 the distance ofthe difference between the divisions of the main scale, the next mark of the vernier coincideswith a mark on the the main scale. Only one mark on the vernier will align with a markon the main scale for a given measurement. See Fig. A.2.

19

20 APPENDIX A. READING VERNIER SCALES

Figure A.2: Reading a Vernier scale. Note how only a single pair of lines on the main scaleand sliding scale line up perfectly.

As you can see accurate measurements of 1/10ths of a main scale division are possible.Consider the scale in Fig. A.3 to have units of millimeters. In Fig. A.3a we see the zero ofthe vernier lies between the 6 and 7 mm marks and the second mark on the vernier coincideswith a mark on the main scale. The reading is therefore 6.2 mm. By similar reasoning thereading in Fig. A.3b is 3.7 mm.

(a) (b)

Figure A.3: Reading a Vernier scale. The left reading is 6.2mm, while the right is 3.7mm.

Nearly all verniers have N divisions of the vernier scale equal to N −1 divisions of the mainscale and the method of determining the reading is similar to that described above. Theleast count is always 1/N of the length of the smallest main scale divisions. In the exampleabove, the least count is 0.01 cm.

APPENDIX B

Statistics

Statistical analysis plays a very important role in experimental work. In this exercise we areconcerned with the connotations of terms such as average, standard deviation, and normaldistribution, as these terms have particular meanings when associated with experimentalwork. In addition, we will develop quantitative tools for judging the validity of data.

Throughout the semester we will associate the term average or mean with the "best estimateof the actual value". Standard deviation will represent the "average value of uncertainty inthe average" or probable error, and a normal distribution will be viewed as a probabilitydistribution. We will use techniques such as linear regression and Chauvenet’s criterion toprovide methods for assessing the validity of data gathered during experiments.

The following sections are brief overviews of the important points as they relate to statisticalanalysis.

Gaussian Distributions

The Gaussian curve, named after Karl Friedrich Gauss, was devised in 1795. Gauss foundthat his experimental data would not repeat itself exactly from trial to trial during exper-iments. After plotting the results of a particular experiment he developed his now famouscurve.

Gaussian distributions are also called normal distributions or bell curves. The term bellcurve obviously arose from the symmetrical curved shape of the Gaussian distribution whilethe name normal distribution is related to one of the mathematical requirements of a Gaus-sian distribution; that it be normalized. In practice all three terms are used interchangeablyand refer to a symmetrical distribution of values caused by random error. There are othertypes of distributions used in statistical analysis which differ from Gaussian distributions(e.g. Poisson distributions or binomial distributions). We will confine our interest solely to

21

22 APPENDIX B. STATISTICS

Gaussian distributions as they are applicable to the type of experimental work we will dothis semester.

Any time measurements are made they are subject to some degree of uncertainty. Thisis known as experimental error. When the error is small and random (as opposed to sys-tematic) then it can be shown that the repeated measurements of the same quantity willcluster around the actual or true value of the measurement. Further, it can be shown thatdistribution of values around the true value will be symmetrical and have a curved shaped(the bell curve).

The "actual", "best", or "true" value of a set of measurements is, as you probably alreadyknow, called the average or mean. It can be shown mathematically that this is the case,however, we will be content at this time to simply accept this fact.

If we denote a measurement as x, then the average of a set of x’s can be calculated as:

Average = x̄ =N∑i=1

xiN

(B.1)

The width of the distribution curve about the average is also important. If it is narrowthen most values are close to the average. If it is wide then the values are not as closeto the average. A narrow distribution indicates higher precision (lower error) than a widedistribution.

The width of a Gaussian distribution is denoted by its standard deviation. Standard devi-ation represents the average difference of a given measurement from the value of the mean.

23

If you look closely at the equation for calculating standard deviation you can see that youare taking the average of the square of the differences between the data points in the set andthe average of the set. If the standard deviation is numerically large then the distribution iswide. If it is numerically small then the distribution is narrow. Naturally then the standarddeviation is a measure of experimental error.

A Gaussian distribution (shape of the curve) is expressed mathematically as

N = Nmax exp[−(x− x̄)2

2σ

], (B.2)

where N = the number of times a given measurement is observed,Nmax = the greatest number of times a single measurement is observed,

x̄ = average value of all measurements,σ = standard deviation, defined below.

σ2x = limN→∞

1N

N∑i=1

(x̄− xi)2, (B.3)

where N = total number of measurements,xi = results of the ith measurement.

There is an interesting and useful relationship between a Gaussian distribution and itsstandard deviation. In any Gaussian distribution 68% of the total area under the curvefalls in the region from x̄−σ to x̄+σ. 95% of the total area falls between x̄− 2σ to x̄+ 2σ.This tells us that if an experiment is done to measure some quantity, although the result isnot duplicated exactly each time, there will be a definite pattern in the results. First therewill be a tendency toward some central value (x̄). Second, 68% of all values will fall within

24 APPENDIX B. STATISTICS

one standard deviation of x̄ and 95% of all values will fall between 2 standard deviations ofx̄. Thus, it is reasonable to associate σ with the uncertainty in a given set of measurementsand to view the Gaussian distribution as a probability curve.

For experiments done in the lab, use the following definition of standard deviation σx. (Onyour calculator this function is usually denoted as σn−1.)

σ2x =1

N − 1

N∑i=1

(x̄− xi)2 (B.4)

It can be shown that the standard deviation of a data set is related to the probable errorin the value of the average for the data set. The derivation of this is too complex to beconsidered at this time, but the result is worth knowing. The standard deviation of themean σm is given by

σm =σx√N

(B.5)

Note the factor of 1/√N present. This tells us that the more measurements we take, the

lower the standard deviation of the mean, and therefore the more confident we are in thatmean. This leads us to a very simple conclusion: More measurements = More accurateresults. All of your results should be presented with uncertainties (where possible), withthe uncertainty being represented by the standard deviation of the mean. Thus

x̄± σm (B.6)

An example: Consider the measurement of a steel rod using a micrometer (accurate to .001cm). The results are listed below.

25

length xi (cm) (x̄− xi)× 10−3 (cm) (x̄− xi)2 × 10−6 (cm2)10.351 0 010.354 -3 910.350 1 110.351 0 010.350 1 110.352 -1 110.352 -1 110.349 2 410.352 -1 110.349 2 4

10∑i=1

xi10 = 10.351 cm

10∑i=1

(x̄− xi)2 = 22× 10−6 cm2

The standard deviation is: σx = [2.2×10−5

10−1 ]12 = 1.6× 10−3cm

The standard deviation of the mean is: σm = σx√N =0.002√

10 = 5× 10−4

Therefore, the length of the rod is: 10.3510 ± .0005 cm.

Chauvenet’s Criterion

Another important consideration in experimental work is judging the validity of data. In-evitably when doing experiments data is generated which upon inspection seems to beincorrect. If mistakes have been made during the experiment then it is easy to discard thedata, however, if the experimental process is found to be valid then one must treat the datavery carefully. More than one Nobel prize in physics has been awarded because someonewould not throw away what was considered to be "bad data" at the time.

We will use Gaussian distributions as probability distributions in assessing the validity ofdata. As an example consider the following set of measurements of some quantity:

3.9, 3.5, 3.7, 4.0, 3.4, 4.1, 1.9

At first glance the value 1.9 appears to be incorrect when compared to the other values.Can the value of 1.9 simply be disregarded because it differs from the rest? If no evidence ofmalfunction in equipment and/or mistake in procedure can be found then one must providea justification for the rejection of data. One school of thought suggests that rejection ofdata is never justified. A somewhat more practical solution is to assess the probability thatthe suspect data is correct (Chauvenet’s criterion).

In order to use Chauvenet’s criterion the student must be comfortable with the conceptsof average and standard deviation and have realized that a Gaussian distribution is in fact

26 APPENDIX B. STATISTICS

a probability curve. The first step in assessing the data is to calculate the average andstandard deviation of the data (including the suspect value of 1.9).

x̄ = 3.50 σx = 0.75

Knowing the magnitude of the standard deviation makes it possible to calculate the dif-ference between the suspect value (xsus) and the average value in terms of the number ofstandard deviations (z).

z = |xsus − x̄|σx

= |1.9− 3.5|0.75 = 2.1

Recall that in a Gaussian distribution 68% of all measurements fall within 1 standarddeviation of the mean, 95% within 2 standard deviations of the mean and so on. Anotherway of saying this is that 32% of all values will be outside one standard deviation fromthe average value and only 5% of all values will be farther away from the average than twostandard deviations. In other words the farther away a value is from the average value, theless probable it is to be present in the data set.

It is possible to calculate the exact probability if one has a table or a computer to providethe values. Table 1 lists the area under a Gaussian curve as a function of the width measuredin standard deviations. Let’s call the probability of being part of this area, Prob(inside).

Since all of the area under the curve represents 100% of the possibilities then the proba-bility of being outside a certain portion of the curve (Prob(outside)) would simply be 1 -Prob(inside). For our example we get 96.43% or .9643 from Table 1, meaning that 96.43%of all our data should be within 2.1 standard deviations of the average.

Prob(outside 2.1σx) = 1− Prob (inside 2.1σx)= 1− .9643 = .0357 ≈ .036.

Therefore only 3.6% of all measurements would be farther away from the average.

Now the important part. In experimental work there is no such thing as part of a mea-surement. You either make a measurement or you don’t make a measurement. How manymeasurements do you need to make in order that one measurement represents 3.6% of thetotal number of measurements?

The number .036 suggests that if there were 28 (1/.036) measurements we should expectto see a value of 1.9, however, in a set of 7 measurements we would expect only .25 of ameasurement (.036 × 7) to have a value around 1.9. We must therefore decide whether.25 is sufficient to indicate that 1.9 is an "improbable" value for this set of measurements.Chauvenet’s criterion sets the value of "improbability" at .5. In other words if the probabilityof a value falling outside a given number of standard deviations (Prob (outside)) multipliedby the total number of measurements is less than .5 then the data may be rejected.

27

The process for using Chauvenet’s criterion is summarized as:

1. Calculate x̄, σx using all the data.

2. Find the difference d between your suspect point xsus and x̄.

d = |xsus − x̄|

3. Express this difference d as a multiple z of the standard deviation σx.

d = zσ, z = d/σ

4. Find z on the table provided, and record the percentage shown. Call this percentageP(inside).

5. Calculate P(outside), where P(outside) = 1 - P(inside).

6. Multiply P(outside) by the number of data points in the set (N).If this number is ≥ .5 the data point(s) is good.If this number is < .5 reject the data point(s).

NOTE: If data is rejected a new x̄, σx must be calculated based on the remaining data.

An Alternate Approach......

Suppose you have 500 data points of which 35 are suspicious. Do you need to use Chau-venet’s Criterion on each point? No, Chauvenet’s Criterion is based on the number of datapoints in the set. Begin by calculating P(outside) from

P(outside) = .5number of data points

and then work backwards to calculate the bounds of σx for valid data. The steps for thisprocedure are summarised below.

1. Calculate x̄, σx using all the data.

2. Calculate P(outside), where P(outside) = 0.5N .

3. Calculate P(inside) = 1 - P(outside).

4. Find P(inside) on the table provided, and record the corresponding z. P(inside) willtypically lie between 2 values on the table. In these cases, choose the value thatcorresponds to the higher z score.

5. Find the range of acceptable values of x:Minimum x: x̄− zσxMaximum x: x̄+ zσxAny measurements outside this range of acceptable values can be rejected.

28 APPENDIX B. STATISTICS

Table B.1: Gaussian Probability: Area under the Gaussian Curve

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.00 0.80 1.60 2.39 3.19 3.99 4.78 5.58 6.38 7.170.1 7.97 8.76 9.55 10.34 11.13 11.92 12.71 13.50 14.28 15.070.2 15.85 16.63 17.41 18.19 18.97 19.74 20.51 21.28 22.05 22.820.3 23.58 24.34 25.10 25.86 26.61 27.37 28.12 28.86 29.61 30.350.4 31.08 31.82 32.55 33.28 34.01 34.73 35.45 36.16 36.88 37.590.5 38.29 38.99 39.69 40.39 41.08 41.77 42.45 43.13 43.81 44.480.6 45.15 45.81 46.47 47.13 47.78 48.43 49.07 49.71 50.35 50.980.7 51.61 52.23 52.85 53.46 54.07 54.67 55.27 55.87 56.46 57.050.8 57.63 58.21 58.78 59.35 59.91 60.47 61.02 61.57 62.11 62.650.9 63.19 63.72 64.24 64.76 65.28 65.79 66.29 66.80 67.29 67.78

1.0 68.27 68.75 69.23 69.70 70.17 70.63 71.09 71.54 71.99 72.431.1 72.87 73.30 73.73 74.15 74.57 74.99 75.40 75.80 76.20 76.601.2 76.99 77.37 77.75 78.13 78.50 78.87 79.23 79.59 79.95 80.291.3 80.64 80.98 81.32 81.65 81.98 82.30 82.62 82.93 83.24 83.551.4 83.85 84.15 84.44 84.73 85.01 85.29 85.57 85.84 86.11 86.381.5 86.64 86.90 87.15 87.40 87.64 87.89 88.12 88.36 88.59 88.821.6 89.04 89.26 89.48 89.69 89.90 90.11 90.31 90.51 90.70 90.901.7 91.09 91.27 91.46 91.64 91.81 91.99 92.16 92.33 92.49 92.651.8 92.81 92.97 93.12 93.28 93.42 93.57 93.71 93.85 93.99 94.121.9 94.26 94.39 94.51 94.64 94.76 94.88 95.00 95.12 95.23 95.34

2.0 95.45 95.56 95.66 95.76 95.86 95.96 96.06 96.15 96.25 96.342.1 96.43 96.51 96.60 96.68 96.76 96.84 96.92 97.00 97.07 97.152.2 97.22 97.29 97.36 97.43 97.49 97.56 97.62 97.68 97.74 97.802.3 97.86 97.91 97.97 98.02 98.07 98.12 98.17 98.22 98.27 98.322.4 98.36 98.40 98.45 98.49 98.53 98.57 98.61 98.65 98.69 98.722.5 98.76 98.79 98.83 98.86 98.89 98.92 98.95 98.98 99.01 99.042.6 99.07 99.09 99.12 99.15 99.17 99.20 99.22 99.24 99.26 99.292.7 99.31 99.33 99.35 99.37 99.39 99.40 99.42 99.44 99.46 99.472.8 99.49 99.50 99.52 99.53 99.55 99.56 99.58 99.59 99.60 99.612.9 99.63 99.64 99.65 99.66 99.67 99.68 99.69 99.70 99.71 99.72

3.0 99.73 - - - - - - - - -3.5 99.95 - - - - - - - - -4.0 99.994 - - - - - - - - -4.5 99.999 - - - - - - - - -5.0 99.999 - - - - - - - - -

29

Linear Regression

Linear regression or least linear squares fitting provides a method by which the "best"straight line fit for a set of data may be calculated. Linear regression does not determine ifthe relationships in a data set are linear. That must be established by some other means(usually from theory). For example the speed of an object undergoing constant accelerationwill change linearly with respect to time. A plot of speed vs. time would therefore be astraight line. Linear regression could be used to determine the slope and intercept of theplot. The fact that the plot is linear is a consequence of the physics of the situation, not theuse of linear regression. If the plot where a curve rather than linear it would be meaninglessto find a slope and intercept using linear regression.

If we wish to fit a straight line through a set of data in which x is the independent variableand y the dependent variable the following must be true:

• each y value must be governed by a Gaussian distribution with standard deviation ofσy.

• the uncertainty in x must be small.

• the uncertainties in y must be uniform.

• the relationship between x and y must be linear, i.e. y = mx+ b.

Deriving the equations for linear regression is beyond the scope of Physics 1000. They aredeveloped by applying probability techniques to Gaussian distributions. The results of thederivation, called the normal equations, allow the calculation of the best fit slope and theintercept of the plot.

INTERCEPT = (∑xi

2)(∑yi)− (

∑xi)(

∑xi yi)

∆ (B.7)

SLOPE = N (∑xi yi)− (

∑xi)(

∑yi)

∆ (B.8)

where∆ = N

∑xi

2 − (∑

xi)2 (B.9)

APPENDIX C

Error Analysis

Error analysis is the evaluation of precision in measurement. As such, the word "error"takes on the connotation of "uncertainty" rather than "mistake". Since no measurement canbe completely free of error it becomes the experimenter’s task to minimize all uncertaintiesas far as reasonably possible. In our experiments this will be a three-fold process:

• Minimization of uncertainties in experimental apparatus.

• Optimization of experimental methods and procedures

• Calculation of uncertainties in final results.

Every experiment should begin with a thorough assessment of the apparatus including:condition, resolution, precision, calibration, etc. The apparatus should then be assembled,and procedures developed that will provide the most accurate results possible. Finally,the precision of the outcome should be calculated. Following this procedure allows theexperimenter to assess the worth of the experiment even before it is actually done.

The experimenter must have a reasonable knowledge of the types of uncertainties that existand how they will propagate in a given experiment. The remainder of this section will dealwith those two subjects.

Types of Errors

Systematic Errors

A systematic error is one which repeats itself to the same degree in each trial of an exper-iment. As such it cannot be discovered simply by repeating a measurement several times.Due to their reproducibility, systematic errors are very difficult to trace.

Systematic errors usually involve some aspect of the experimenter’s apparatus or methodol-

30

SUMMARY 31

ogy (improperly calibrated equipment, poor experimental design, and/or procedure are themost likely places to find systematic errors). Consider an experiment to measure voltagein an electrical circuit. If the voltmeter used in the experiment reads 1.5 volts when novoltage is applied (calibration error), each reading of the voltage will be 1.5 volts too large.This is typical of the most common types of systematic errors.

Random Errors

Simply stated, random errors are errors which can be found by repeating a measurementseveral times. Random errors can be treated statistically (systematic errors cannot) andtherefore repeated measurement allows one to minimize the effect of random error in exper-iments. Random error arises from our inability to hold all the variables constant in additionto the random character inherent in nature.

Reading Errors

Reading error is associated with any directly measured quantity and reflects the limits ofthe devices used in the measurement process. For example, a ruler graduated in millimetreshas much less reading error than a ruler graduated in centimetres. The resolution of thescale plus the interpolation of the measurement between the finest graduations of the scaleconstitute the major contributors to reading error.

In the case of digital readings the error is considered to be ½ of the final digit. For example,if a display reads 1.09, and there is no other information, the reading would be consideredto be 1.09 ± .005.

Summary

All the factors discussed, systematic, random, and reading errors combine to yield theoverall error in a measurement. Experimenters strive to minimize error through carefuldesign and thoughtful execution of experiments.

Data gathered from experimental work must carry with it some indication of uncertainty.This estimate of uncertainty follows the data through any calculation or manipulation andis always quoted with the final results. The next section deals with the propagation of errorduring calculations. It is more involved than the previous discussion and thus demandsmore attention.

32 APPENDIX C. ERROR ANALYSIS

Propagation of Errors

Definitions

Absolute Error: The total uncertainty in a quantity, x; usually expressed as δx (i.e.x± δx).

Relative Error: The uncertainty in a quantity, x in fractional form. Usually expressed asδx/x, where δx/x is dimensionless. The relative error can be expressed as percentageerror by multiplying δx/x by 100%.

Example: An iron block is found to weigh 3.44 ± .03 kg.Absolute error δm = .03 kgRelative error δm/m = .03/3.44 = .01 = 1%

Possible Error: The maximum possible uncertainty in the outcome of a mathematic op-eration involving values and their uncertainties.

Example: Consider the addition of two values A and B.

If A is known to a precision of ±δA, and B to a precision of ±δB, the sum of the twovalues could be as large as (A+ δA) + (B + δB), or as small as (A− δA) + (B − δB).If C = A+B then the possible error in C, δC would be δC = δA+ δB.

Possible error assumes the most extreme values in the uncertainty combine in theworst possible combination. For example, if A = 10±1 and B = 20±2 then for A+Bthe largest possible value would be 11 + 22 = 33 while the smallest value would be9 + 18 = 27, in other words 30 ± 3. Using the rule δC = δA + δB we arrive at thesame value for error in C namely, δC = 1 + 2 = 3.

Probable Error: A more conservative estimate than possible error. Probable error as-sumes that quantities used in calculations do not combine in the worst possible fash-ion (applies to mutually exclusive events) but rather with a statistical probability.Estimates of probable error will be lower than those of possible error.

PROPAGATION OF ERRORS 33

Rules for the Propagation of Errors in Calculations

Addition / Subtraction: A+B = C, A−B = C, where A± δA,B ± δB

Possible Error: δC = δA+ δB

Probable Error: δC =√

(δA)2 + (δB)2

Multiplication / Division: A×B = C, A/B = C, where A± δA,B ± δB

Possible Error: δC/C = δA/A+ δB/B

Probable Error: δC/C =√

(δa/A)2 + (δb/B)2

Exponents: C = An , where A± δa

Error: δc/C = n δa/A

General Expression for Error Propagation

The relationship for error propagation can be expressed and manipulated as the total dif-ferential for a dependent variable which is a function of one or more independent variables.

Suppose we had a function f(x, y, z), its total differential would be expressed as

df = ∂f/∂x dx+ ∂f/∂y dy + ∂f/∂z dz,

if we exchange the differentials df, dx, dy, dz for the uncertainties δf, δx, δy, δz we havean expression which allows us to calculate error propagation for more than the simplearithmetic operations shown previously.

δf(x, y, z) =

√(∂f

∂xδx

)2+(∂f

∂yδy

)2+(∂f

∂zδz

)2