A LABORATORY MANUAL

FOR

PHYSICS 117.3 &

PHYSICS 125.3

REVISED 2013

Department of Physics & Engineering Physics

University of Saskatchewan

PHYSICS 117.3/125.3 Laboratory Manual

TABLE OF CONTENTS

INTRODUCTION Page A: General Instructions ......................................................................... i B: Laboratory Reports .......................................................................... iii C: Graphical Representation and Interpretation of Experimental Data ....................................................................... vii D: Discussion of Errors ......................................................................... xii E: Significant Figures ........................................................................... xiv F: Estimating Absolute Errors from Random Causes .......................... xvi G: Propagation of Errors in Calculations ............................................. xvii H: Sample Calculations ........................................................................ xxiv I: Comparing Two Quantities .............................................................. xxv

Experiment MECHANICS

M39 Static Equilibrium (Non-concurrent Forces) …………………………………. 1 M19 Hooke's Law and Simple Harmonic Motion ...................................................... 5

LIGHT L11 Dispersion of Light by a Prism .......................................................................... 8

MODERN PHYSICS MP6 Hydrogen Spectrum ........................................................................................... 13 MP9 Radioactive Decay - Measurement of Half-life ................................................. 17

APPENDIX The Vernier Caliper ........................................................................................... 21 The Micrometer Caliper .................................................................................... 22 The Spectrometer ............................................................................................... 23 Beckman 310 Multimeter .................................................................................. 26 SI Units .............................................................................................................. 28 Data and Graph sheets

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A: GENERAL INSTRUCTIONS

Safety in the Laboratory The safety issues related to the equipment and procedures used in the Physics 117 laboratory are as follows:

thermal hazards due to use of mercury light source (experiment L11) electrical hazards, including high current and high voltage

General Safety Policies

No food or drink is to be consumed in the laboratory, nor be on the laboratory benches. Liquids brought into the lab room must be kept in sealed containers, which are not to be opened in the lab room and which must be placed on the floor (i.e. below the lab bench).

Follow all instructions In addition to the instructions in this laboratory manual, follow all verbal and written instructions provided by the instructional staff.

Come Prepared to the Laboratory

A proper understanding of your first year physics class requires a combination of studying the lecture material, solving problems, and performing experiments in the laboratory. An attempt has been made to relate the experimental work directly to the lecture material. Due to space and equipment limitations, however, experimental work may, in some cases, have to be carried out before the topic has been covered in the lectures. To compensate for this, before coming to the laboratory you should carefully read the sections of your text which are covered in the assigned experiment. If the material has been taken in lectures, the lecture notes should be studied as well. After having done this preparatory reading, carefully review the object, theory and procedure outlined in the lab manual so that you have an understanding of the purpose of the experiment, what data will be required to accomplish that purpose, and how that data will be collected. Use the material presented in this manual to plan in advance the way the experimental work should be organized.

Partners

Although students are allocated partners during the first laboratory period of the term, the laboratory reports should be written independently. This does not imply that partners should not cooperate; time and effort can be saved by working together in sharing the numerical calculations, checking results, and discussing the theory and performance of each experiment. Partners should share the work so that both partners understand the whole of the experiment and become familiar with the equipment. Partners may, and should, discuss their conclusions and sources of error with each other, BUT they must each write these sections in their own words.

Laboratory Assignments

A schedule of assignments, indicating who which experiments, when, and where, will be posted outside rooms 112 and 117, before the second laboratory session.

Grades

Marks obtained for the laboratory work account for 15% of the final grade, unless otherwise indicated by the professor of the class.

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Materials

Each student should bring to the laboratory an ACCO folder and 3-hole-punched sheets of ¼" grid paper, an inexpensive set of geometrical instruments, and an approved calculator (Texas Instruments TI-30X series or Hewlett Packard HP 10s or HP 30S). Specific instructions about the materials required for the laboratory classes will be given in the first meeting of the class.

Absenteeism

Students who are absent from a laboratory period for a justifiable reason (illness, Huskie Athletics…) may be able to arrange with the laboratory instructor to perform the experiment at another time. Documentation explaining the absence should be presented to the laboratory instructor whenever possible.

Assistance in the Laboratory The demonstrator in the laboratory is there to assist you. When something arises which you do not understand, and which cannot be resolved with the help of your partner, please consult the demonstrator.

BEFORE LEAVING THE LABORATORY, RETURN EQUIPMENT TO ITS ORIGINAL LOCATION SO THAT THE NEXT GROUP WILL NOT START THEIR EXPERIMENT AT A DISADVANTAGE. DISCONNECT ELECTRICAL CONNECTIONS. TURN OFF GAS, WATER AND AIR USED IN THE EXPERIMENT.

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B: LABORATORY REPORTS One of the aims of this laboratory is to teach you the methods that professional physicists have found satisfactory for recording the performance of experiments. Two criteria should be met by a satisfactory laboratory report:

1. A person with an educational background similar to yours should be able to easily follow through the experiment by reading your report, and if desired should be able to easily perform the experiment, obtaining results similar to yours.

2. The results of your study should stand out clearly and the conclusions drawn from these results should follow logically.

A suggested approach follows on the organization of your work to meet the above criteria. Details, such as correct procedures for drawing graphs, calculating errors, etc. are covered in later sections.

A log book format, rather than a formal lab report format will be used. The emphasis is on whether or not you understood and properly performed the experiment. The information contained in the log book should clearly indicate how the experiment was performed, what measurements were made, how these measurements were analyzed, and what conclusions were drawn. It is essential that you read the laboratory manual before coming to the lab so that you have a basic understanding of what you will be doing.

The report for each experiment will contain the following sections:

Heading title of experiment, date experiment performed, your name, partner's name overall format of report worth ½ mark

Object one or two sentences describing purpose of experiment worth ½ mark

Theory For each part of the experiment, a well-labelled diagram must be included. list of equations to be used in the lab, with definitions of symbols derivations of error equations should be included in this section worth 1 mark

NOTE: Heading, Object, and Theory sections must be completed before coming to the lab.

Data data should be tabulated whenever possible Tables must be titled, and as much information as possible is to be put in the column

headings rather than included with each entry. The heading of a column contains a one- or two-word descriptive label, the symbol for the quantity being tabulated, the units, the power of ten if scientific notation is being used, and the experimental error (if constant for all values in column). Individual column entries will be numbers or numbers ± errors. See Table 1 further in this introduction for an example of a proper table.

data must be recorded, with their errors and units, directly into the lab notebook (transfer of measurements recorded on scrap paper wastes time and can introduce mistakes)

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incorrect data are to be neatly crossed out with a single line in such a way that they are still legible

worth 1½ marks

Analysis this section involves manipulation of the raw observational data by graphing, making

numerical calculations, or in some other way specified in the instructions sample calculation (and error calculation if required) for each different equation units must be carried through calculations use a separate, complete page for each graph analysis results should be presented in tabular form whenever possible if possible, compare experimental results with theoretical or accepted values. The primary

criterion to be used when comparing values is whether or not the values agree within experimental error (i.e. do the error ranges overlap?). Although it is not the preferred comparison criterion, percentage difference between experimental and accepted/theoretical values may be used on occasion.

worth 2½ marks marks are lost in ½ mark increments for the following: calculation errors graph format errors units not carried through calculations poor presentation of results

Conclusion state results (with experimental error) and state whether or not there is agreement within

experimental error with theoretical or accepted values must include a discussion of the physical concepts examined in the experiment worth 2½ marks

Sources of Error state as many factors as you can think of which might have affected the outcome of the

experiment, and which were not accounted for in any way explain how each of the above factors would be expected to affect your calculated results,

and whether the effects would be large or small random measuring error need not be mentioned (it is usually accounted for in error

calculations) The possibility of incorrectly calibrated measuring instruments should not be mentioned

unless there is a particular reason for doing so. The same holds for the possibility of incorrect calculations.

worth 1½ marks Point-form may be used for the written sections of the report, but you must ensure that the meaning of your comments is clear. Reports are to be done in pen, although pencil may be used for diagrams. Corrections are to be made by drawing a single line through the incorrect values and/or statements, then writing the correct value/statement.

At the discretion of the marker, a report may be worth an additional 1 mark (max. mark = 10). To receive this bonus mark the report must contain work in addition to that required by the

v

manual. For example, provide constructive criticism and suggestions for improvement of the experiment or obtain additional data from which further conclusions relating to the experiment can be drawn.

NOTE: An average lab report is worth 7.0 marks. A good lab report is worth 8.0 marks. A very good lab report is worth 8.5 to 9.0 marks. An excellent lab report is worth 9.0 to 10 marks.

LOG BOOK STYLE SAMPLE The following is intended to show by example the style that is expected for your laboratory reports. The example is for the period measurement done in an experiment examining the simple harmonic motion of a mass on a vertical spring. The sample report will be written as if the period measurement comprised a complete experiment. 5 MAY 95 M19 – HOOKE’S LAW AND SIMPLE HARMONIC MOTION BRIAN ZULKOSKEY Object: to measure the period of the oscillations of a vertical mass-spring system Theory: A 200 g load was placed on the pan and the oscillation period, T, was measured five times using a photogate timer. Equation for Average Period:

554321 TTTTT

Tave

where T1, T2, T3, T4, T5 are the individual period measurements.

Equation for Average Deviation of Period values (error in Average Period):

Tave = 5

||||||||||.ave.dev 54321 aveaveaveaveave TTTTTTTTTT

photogate timer

spring

mass on pan

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Data:

Trial 1 2 3 4 5 Average Period, T (± 0.0001 s)

0.5731 0.5750 0.5737 0.5745 0.5739 0.57404 ± 0.00057

Analysis: Calculation of Average Period:

T1 = 0.5731 s

T2 = 0.5750 s

T3 = 0.5737 s

T4 = 0.5745 s

T5 = 0.5739 s

554321 TTTTT

Tave

5

s5739.0s5745.0s5737.0s5750.0s5731.0 aveT

s57404.0aveT

Calculation of error in Average Period (Average Deviation of Period values):

Tave = 5

||||||||||.ave.dev 54321 aveaveaveaveave TTTTTTTTTT

Tave = ave.dev. = |0.5731 s – 0.57404 s| + |0.5750 s – 0.57404 s| + |0.5737 s – 0.57404 s|

+ |0.5745 s – 0.57404 s| + |0.5739 s – 0.57404 s| ÷ 5

Tave = ave.dev. = 0.00057 s

Tave = 0.57404 ± 0.00057 s

Conclusion: The average period of the oscillating vertical mass-spring system was determined to be 0.57404 ± 0.00057 s. (In a regular experiment you would normally compare your experimental result to a theoretical or accepted value and discuss the significance of the result of the comparison.)

Sources of Error: it was difficult to avoid sideways motion of the mass-spring system, which would lengthen

the period (moderate effect) air resistance will damp the motion (small effect)

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C: GRAPHICAL REPRESENTATION AND INTERPRETATION OF EXPERIMENTAL DATA

In order to understand a physical problem, one usually studies the dependence of one quantity upon another. Whereas sometimes such a dependence is obtained theoretically, at times it must be arrived at experimentally. The experimental determination involves obtaining experimentally a series of values of one quantity corresponding to the various arbitrary values of the other and then subjecting the data to some kind of analysis. One of the most convenient and useful means of treating the experimental data is by graphical analysis.

A graph shows the relation between the two quantities in the form of a curve. Suppose we are interested in looking at the relation between the volume of a piece of iron and its mass. If we make measurements on pieces of iron, we find that 1 cm3 has a mass of 7.87 g, 2 cm3 has a mass of 15.74 g, 3 cm3 has a mass of 23.61 g, and so on. This kind of relation in which doubling the volume doubles the mass, tripling the volume triples the mass, is called direct proportion. In Physics you will encounter many cases of such relations. In describing this relation we say:

(i) Mass ‘is directly proportional to’ volume of iron or mass ‘varies directly as’ the volume of iron.

(ii) Mathematically we can write the relation as M V, where M is the mass of a piece of iron and V is its volume and the symbol means ‘is proportional to’. If we have two different volumes of iron, V1 and V2, their masses M1 and M2 may be expressed as

2

1

2

1

V

V

M

M

Another form of this relation expresses the fact than when mass and volume are related by direct proportion, they have a constant ratio. Thus

V

M of one sample =

V

M of another sample = k

The constant k is called the proportionality constant. In our example of iron

k = 7.87 g for each cubic centimetre

We now express this relation as an equation for any piece of iron:

V

M = k or M = kV, where the value of k as determined experimentally is 7.87 g/cm3.

An equation established in this way is an empirical equation.

We can illustrate this relation between mass and volume for iron by a graph. In trying to study the mass of a piece of iron as a function of its volume, we chose certain values for the volume and determined the corresponding values for the mass. In such a situation volume is said to be the independent variable and the mass is the dependent variable. Generally, the independent variable should be the variable that is altered in regular steps in the experiment, and the dependent variable is the quantity which is measured for each regular step of the independent variable. The independent variable (in our case the volume of a piece of iron) is plotted along the x axis and the dependent variable (in our case, the mass of the piece of iron) is plotted along

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the y axis. To plot the data we must choose scales – one for the vertical direction, marking off some suitable number of grams for each vertical division of the paper, and one for the horizontal direction, marking off volumes in cm3. Now we can place a point on the graph for each pair of values that we know.

Volume, V (cm3)

Mass, M (g)

1 7.87

2 15.74

3 23.61

Note that a straight line can be drawn through the data points: the graph is linear. The ratio

V

M is the same for all points on the line, i.e. as already noted,

V

M = k or M = kV where k is

a constant. As seen here and discussed in the next section, M = kV is the equation of a straight line passing through the origin if M and V are plotted against each other. k is the slope of the line.

General Linear Relationship

A straight line plot of y versus x indicates that the relationship is linear and is of the form

y = mx + b

where b is the intercept on the y-axis (the value of y when x = 0) and m is the slope of the line.

As an example, consider an examination of the relation between the strain (x) produced in a spring as a function of the applied force (F). A linear relation implies

F = kx + b

where k and b are constants. For this example refer to the data in Table 1.

Table 1: Spring Constant, Spring #1

Force, F (± 0.05N)

Average Strain, x (± 0.1 cm)

0.19 3.0 0.40 6.2 0.71 9.6 1.00 12.6 1.26 16.0 1.50 19.2

5

10

15

20

25

0 1 2 3

Volume (cm3)

Mas

s (g

)

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(x4, y4) = (19.2 cm, 1.55 N) (x2, y2) = (17.8 cm, 1.40 N)

N/m8.33N/cm0833.0

cm0.12

N00.1

cm8.5cm8.17

N40.0N40.1

slope12

12

xx

yy

N/m8.70N/cm0870.0

cm2.16

N41.1

cm0.3cm2.19

N14.0N55.1

slopemax34

34

xx

yy

(slope) = max slope – slope slope = 8.33 ± 0.37 N/m = 8.70 N/m – 8.33 N/m = 0.37 N/m (x1, y1) = (5.8 cm, 0.40 N) (x3, y3) = (3.0 cm, 0.14 N)

Average Strain, x (cm)

Graph 1: Force vs. Average Strain to determine Spring Constant

For

ce, F

(N

)

x

The experimentally obtained points usually do not lie exactly on a straight line; this is expected, because experimental data are never exact. In a straight line plot of experimental results, the straight line goes above some points and below others and represents an honest attempt on the part of the plotter to show the trend of the data. See Graph 1.

If the plot of y versus x yields a straight line, the relationship is linear and is of the form

y = mx + b

If m = 0, y does not depend on x. If the plot is a straight line passing through the origin, b = 0.

When drawing graphs, the following rules are to be observed in order that the graphs neatly and clearly represent the experimental data.

1. Draw the axes. Axes are not normally placed along the boundaries between the graph paper and the margin although in some cases it may be necessary to do so because of the extent of the graph. Axes are usually drawn one or two large divisions in from the left and up from the bottom margin respectively. Label each axis with the variable being plotted (e.g. mass, period, time, etc.) and the units.

2. Mark the scales along the axes. The following points should be considered in choosing the scale:

a) the graph should be easy to read and it should be possible to easily interpolate values without the need for a calculator. Make each large division equal to 1, 2, 4, 5 or 10 units. Do NOT use scales of 3, 6, 7, 9, ... units per division;

b) the resultant curve should fill the whole graph and should not be confined to a small area of the graph paper;

c) the geometrical slope of the curve (if it is a straight line) should be approximately unity;

d) the point (0, 0) should not appear on the scales unless the point (0, 0) is a significant point on the graph.

3. Plot the points carefully. Mark each point so that the plotted point stands out clearly. When two sets of different data are plotted on the same set of axes use different symbols (e.g. circles and crosses) for each set of data.

4. Fit a curve to the plotted points. Most of the graphs drawn will be for the purpose of illustrating a law or determining a relationship. Hence it is reasonable to assume that the graph should be a uniformly smooth curve, possibly a straight line. Owing to the limits of experimental accuracy not all the plotted points will be exactly on the smooth curve or straight line. Use a transparent French curve or straight edge and draw a curve through the plotted points so that:

a) the curve is smooth;

b) the curve passes as close as possible to all the data points;

c) when taken in moderately sized groups, as many points fall on one side of the curve as on the other.

Note that the curve need not pass through any one point, and certainly need not pass through the end points as these end points are usually obtained at the limits of the accuracy of the

xi

measuring instrument. Use dashes to show the extrapolation of the curve past the range of the data points.

5. Give the completed graph a descriptive title (what is being plotted and why). This should be placed in a position where it does not interfere with the curve.

6. If a slope calculation is required it should be done on the graph page or on the page immediately following the graph page. This reduces the possibilities of error in slope calculations and makes interpretation of the graph easier. A LARGE triangle should be drawn on the graph with vertices (x1, y1) and (x2, y2) as shown in Graph 1. The vertices are then used to calculate the slope according to the formula:

x

y

xx

yy

12

12

run

riseslope

Data points should not be used in slope calculations.

The experimental error in the slope can be calculated by performing an error calculation (discussed in a later section) on the slope equation.

7. It is sometimes desirable to indicate the experimental error in the plotted data points (see the section Discussion of Errors). This is done with ERROR BARS, which are shown in Graph 1. Vertical error bars are drawn by going above and below a data point a distance, determined by the vertical scale, corresponding to the error in the y-value. Horizontal error bars are drawn by going on either side of the data point a distance corresponding to the error in the x-value.

When error bars are drawn, the experimental error in the slope of a straight line is determined by one of the following methods:

Method 1

a) as already discussed, the best-fit straight line is drawn and its slope is calculated;

b) another line is drawn on the graph, this EXTREME-FIT line is the line passing through or near the error bars of all the data points and having the maximum or minimum possible slope;

c) the error in the best-fit slope is then taken as the absolute value of the difference between the extreme-fit slope and the best-fit slope.

Method 2

a) both the maximum-slope and the minimum-slope EXTREME-FIT lines are drawn on the graph;

b) the average slope is taken as the average of the slopes of the maximum-fit and minimum-fit lines;

c) the error in the average slope is taken as half the difference between the maximum-fit slope and the minimum-fit slope.

2

min)(max)(;

2

min)(max

slopeavgslopeavg

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D: DISCUSSION OF ERRORS Accuracy of Physical Measurements

Whether an experiment is of a precision type in which the answer is the magnitude of a quantity or of a qualitative nature in which the main aim is to substantiate a qualitative conclusion, it always involves taking measurements. Every measurement of a physical quantity is uncertain by some amount. When an observation or a result is recorded, one may ask the question - how precisely do we know this quantity? What we are asking for is a statement of some range of confidence in the quoted value or an expression of the uncertainty, sometimes called the error, in the quantity. By the word error we do not mean mistake but rather the uncertainty in the quantity.

There are three types of errors that may occur in experimental work:

1. Blunders or Mistakes on the part of the observer in reading the instruments or recording observations can be avoided by the exercise of reasonable care and therefore need no discussion. Some simple rules that will reduce the possibility of such mistakes are:

a) consider whether or not the reading taken is reasonable. Is it of the order of magnitude expected?

b) record the reading immediately and directly in the laboratory notebook. Do not attempt to remember it even for a few minutes nor record it on a loose sheet;

c) whenever possible have your partner check your reading.

2. Systematic Errors refer to uncertainties that influence all the measurements of a particular quantity equally. These errors may be inherent in the calibration of a particular instrument or in the particular design of the apparatus used. For example, the reading on a certain thermometer may be consistently too high because the thermometer was incorrectly calibrated. Systematic errors can usually be eliminated either by calibrating the particular instrument under conditions similar to those of the experiment or by applying a correction term in the calculations. In those cases where it is known to be necessary to apply a correction for a systematic error in the experiments performed in this laboratory, the instructors will give the details of the procedure to be followed.

3. Experimental or Random Errors remain even when all mistakes and systematic errors have been avoided or accounted for. Thus when a given measurement is repeated the resulting values, in general, do not agree exactly. No one has so far designed a perfect experiment: imperfections in our senses; in the instruments we use; and chance variations in conditions assumed to remain constant all combine to cause uncertainties in our measurements. These random errors are unavoidable and their exact magnitude cannot be determined.

A good example of these random deviations or accidental errors is provided by the shot pattern obtained when an accurately zeroed and solidly clamped rifle is fired at a target. The pattern of fire will show a scatter of shots about the centre of fire, the number of shots in a given annular ring of constant width being progressively less as the distance from the centre of fire increases. The same effect may be demonstrated by repeatedly dropping a dart or sharp-pointed pencil vertically onto a mark on a piece of paper. The scatter of the shots about the aiming mark is analogous to the spread of experimentally obtained values about their mean value, due to accidental errors.

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It is often difficult to determine the magnitude of systematic errors in an experiment. For example, in the case discussed of a systematic error introduced by a poorly calibrated thermometer one doesn't know if the error is 1° or 10° unless the thermometer is compared to a reliable thermometer. In the case of random errors, however, the data taken in the experiment is enough to indicate how large these errors are.

A common way of approximating the magnitude of random errors is the use of significant figures. For a more reliable estimate of these errors, statistical methods and error calculations are employed.

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E: SIGNIFICANT FIGURES

Numerical quantities have a different meaning in physics than they do in mathematics. The term ‘significant figures’ refers to the fact that while to a mathematician numbers are exact quantities, to a physicist a number is often a measured quantity and therefore inexact. A rough approximation of the uncertainty or experimental error in a number is indicated by the physicist by only writing those figures of the number in which he/she has confidence, i.e. by only writing the figures that are significant. For example, to a mathematician, 16 is the same as 16.0 or 16.00, but to a physicist, if x is a length measurement then x = 16 cm means 15.5 cm < x < 16.5 cm; x = 16.0 cm means 15.95 cm < x < 16.05 cm; and x = 16.00 cm means 15.995 cm < x < 16.005 cm.

Note that zeros used as place holders between the decimal point and the number are not significant and that zeros after the number may or may not be significant. To avoid this possible source of confusion very large and very small numbers are often written in scientific notation (also called power of ten notation). In this notation the number is written with one figure to the left of the decimal point and then multiplied by the appropriate power of ten.

e.g. .001804 = 1.804 × 10–3 (4 sig. fig.)

These conventions regarding significant figures are illustrated in the following table:

Two Significant Figures

Three Significant Figures

Four Significant Figures

32 32.0 32.00 .0032 .00320 .003200

3.2 × 104 3.20 × 104 3.200 × 104 When using significant figures in calculations remember that the result of a calculation cannot be expressed to a greater degree of accuracy than the numbers from which it is calculated. For example, consider a case in which the length of an object is 20.5 cm and the width is 14.23 cm, and we are interested in finding the sum of the length and width of the object. Representing unknown figures by a question mark, we may write the sum as: 20.5?? 14.23? 34.7? Notice that the sum of a known figure and an unknown one is an unknown figure. The result of this example can be generalized to the following statement: the result of an addition or subtraction calculation is rounded off to the same number of DECIMAL PLACES as the number with the least decimal places that is used in the calculation. (Note that before applying this rule all the numbers involved in the calculation must be expressed to the same power of ten if scientific notation is being used.)

Similarly, when multiplying or dividing, a result can have no more SIGNIFICANT FIGURES than does the number with the fewest significant figures that is used in computing the result.

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Exercises on Significant Figures

1. How many significant figures does each of the following numbers have? a) 6.3 × 103 b) .00370 c) 6700 d) 670.0 × 103

2. Round off to 2 significant figures and express in scientific notation: a) 3.95 b) .02862 c) 219 d) 4326

3. Calculate to the appropriate number of significant figures and express in scientific notation: a) 63400 + 82 b) 480 ÷ .060 c) 379 – (6 × 103) d) 8900 ÷ 30 e) 290 + 6.7 f) .030 – .003 g) 321 × 3

4. Suppose that you are instructed to draw a line ten centimetres long using an ordinary ruler (one which has millimetres marked on it). How will the length of the line be best expressed?

a) 1 × 101 cm b) 10 cm c) 10.0 cm d) 10.00 cm

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F: ESTIMATING ABSOLUTE ERRORS FROM RANDOM CAUSES The ABSOLUTE error in a quantity is the estimate of the range of values to be expected due to experimental errors. The absolute error has the same units as the quantity and is denoted by the Greek symbol lower-case delta () in front of the symbol for the quantity. For example, if a length measurement is denoted by x then x is the absolute error in this length. Experimental error can also be expressed relative to the quantity. This RELATIVE error is defined as the absolute error divided by the quantity. In symbols the relative error would be x/x for the example mentioned above. If expressed as a percentage, the relative error becomes a PERCENTAGE error, and is calculated by (x/x) × 100%.

The correct use of significant figures gives a rough estimate of the magnitude of random errors involved in taking readings. A more precise and reliable method of determining random errors is to determine absolute errors by estimation or calculation and then to use these values in subsequent calculations. Generally speaking, calculated values can be no more precise than the individual measurements. In fact, the errors accumulate so that the calculated value is usually less precise than the measurements on which it is based. The errors in a calculated quantity can be determined from the errors in each of the directly measured quantities on which the calculation is based. Before proceeding to calculate the experimental error, however, we must decide what numerical value shall be given to the absolute error in any measurement. The following rules may be used for assigning absolute errors:

1. When a quantity is just measured once, the measured value is taken as being the ‘true’ value and the absolute error due to random causes is estimated as the MAXIMUM INSTRUMENTAL ERROR in the measuring instrument. The maximum instrumental error will usually be the least count of the instrument (the size of the smallest scale division). For example in measuring the length of an object with a meter stick, the error would be ±1 mm since 1 mm is the smallest scale division on the meter stick.

2. A more accurate estimate of error can be obtained when a quantity is measured several times. The arithmetic mean of the measurements is taken as the ‘true’ value and the average deviation from the mean is taken as the absolute error unless it is less than the maximum instrumental error as described in 1. The average deviation is given by

ave.dev. =x x x x x x

nn1 2

where x1, x2, ..., xn are the individual measurements,

xn

xx x x x

nii

nn

1

1

1 2 3 is the average of these readings,

and n is the number of measurements.

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G: PROPAGATION OF ERRORS IN CALCULATIONS In a number of cases the quantity of interest cannot be measured directly – it is calculated by using two or more directly measured quantities. When uncertain quantities are used in numerical calculations one usually wishes to know the error in the result. The simplest method, although not the most accurate, is to determine the maximum possible error. MAXIMUM POSSIBLE ERROR METHOD

Bottom Line: The maximum absolute uncertainty in the addition or subtraction of measured values is the sum of the absolute errors in the measured values. The maximum percentage (relative) uncertainty in the multiplication or division of measured values is the sum of the percentage (relative) errors in the measured values. The maximum percentage (relative) uncertainty in a measured value raised to a power is the product of the power and the percentage (relative) error in the measured value.

Details: Addition and Subtraction

Consider the sum of two measurements, 25.4 ± 0.1 cm and 7.5 ± 0.2 cm. The result is 32.9 cm. To find the error in this result, consider that the first measurement lies between 25.3 and 25.5 cm. The second measurement lies between 7.3 and 7.7 cm. Therefore the sum lies between 32.6 and 33.2 cm, which may be written as 32.9 ± 0.3 cm. The absolute error in the result is the sum of the absolute errors of the two measurements. In symbolic form this may be written:

(a ± a) + (b ± b) = (a + b) ± (a + b)

To generalize:

if P = Q1 + Q2 + ... + Qn

where Q1, Q2, ... Qn are all measured quantities,

then P = Q1 + Q2 + ... + Qn

Now consider the subtraction of the two measurements from the previous discussion. The result is 25.4 cm – 7.5 cm = 17.9 cm. The limits of the range of the result are obtained by calculating

(25.4 cm + 0.1 cm) – (7.5 cm – 0.2 cm) = 25.5 cm – 7.3 cm = 18.2 cm

and

(25.4 cm – 0.1 cm) – (7.5 cm + 0.2 cm) = 25.3 cm – 7.7 cm = 17.6 cm

Thus the result is expressed as 17.9 ± 0.3 cm. Thus when quantities are subtracted the absolute error in the result is the sum of the absolute errors in the quantities.

Combining this with the result for addition, if P is calculated from the addition and/or subtraction of a number of measured quantities Qi,

P = Q1 + Q2 – Q3 – Q4 + ... + Qn

then P = Q1 + Q2 +Q3 + Q4 + ... + Qn

xviii

WHEN QUANTITIES ARE ADDED OR SUBTRACTED THE MAXIMUM POSSIBLE ABSOLUTE ERROR IN THE RESULT IS THE SUM OF THE ABSOLUTE ERRORS IN THE QUANTITIES. Multiplication

Consider a rectangle measured to have sides a ± a and b ± b. The area A and its absolute error A are calculated as follows:

A±A = (a ± a)( b ± b) = ab ± (a)b ± ab) ± (a)b)

Factoring the product ab yields:

A A aba

a

b

b

a b

ab

1( )( )

Noting that ab = A, and ignoring the second order term (a)(b)/ab since it will be small compared to the other terms in the square brackets, yields:

A Aa

a

b

b

i.e. A

A

a

a

b

b

In general, if P is the product of measured quantities Q1, Q2, ... Qn, then the relative error in P is

P

P

Q

Q

Q

Q

Q

Qn

n 1

1

2

2

PQ

Q

Q

Q

Q

QP

n

n

2

2

1

1

THE MAXIMUM POSSIBLE RELATIVE ERROR IN A PRODUCT IS THE SUM OF THE RELATIVE ERRORS IN THE FACTORS. Noting that relative and percentage errors are related by % error = 100 × (relative error), the maximum possible percentage error in a product is the sum of the percentage errors in the factors. Division

Let P be the result of the division of Q by R.

i.e. PQ

R

Then

P PQ Q

R R

QQ

Q

RR

R

Q

R

Q

QR

R

1

1

1

1

Applying the binomial expansion and ignoring 2nd order terms yields:

xix

P PQ

R

Q

Q

R

RP

Q

Q

R

R

1 1

so

P PQ

Q

R

R

i.e. P

P

Q

Q

R

R and P

R

R

Q

QP

THE MAXIMUM POSSIBLE RELATIVE ERROR IN A QUOTIENT IS THE SUM OF THE RELATIVE ERRORS IN THE DIVISOR AND DIVIDEND. Powers and Roots

Using the result obtained for multiplication and realizing that the expression Qn means to multiply Q by itself n times, if

P = Qn then

P

Pn

Q

Q

THE MAXIMUM POSSIBLE RELATIVE ERROR IN RAISING A BASE VALUE TO AN EXPONENT IS THE PRODUCT OF THE EXPONENT AND THE RELATIVE ERROR IN THE BASE.

For a more general example, consider the quantity N given by the following expression:

N = (AB4)/(CD½)

where A, B, C, D are all measured quantities. The maximum possible relative error in N is

N

N

A

A

B

B

C

C

D

D 4 ½

Note that the error in N is influenced more by errors made in measuring B than any of the other quantities. In designing and performing such an experiment great care should be taken to measure B with the highest degree of precision attainable. SUMMARY 1. All measured quantities have associated with them an uncertainty which can be expressed as

an absolute or relative error and which limits the number of significant figures of these quantities and any results calculated from them.

2. When uncertain quantities are added or subtracted the maximum possible absolute error of the result is determined by adding the absolute errors of the separate quantities.

3. When uncertain quantities are multiplied or divided the maximum possible relative error of the result is determined by adding the relative errors of the separate quantities.

xx

4. When an uncertain quantity is raised to a power (including a fractional power) the maximum possible relative error in the result is determined by multiplying the relative error in the quantity by the exponent.

5. When uncertain quantities are related in more complicated ways (e.g. log functions, trigonometric functions) the result must be calculated twice, once with the experimental values and then with the extreme values of the quantities. The maximum possible absolute error is the difference between these two values of the complicated function.

These basic error propagation rules can be applied to elementary trigonometric functions as well, using the following sequence: (Be sure to perform all three steps)

First, derive the relative error equation in the usual way, treating the trigonometric function and its argument as a single variable:

i.e., if the equation is x = () (cosθ), the relative error equation would be

)(cos

)(cos

x

x

Therefore xx

)(cos

)(cos

Second, replace the relative error in the trigonometric function with its mathematical equivalent from the set given below:

δ(cosθ) = (δθ)(sinθ), δ(sinθ) = (δθ)(cosθ), δ(tanθ) = (δθ)/(cosθ)2

(These are derived using differential calculus, and cannot be directly obtained using our basic rules. They also require the ERROR in the angle (not the angle itself) to be expressed in Radians)

Then, in our example, xx

)(cos

))(sin(

Normally, an error equation in this form would suffice. However, the trigonometric function in the denominator could potentially equal zero, causing a division by zero.

Third, back-substitute the original equation into the error equation, expand and simplify:

i.e., since x = () (cosθ), replace x in the above equation with () (cosθ):

)(cos)(cos

))(sin()(cos)(cos

)(cos

))(sin(

x

which simplifies to ))(sin()(cos x

NOTE: For the units to work out correctly in this type of equation, (δθ) [and only (δθ)], must be expressed in RADIANS. (δθ° x 2π/360 = δθrad)

xxi

Examples

1. P = (m + M) (1)

There is both addition and multiplication in this equation. Make the substitution

a = m + M (2)

As this is addition only, we add absolute errors.

a = m + M (3)

Substituting equation (2) into equation (1) yields

P = a

which is multiplication only so we add relative errors:

PMm

Mm

a

aP

Mm

Mm

a

a

P

P

+

+so,

+

+

2. When resistances R1, R2, and R3 are connected in parallel the total resistance, R, is given by the equation

1 1 1 1

1 2 3R R R R (1)

Make the substitutions aR

bR

cR

dR

1 1 1 1

1 2 3, , ,

Equation (1) becomes a = b + c + d which is addition so we add absolute errors.

a = b + c + d (2)

From aR

=1

, a can be found. This is division so we add relative errors,

a

a

R

R

R

R

R

R

1

10

but absolute errors are needed to substitute into equation (2)

a aR

R R

R

R

R

R=

12

Similarly for b, c and d we get

bR

Rc

R

Rd

R

R= = =1

12

2

22

3

32, ,

Substituting these values into equation (2):

R

R

R

R

R

R

R

R21

12

2

22

3

32

xxii

or

R RR

R

R

R

R

R

2 1

12

2

22

3

32

ERROR CALCULATION QUESTIONS

These questions are designed to help you in reviewing error calculations. In each of the questions you will be given an equation and some data to use in the equation. Each question consists of four parts.

(a) Give the relative error equation. The relative error equation should be in symbols only.

(b) Use the data to do the calculation. Watch your significant figures and the units.

(c) Calculate the relative error and the percentage error.

(d) Calculate the absolute error.

Sample

The equation of kinetic energy is KE = ½m2.

m = 5.32 ± 0.03 kg

= 2.37 ± 0.07 m/s

(a)

2m

m

KE

KE

(b) KE =(5.32 kg)(2.37 m / s)

J2

214 9 .

(c) KE

KE

0.03 kg

5.32 kg

0.07 m / s

2.37 m / s2 0 065 6 5%. .

(d) KE = .065 KE = .065(14.9 J) = 1.0 J

KE = 14.9 ± 1.0 J 1. The equation of an ellipse is

y bx

a= 1

2

2 ,

a = 22.6 ± 0.1 cm, b = 46.9 ± 0.1 cm, x = 12.2 ± 0.1 cm: y = 39.48 0.29 cm

xxiii

2. From Kepler's third law of planetary motion, the orbital period T of a satellite in a circular orbit at height h above a massive body of radius R and mass M is:

TR h

GM= 2

3

( ),

R = (1.738 ± 0.001) × 106 m; h = (1.1 ± 0.1) × 106 m

G = 6.67 × 10–11 N·m2/kg2; M = (7.35 ± 0.01) × 1022 kg

T = (1.357 0.073) 104 s

3. Consider an Atwood's machine in which the pulley is neither frictionless nor massless. In that case the acceleration is

am m g

I R m m m

( )

( / )2 1

21 2

where I is the moment of inertia of the pulley, R is its radius and m is a mass that is used to counteract the friction. Given g = 980 cm/s2, m1 = 100 ± 1 g, m2 = 200 ± 2 g,

I = 950 ± 50 g·cm2, R = 3.5 ± 0.1 cm and m = 5.0 ± 0.1 g: a = 256 15 cm/s2.

xxiv

H. SAMPLE CALCULATIONS For each different equation used in the analysis of an experiment a sample calculation (and possibly a sample error calculation) are required. The following discussion explains what is required in a proper sample calculation.

1. Label the calculation descriptively. i.e. Explicitly state, in words, what is being calculated. If this is one of many similar calculations, specify which one.

2. State the values, with their units and experimental errors, that will be used in the calculation. 3. State the equation in symbols. 4. Show the substitution of values (with their units but not experimental errors) into the

equation. 5. Give the result of the calculation, with correct units. (Don’t round off. Keep extra

significant figures at this point, especially trailing zeros. Rounding will be done later.) 6. State the error equation in symbols. (This was derived as part of your Theory section) 7. Show the substitution of values (with units) into the error equation. 8. Give the unrounded result of the error calculation, with units. 9. State the result and its absolute error with the appropriate number of significant figures.

The following rule is to be used in rounding off the results of calculations: ROUND OFF THE ABSOLUTE ERROR IN THE RESULT TO TWO SIGNIFICANT FIGURES, THEN ROUND OFF THE RESULT TO THE SAME NUMBER OF DECIMAL PLACES AS THE ABSOLUTE ERROR. If scientific notation is being used, express both the result and its absolute error to the same power of ten before applying the second part of this rule.

As an example, suppose the magnitude of the momentum of a car on an air track whose mass and speed have been measured is to be calculated. The following is the correct sample calculation:

1. Sample Calculation of Momentum of Car (label) on Air Track for Trial #1.

2. m = 0.675 ± 0.001 kg (statement of values) = 0.123 ± 0.005 m/s

3. p = m (equation in symbols) 4. p = (0.675 kg)(0.123 m/s) (substitution of values) 5. p = 0.083025 kg·m/s (calculation result)

6. pm

mp

(error equation in symbols)

OR p = m + m (if back-substitution was employed)

7. p

0 001

0 675

0 005

01230 083025

.

..

kg

kg

m / s

m / skg m / s

.

. (substitution of values)

8. p = 0.003498 kg·m/s = 0.0035 kg·m/s (error calculation result)

9. p = (0.0830 ± 0.0035) kg·m/s OR (round off) p = (8.30 ± 0.35) × 10–2 kg·m/s if scientific notation is being used

xxv

I: COMPARING TWO QUANTITIES Experiments sometimes involve determining a physical quantity by two different methods to see if the results of the two methods agree. Sometimes one wishes to compare the experimental results with the predictions of a theory. In either case it is highly unlikely that there will be exact numerical agreement between the values to be compared. To compare the quantities, and decide if there is agreement, their experimental errors must be calculated. If the ranges of the two values overlap the values are said to agree within the limits of the error of that experiment. If the ranges do not overlap, then the values do not agree.

For example, if one determines the value of g to be 979 ± 3 cm/s2 and the known value is 981 cm/s2 then the two values agree because the range of the experimental value (976 cm/s2 to 982 cm/s2) includes the known value.

Bar Graph Comparison gexp = 979 3 cm/s2

975 976 977 978 979 980 981 982 983 cm/s2 | | | | | | | | | | |

Algebraic Comparison

The test for error range overlap can be expressed quantitatively as follows. Let x1 x1 and x2 x2 be two values that are to be tested for agreement within experimental error. If the magnitude of the difference between the values is less than the sum of their absolute errors then the values agree.

i.e. if | x1 – x2 | x1 + x2 is a true statement,

then x1 and x2 are said to be equal, “within experimental error”. When experimental errors have not been computed, the experimental value is compared with the standard known value by computing the percentage difference:

%100valueknown

valueknownvaluealexperimentdifference%

When a physical quantity does not have an accepted value, the following equation is used to calculate the percentage difference between values determined by two experimental methods:

%100valuesmaller

valuesmallervaluelargerdifference%

When the percentage change in a value is to be calculated, the following equation is used:

%100valueinitial

valueinitialvaluefinalchange%

range of gexp values

gacc

1 M39 M39.1

STATIC EQUILIBRIUM (NON-CONCURRENT FORCES)

OBJECT The object of this experiment is to examine the conditions that apply to a rigid body in static equilibrium.

THEORY

Reference: Sections 8.1, 8.2, and 8.4, College Physics, Serway and Vuille

Let F

be a force acting on an object and let r be a position vector from a chosen point O (the axis of rotation) to the point of application of the force. The torque exerted by the force (1) τ = ± rF sinθ

where r is the length of the position vector, F the magnitude of the force, and θ the angle between r and F

.

The torque is positive if the force tends to produce a counter-clockwise (CCW) rotation about the axis, and negative if the force tends to produce a clockwise (CW) rotation.

An alternative expression for torque uses the concept of lever arm. The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force (the line of action). Since d = r sinθ,

(2) τ = ± F d

See figure 8.4 on page 238.

EXPERIMENT

Equipment: Geometry set, 12” ruler, large right triangle (provided by the student)

Parallel rulers, masses, electronic balance, metal bar, pulley board (provided in lab)

Three different masses are used to supply three forces on a bar as shown in Figure 1. Note that the weight of the bar produces a fourth force acting down at the centre of the bar.

Figure 1

O

2 M39.2

Procedure:

Secure a large piece of paper to the panel. Hang three different masses and the bar over the pulleys as shown in Figure 1, such that the bar lies near the centre of the sheet. Each mass should be greater than or equal to 100 grams and less than 350 grams. Using two triangles trace the corners of the bar, the four holes on the bar, and points at the ends of the three strings onto the paper (using the method employed in lab M3). The appropriate force designation (F1, F2, F3) should be written alongside the appropriate line.

Measure the mass of the bar and of each of the hanging masses using an electronic balance, OR use the stamped values on each mass (as in Lab M3) and the given mass of the bar. Record these masses, with error, (in grams), on the sheet of paper at the appropriate location.

Remove the paper from the board. Draw lines from the points indicating the position of the holes on the bar to the points indicating the position of the ends of the three strings. Draw a line connecting the centres of each of the holes, extending in both directions, and define this line as the x axis of an x-y reference frame. Choose the origin O to be the geometric centre of the bar and draw the y axis perpendicular to the x axis. Draw the direction of the force vector, W, due to the weight of the bar, parallel to F2. (Remember that the weight of an object acts through its centre of mass, assumed to be the centre of the bar in this case).

Measure, and record on the sheet, the acute angles θ, that the directions of the force vectors make with respect to the x-axis, for each of the four force vectors.

Construct the lever arm distances, d, (see figure 8.4 in your text) as follows: Extend the line of action of each force. (A dotted/dashed line is recommended). Using either a triangle and a straight-edge, or a compass and straightedge, construct a perpendicular line from the Origin to the line of action of the force. This line is the “lever arm distance”, d. Measure and record these distances on the sheet, in cm. (Note that d for the weight force is 0.0 cm)

Transfer the data from the sheet to the Data Table provided. Convert the masses (and their errors) to kilograms and the lever arm distances (and their error) to metres. For each angle specify, in the data box, whether this angle was measured above or below the positive or negative x axis

ANALYSIS

Calculate the magnitude of the forces acting on the bar, (which are simply the weights of the masses), with their experimental errors, in Newtons. (in the lab, g = 9.811m/s2 “exactly”): (3) F = mg.

Resolve each of the forces into x and y components, with errors. Give the appropriate sign (ie. direction) to each of the components:

(4) Fx = ± F cosθ

(5) Fy = ± F sinθ

Calculate the magnitude of the net force on the bar, with error, for both x and y direction:

(6) Fnetx =ΣF x = F1x + F2x + F3x+ Fwx

(7) Fnety =ΣF y = F1y + F2y + F3y+ Fwy

3 M39.3

In each case, if the sum of the force components (ie the “net” force) is LESS than the sum of the errors in the components, then that net force can be considered “0”, within experimental error.

Calculate the torque about the origin O due to each force, with error, using equation (2). Assign each torque as positive or negative using the sign convention given in the Theory.

Calculate the resultant net torque acting on the bar due to the forces, with error:

(8) τnet = Στ = τ1 + τ2 + τ3 + τw

If the sum of the torques (ie the “net” torque) is LESS than the sum of the errors in the torque, then the net torque can be considered “0”, within experimental error.

CONCLUSION

What was the state of motion of the metal bar when it was freely hanging on the pulley board?

Given this state of motion, what do you expect to obtain for the sum of the external forces acting on the metal bar?

Discuss the results of your force calculations. Explain clearly whether or not your results confirm your expectation for the net external force on the bar.

Given the state of motion of the bar, what do you expect to obtain for the sum of the external torques acting on the metal bar?

Discuss the results of your torque calculations. Explain clearly whether or not your results confirm your expectation for the net external torque on the bar.

In your own words, state the conditions that are satisfied by an object in static equilibrium.

SOURCES OF ERROR

In addition to discussing all factors that likely affected your experiment, consider the following:

• Are there additional forces affecting the system that have not been considered?

• Are there errors that may arise from the method of transferring the string positions to the paper?

4

M39 STATIC EQUILIBRIUM (NON-CONCURRENT FORCES)

DATA & RESULTS Force and Torque Analysis for Object in Static Equilibrium

Force

Mass, m (kg)

Force, F (N)

Angle with x-axis, θ

(° ± ____°)

Fx (N)

Fy (N)

Lever arm, d (±______m)

Torque, τ (N·m)

1 ± ±

± ±

±

2 ± ±

± ±

±

3 ± ±

± ± ±

W (bar) ± ±

± ± 0.000 ±

Σ

± ±

±

5

M19 M19.1 HOOKE'S LAW AND SIMPLE HARMONIC MOTION

OBJECT The object of this experiment is to determine whether a vertical mass-spring system obeys Hooke's Law and to study simple harmonic motion.

THEORY

References: Sections 13.1 and 13.3, College Physics, Serway and Vuille

The restoring force, F, of an ideal spring is said to obey Hooke’s law:

xkF −= (1)

where F is the restoring force exerted by the stretched or compressed spring; k is the spring constant of the spring; x is the displacement of the system from its unstrained length.

In the case of the vertical spiral spring and pan used in this experiment, when a mass is added to the pan and the system is lowered to its new equilibrium position, the spring stretches until the restoring force due to the additional stretch of the spring equals the weight of the added mass. The displacement, x, is the displacement of the system from its position before any mass was added, i.e. the displacement from the original equilibrium position.

If the mass at the end of the spring is pulled down and released, the system oscillates. For a system in which the restoring force obeys Hooke’s Law, the oscillatory motion is called simple harmonic motion. It can be shown that for a body moving in simple harmonic motion, T, the period of the motion (the time for one complete oscillation) is given by

kmT π2= (2)

EXPERIMENT

Equipment: spring-pan assembly, masses, photogate timer, electronic balance

The apparatus consists of a hanging spring, with a pan suspended from it on which masses may be placed. The position of the pan can be read on a millimetre scale printed on a reflective metal plate mounted on the post supporting the spring-pan assembly.

Procedure:

Remove the spring-pan assembly from the post. Detach the spring and measure the mass of the pan.

Re-attach the spring and hang the spring-pan assembly on the support. Set the spring displacement scale to read zero with no mass on the pan. Add a mass of 50 grams and record the scale reading. The pointer on the pan should be aligned with its reflected image to eliminate any error due to parallax when taking readings. Record the displacement scale readings for load masses of 100 to 350 grams in 50 g increments.

With the load mass of 350 grams, start the mass-spring system oscillating in the vertical direction (try to avoid any pendulum-like swing) and using the photogate as a stopwatch (set it to the PULSE position) measure the time for 10 complete oscillations.

Repeat the above measurement (time for 10 oscillations) for masses of 300, 250, 200, and 150 grams.

6

M19.2

ANALYSIS

Calculate the force exerted by the spring (= mload g) and the error in the spring force.

Plot a graph of the force exerted by the spring versus the displacement x.

Determine the spring constant, k, of the spring from the graph.

In equation (2) the mass m is the total mass that is undergoing oscillations.

Thus we will use m = mpan + mload (3)

Referring to equation (2), plot the measured period versus an appropriate function of m to show that your mass-spring system is undergoing simple harmonic motion.

Using the results from your graph of period as a function of oscillating mass, perform the necessary calculations to attempt to verify equation (2). Be sure to clearly state what you are calculating and why.

CONCLUSION

Do your results confirm that your mass-spring system obeyed Hooke’s Law? Explain.

Explain clearly in your own words whether you feel justified in concluding that your mass-spring system is an example of a system that can undergo simple harmonic motion?

SOURCES OF ERROR

Consider the effects of any assumptions made regarding the spring.

Consider the effects of additional forces on the pan.

Was the motion of the pan purely vertical? If not, consider the effects of the observed motion.

The mass of the pan was added to the load mass to obtain the total mass that was oscillating. Was any other mass ignored in this calculation? If so, specify and discuss.

7

M19 HOOKE'S LAW AND SIMPLE HARMONIC MOTION

DATA & RESULTS

mass of pan, mpan = _______________ ± ______________ kg

Force, Displacement, and Oscillation Data for Spring

Load Mass mload (kg)

Spring Force (mload g)

(N)

Scale Reading x (cm)

(±______cm)

Time for 10

oscillations (s ± ______

s)

Period, T (s)

(± ______ s)

Oscillating Mass

(mload + mpan) (kg)

0 0 0.0

0.050 ±

_______ ±

0.100 ±

_______ ±

0.150 ±

_______ ± ±

0.200 ±

_______ ± ±

0.250 ±

_______ ± ±

0.300 ±

_______ ± ±

0.350 ±

_______ ± ±

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

Spri

ng F

orce

(N)

Displacement from Equilibrium, x (cm)

Spring Force versus Displacement

0.500

0.550

0.600

0.650

0.700

0.750

0.800

0.400 0.450 0.500 0.550 0.600 0.650

Peri

od (s

)

__________________________________ (_________)

Measured Period versus ________________________

8 L11 L11.1

DISPERSION OF LIGHT BY A PRISM

OBJECT

The object of this experiment is to use a spectrometer to observe the dispersion of light passing through a glass prism and to verify the Cauchy Dispersion relation.

THEORY

Light of a given wavelength passing through a glass prism will be deviated by an angle D, as shown in Fig. 1. The angle of deviation, D, depends on the angle of incidence. The smallest value of D obtained as the angle of incidence is varied is called the angle of minimum deviation, Dm, for this wavelength. If A is the apex angle of the prism, the index of refraction of the glass of the prism, n, for the wavelength used is given by:

+

=

2sin

2sin m

A

DA

n (1)

Since A = 60.0° ± 0.1° for the prisms used in this experiment, equation (1) can be simplified to:

+=

20.60sin2 m

o Dn (2)

Different wavelengths have different corresponding values of Dm since the refractive index of the material of the prism depends on the wavelength. This variation of n, the refractive index, with λ, the wavelength, is called dispersion. The first successful attempt to represent the dispersion relation was made by Cauchy. A simple form of Cauchy’s dispersion relation may be written as

bmn +

= 2

1λ

where m and b are constants.

PROCEDURE

Caution: 1. Optical surfaces are easily marked. DO NOT TOUCH the spectrometer lens surfaces. Handle the prisms by touching only the top and bottom faces.

2. The lamp becomes very hot during operation. DO NOT TOUCH the lamp.

3. The arc in the gas lamp will not strike when hot. Therefore, do not switch the lamp off until you have finished the experiment.

Before starting the experiment, familiarize yourself with the spectrometer. If in doubt as to the function of any part of the instrument, refer to the description of the spectrometer in the Appendix or consult your demonstrator.

9 L11.2

Examine the circular scale and learn to read the angular settings to the limit of the precision of the instrument (± 0.1°). Also, note the direction in which the screw is turned to adjust the width of the slit on the collimator. Do not make any other adjustments on the collimator.

Carefully remove the prism table from the spectrometer. Direct the collimator slit toward the lamp and move the telescope until the slit image is in view. Adjust the eyepiece until the crosshairs are in sharp focus. The image of the slit as viewed through the telescope should be well-focused and free of parallax with reference to the crosshairs.

Once a sharp, well-focused image of the slit is obtained in the field of view of the telescope, adjust the slit width as narrow as convenient. Note that a wider slit will be a source of error in the experiment; a very narrow slit on the other hand will reduce the amount of light, making it difficult to locate the image of the slit after dispersion.

Align the telescope and the collimator so that the telescope crosshairs exactly coincide with the undeviated image of the slit. The tangent screw may be used for fine adjustment. The vernier should now read zero. If it does not, unlock the spectrometer table using the clamping screw under the collimator, adjust the spectrometer table to read zero and lock it into position for the remainder of the experiment. Now place the prism on the spectrometer table so that the light passes through the prism as shown in Fig. 1. Ensure that the frosted side of the prism is against the post of the prism table. With the unaided eye it should now be possible to see the deviated images of the slit by looking into the ‘telescope’ side of the prism.

Figure 1

The images will appear as a pattern consisting of a number of parallel coloured lines. If the telescope is now brought into the line of sight, a well-focused line spectrum of different colours should be visible. Select one line in the spectrum. While viewing this line, slowly rotate the prism table. The line will either move toward or away from the undeviated direction (i.e. D will either decrease or increase). If D is increasing, rotate the prism table in the opposite direction. While slowly rotating the prism table so that D is decreasing, move the telescope so as to keep the selected line in view. Eventually the selected line will ‘come to rest’ and then reverse its motion as the rotation of the prism table is continued in the same direction. The prism orientation corresponding to the ‘rest’ or ‘reversal’ position is the position of minimum deviation. It will be necessary to fine-tune the ‘reversal’ position for each spectral line.

light source

collimator

telescope

spectrometer table

frosted side of prism

0° D

10 L11.3

For each of the eight spectral lines identified in the table of mercury wavelengths, adjust the telescope position until the crosshairs coincide with the line. Record the telescope angular position, which corresponds to the minimum deviation angle for that spectral line. Record the wavelength of the spectral line, and compute the index of refraction of the glass prism for this line using equation (1) or (2). The calculations for the index of refraction must be done to four significant figures. (δn = 0.002)

Plot the refractive index (n) of the glass prism versus the inverse square of the wavelength (1/λ2). From the graph, determine the Cauchy m and b constants.

CONCLUSION

Discuss whether or not the shape of the graph of n versus 1/λ2 verifies Cauchy’s Dispersion relation. Discuss the values obtained for refractive index of the prism.

SOURCES OF ERROR

• Discuss possible effects due to the adjustment of the spectrometer. • Discuss possible effects due to the condition of the prism.

TABLE 1

MERCURY SPECTRUM

Relative Wavelength* Colour Intensity (nm) Comment

Violet 1 7 404.7 use for determination of n

" 2 1 407.8

Blue 8 435.8 use for determination of n

Blue-Green 1 6 491.6 use for determination of n

" " 2 3 496.0

Green 10 546.1 use for determination of n

Yellow 1 9 577.0 use for determination of n

" 2 9 579.1 Blend† (579.1, 579.0)

Orange-Red 1 1 607.3 use for determination of n

" " 2 1 612.3

Red 1 3 623.4 use for determination of n

" 2 3 690.7 use for determination of n *M.I.T. Wavelength Tables. †A blend occurs when two or more spectral lines appear as one line due to poor resolution in the equipment.

11 L11.4

TABLE 2

SODIUM SPECTRUM

Relative Wavelength*† Colour Intensity (nm) Comment

Violet 1 2 449.6 Blend (449.8, 449.4)

" 2 3 466.7 Blend (466.9, 466.5)

" 3 1 475.2

Blue-Green 1 6 498.3

" " 2 4 515.2 Blend (515.3, 514.9)

Green 8 568.8

Yellow 10 589.3 Blend (589.6, 589.0)

Red 1 7 615.8 Blend (616.1, 615.4)

" 2 1 696.5? Potassium Line ?

12 L11.5

DISPERSION OF LIGHT BY A PRISM

DATA & RESULTS

Source (Na or

Hg)

Colour Wavelength λ (nm)

1/λ2 (× 10–6 nm–2)

Deviation Angle, Dm

(° ± _____°)

Refractive Index, n (± 0.002)

1.610

1.620

1.630

1.640

1.650

1.660

1.670

1.680

1.690

1.700

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Ref

ract

ive

Inde

x, n

Inverse Square of Wavelength (x 10-6 nm-2)

Refractive Index versus Inverse Square of Wavelength to verify Cauchy's Dispersion Relation

13 MP6 MP6.1

HYDROGEN SPECTRUM

OBJECT

The object of this experiment is to observe some of the lines in the emission spectrum of hydrogen, and to compare their experimentally determined wavelengths with those predicted by the Bohr model.

THEORY

References: Sections 24.8 and 28.3, College Physics, Serway and Vuille

According to Niels Bohr’s model (1913), the hydrogen atom (Z = 1) can only exist in discrete energy states whose energies are given by

eV6.1312 222

42

nnekm

E een −=

⋅−=

where e and me are the electron charge and mass; ke is the Coulomb constant; ħ is Planck's constant divided by 2π; and n is the quantum number (n = 1, 2, 3, ...).

When the atom undergoes a transition from one state (with quantum number ni) into a different state of lower energy (with quantum number nf), it will emit a photon whose energy is

hf = ( )

−=− 222

42 112 if

eefi nn

ekmEE

(1)

Using the relationship between speed, frequency, and wavelength, we may write

λhchf = (2)

From equations (1) and (2), the wavelength of the emitted photon is given by

−×=

−= −

2217

223

42 11m10097.1114

1

ifif

ee

nnnncekmπλ

i.e.

−= 22

111

if nnR

λ (3)

where R = Rydberg constant = 1.097 × 107 m–1 .

The photons produced by all the possible transitions comprise the emission spectrum of the atom.

The lines of the spectrum are classified into several different series, according to the quantum number of the final state.

In the present experiment, we are concerned with the Balmer series. (In 1885 a Swiss schoolteacher, Johann Balmer, found an empirical equation that gave the wavelengths of the visible emission lines in the hydrogen spectrum.) The Balmer series is characterized by nf = 2. The quantum number ni of the initial state is thus equal to 3, 4, 5, ... .

14 MP6.2

The apparatus used to observe the spectrum consists of a hydrogen lamp and a grating spectrometer.

A diffraction grating (transmission type) is a piece of transparent material (in our case, gelatin stuck onto glass) with very fine, closely spaced rulings or grooves cut or moulded into it.

When a monochromatic beam of light is incident on the grating, an interference pattern is formed as the beam passes through each of the grooves. The pattern consists of sharp, bright lines (positions of constructive interference) whose angular positions are determined by

d sinθ bright = mλ (4)

where m is the order of the line (m = 0, ±1, ±2, ...); λ is the wavelength of the light; d is the spacing between rulings on the grating; and θ bright is the deviation angle of the transmitted beam.

If the incident beam is composed of different wavelengths, these wavelengths will be separated into distinct lines.

Figure 1. Diagram of Apparatus

PROCEDURE

A. Theoretical (Do this before coming to the lab)

Using equation (3), calculate the wavelengths of the first 3 lines of the Balmer series (nf = 2; ni = 3, 4, 5). Express the wavelengths in nanometres. (1 nm = 10–9 m).

light source

collimator

spectrometer table

adjustable slit

diffraction grating

2nd order spectrum

1st order spectrum

0th order spectrum (single line)

(line pattern is symmetric about 0th order)

15 MP6.3

B. Experimental (refer to the Appendix at the back of the manual for a description of the spectrometer)

1. Turn on the power supply for the hydrogen gas discharge tube. THE POWER SUPPLY OPERATES AT HIGH VOLTAGE – DO NOT TOUCH THE ELECTRICAL CONTACTS BETWEEN THE SUPPLY AND THE TUBE.

2. Record k, the # of lines/mm for your grating. Note that d, the ruling spacing, equals 1 / k.

3. With the grating removed from the spectrometer, rotate the telescope in-line with the collimator and observe the bright pink line being emitted by the hydrogen source. Lock the telescope and unlock and rotate the spectrometer table so that the spectrometer reads about 180°. Lock the spectrometer table and unlock the telescope. Place the grating on the spectrometer and align it so that it is perpendicular to the incident light from the collimator. Observe the zeroth order spectrum (the bright pink line occurring in the incident direction). Check the focusing of the telescope (the edges of the line should be sharp), and adjust the line to the desired width (fairly narrow) by means of the adjustable slit on the end of the collimator.

4. Swing the telescope left and locate the three prominent visible lines of the first order spectrum: a dark violet line, a bright blue-green (turquoise) line, and a bright red line. Record the angular position of each line under the Left Side Angle column in the data table. Note that the vernier scale allows angles to be read to 0.1° accuracy.

5. Swing the telescope right from the incident direction and again locate the three prominent lines. Record their angular positions under the Right Side Angle column.

6. Repeat steps 4 & 5 for each line in second order.

7. For each line, calculate the difference in angular positions (2θ bright), and the experimental wavelength using equation 4.

8. For each colour, average the experimental wavelengths at each order and compare this average with the theoretical wavelength.

CONCLUSION

Discuss the significance of the observation of discrete spectral lines in the emission spectrum of hydrogen.

SOURCES OF ERROR

• Discuss possible effects due to the adjustment of the spectrometer. • Discuss possible effects due to the orientation of the grating.

16 MP6

HYDROGEN SPECTRUM

DATA & RESULTS

k = (______________ ± ____________ ) lines/mm

d = (_________________ ± ____________ ) nm

Pink line (m = 0): θ = (______________ ± ________)°

COLOUR m Left Side Angle, θL (° ± 0.1°)

Right Side Angle, θR (° ± 0.1°)

Difference, θR –θL(=2θ)

(° ± 0.2°)

θ (=2θ/2) (° ± 0.10°)

λ (nm)

±

±

±

±

±

±

±

±

17 MP9 MP9.1

RADIOACTIVE DECAY - MEASUREMENT OF HALF-LIFE

OBJECT

The object of this experiment is to measure the half-life of the beta decay of Indium-116.

THEORY

Reference: Section 29.3, College Physics, Serway and Vuille

The decay rate, or activity, R, of a radioactive isotope is the rate of change of the number of nuclei of the radioactive isotope, i.e. the number of decays occurring per unit time:

R = tN

∆∆

− or, more precisely, R = dtdN

− (1)

R depends on a property of the radioactive isotope called the decay constant, λ, (a quantity with units of inverse time) and on the remaining number of nuclei of the radioactive isotope.

R = λN (2)

Combining (1) and (2) yields

dtdN

− = λN (3)

which is an equation that can only be solved using calculus.

The solution to (3) is that the abundance, N, of a radioactive isotope decreases with time from its initial abundance, N0, according to the equation

N = N0 e–λt (4)

The half-life, T½, is defined as the time needed for ½ of the original sample to undergo radioactive decay. Evaluating equation (4) at a time of one half-life gives:

½N0 = N0 e–λT½ (5)

Taking the natural logarithm of both sides and solving for T½,

T½ = λλ693.02ln

= (6)

In this experiment, the beta particles (electrons) emitted as the Indium decays are detected by a Geiger-Muller tube and recorded as a count. The Geiger tube will also record counts due to background radiation from the air, the ground, the building, etc. These background counts, Cb, must be subtracted from the total counts, C, to obtain the counts due to the source alone, Ccorr.

i.e. Ccorr = C – Cb (7)

Note that this equation for Ccorr is only appropriate if equal time intervals are used to measure Cb and C. If the time intervals are not equal then the appropriate equation is

∆∆

−=tt

CCC bbcorr

18 MP9.2

where ∆tb is the time interval used to measure the background radiation and ∆t is the time interval used to measure the total counts.

The number of counts from the source, Ccorr, recorded in any time interval, is directly proportional to the activity, R, which in turn is directly proportional to N (equation (2)).

Therefore Ccorr = C0corr e–λt (8)

where C0corr = initial source counts (at time t = 0).

Taking the natural logarithm of both sides of equation (8) yields

ln(Ccorr) = ln(C0corr) – λt (9)

Thus a graph of ln(Ccorr) versus t should be linear and have a slope of –λ. Therefore, the decay constant, λ, = –slope and, as before,

T½ = λ

2ln (6)

PROCEDURE

1. Set the counter knob to 10 minute counts and set the voltage knob to the value indicated on your machine. Press the POWER button.

2. Press the RESET button and then the COUNT button to take a 10 minute background count with no source present. At the end of 10 minutes, the apparatus will cease registering additional counts and the STOP light will be illuminated. Record the displayed value as Cb and reset the counter.

3. Place the Indium source in the top shelf below the Geiger tube.

4. Simultaneously start the clock timer (for measuring elapsed times) and press the COUNT button to make a 10 minute count.

5. After recording each count, reset the counter and start the next count at the starting times specified in Table 1.

ANALYSIS

1. Calculate the error in each count as the square root of the count. ( CC =δ )

2. Calculate the corrected count (equation (7)). Calculate the error in the corrected count.

3. Calculate the natural logarithm of the corrected count, ln(Ccorr), and its error:

δ (ln(Ccorr)) = corr

corr

CCδ

4. Plot ln(Ccorr) versus starting time, t, in minutes and determine the slope and λ (= –slope).

5. Calculate the half-life of Indium-116 using equation (6).

CONCLUSION

Discuss the significance of the shape of your graph.

19 MP9.3

SOURCES OF ERROR

As usual.

20 MP9

RADIOACTIVE DECAY - MEASUREMENT OF HALF-LIFE

DATA & RESULTS

Indium-116 Half-life Data using 10 minute counts

Background Count (Cb) = _________________ ± _________________

Starting Time, t (min)

Count, C

δ C

Corrected Count Ccorr

δ (Ccorr)

ln(Ccorr)

δ (ln(Ccorr))

0.0

11.0

22.0

33.0

44.0

55.0

66.0

77.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

0 10 20 30 40 50 60 70 80

Nat

ural

Log

arith

m o

f Cor

rect

ed C

ount

s, ln

(Cco

rr)

Starting Time, t (min)

Determination of Half-life

21 A1

APPENDIX

EQUIPMENT

VERNIER CALIPER: On a meter stick the graduations are in millimetres so the accuracy of any measurement is no better than the smallest scale division of 1 millimetre. The Vernier caliper can be used to achieve an accuracy of 0.1 mm.

Figure 1

The above diagram illustrates one type of vernier caliper. Normally, the object to be measured is placed between the open jaws at the lower left. By squeezing the thumb release the jaws are then shut on the object and one is ready to read the caliper. (If the inside diameter of a hollow pipe is required, one places the ‘horns’ shown on the top left inside the pipe and the caliper can then be opened further until stopped by the horns.)

The diagram shows that the fixed main scale is calibrated in both centimetres and inches. We will only use the metric scale in the physics lab. The key to the vernier caliper is the vernier scale which slides along as the caliper is opened or closed. This vernier scale contains 20 divisions with 21 reference marks labelled from 0 to 10 (in units of tenths of mm). If you examine it closely you will notice that the length of the 20 vernier divisions equals the length of only 19 millimetre divisions on the main scale. Ideally only 1 of the 21 vernier reference marks falls EXACTLY OPPOSITE a mm mark on the main scale. This vernier principle allows the instrument to be read to 1/20 mm.

The caliper reading is taken in two steps. First the reading on the main scale is taken opposite the 0 reference mark on the vernier scale. (A common mistake is to read the main scale opposite the left edge of the metal slider containing the vernier scale.) In the example shown in the diagram the 0 reference mark is between the 2.1 and 2.2 cm position. Hence the first part of the reading is 2.1 cm.

22 A2 The second step is to closely examine the vernier divisions. Notice that most of the vernier reference marks do NOT exactly line up with the mm marks on the main scale. Ideally only 1 vernier mark lines up exactly. In practice, due to the width of the lines, it is often difficult to tell precisely which one lines up. In the example shown the vernier mark labelled by the number 8 as well as the two unlabelled marks (7 & 7.5) appear to line up exactly with mm marks on the main scale. In this situation it is best to adopt the vernier mark in the middle (i.e. the 7.5 tenths of a mm vernier mark). Hence the actual reading is the sum of this vernier reading and the main scale reading.

reading = 2.1 cm + (7.5/10) mm = 2.175 cm.

Since the next possible higher reading is 2.180 cm, the reading error is ±0.005 cm.

It is good practice to always check the reading of a caliper when the jaws are shut. This reading is called the Zero Point Reading. If the jaws become slightly bent, the zero point reading may no longer be 0.000 cm. The corrected caliper reading is taken as the difference of the initial reading and the zero point reading. Note that there is still a reading error of ±0.005 cm in taking the zero point reading (even if it reads 0.000 cm). Thus the accuracy of the final corrected reading is the sum of the 2 reading errors or ±0.010 cm.

As an example, suppose the zero point reading is 0.015 cm. Using the initial reading shown in the diagram, the final corrected reading would be:

2.175 cm ± 0.005 cm

– 0.015 cm ± 0.005 cm

2.160 cm ± 0.010 cm

MICROMETER CALIPER: The micrometer is an instrument designed for the measurement of small distances like the diameter of a wire or thickness of a thin sheet of paper. The micrometer provided in the laboratory can measure distances to within 1/100 mm. As shown in Figure 2 the micrometer consists of a thimble moving on a hub toward or away from an anvil.

Figure 2

23 A3

The basic principle of the micrometer caliper is this: As the screw is turned by one revolution, it advances a distance equal to the pitch of the screw. For the micrometer in the laboratory the pitch of the screw is 0.5 mm and the scale on the thimble contains 50 divisions. Therefore, one can obtain a reading accurate to 1/50 of 0.5 mm (= 0.01 mm) with such a micrometer.

The markings on the hub give readings to the nearest 0.5 mm, and the reading on the thimble gives the number of hundredths to be added to the hub reading. For example, the micrometer in Figure 3 reads 5.50 mm + 0.18 mm = 5.68 mm.

Figure 3

The object to be measured is placed between the jaws A and B and the thimble is advanced gently until contact is made with the object by both jaws. To apply constant pressure each time, advance the thimble by rotating the ratchet head until it begins to slip. If the micrometer does not have a ratchet, adjust the thimble until the jaws just make contact with the object to be measured. This must be a gentle pressure, otherwise the object's dimensions may be altered or the micrometer damaged. If, when the jaws are closed against each other, the reading on the scale is not zero, a correction must be added to, or subtracted from, each measurement. The zero point reading being other than 0.00 mm is an example of a systematic error which will occur in each measurement unless a correction is made.

SPECTROMETER: The spectrometer is used in the laboratory to investigate the spectrum of light emitted by a source. It consists of three main parts: the collimator, the prism table and the telescope as shown in Figures 4 and 5. The light from a source is allowed to illuminate the slit of the collimator. A prism or grating is placed on the prism table and the dispersed or diffracted light is viewed through the telescope. The angle by which the light is deviated is measured using a 0.1° vernier scale attached to the telescope.

The function of each of the main parts and other knobs and clamps is given below.

Collimator

A tube that contains a convex lens at one end and a slit of adjustable width at the other end, at the focal point of the lens. Light admitted by the slit will be travelling in parallel rays after emerging from the lens.

24 A4

Figure 4

Telescope

A tube that contains an objective lens and an eyepiece. The eyepiece consists of a lens, and crosshairs to permit accurate positioning of the telescope in line with the image of the slit. The entire telescope may be rotated when the telescope position lock is loosened. A focused image of the slit is obtained by adjusting the relative positions of the two lenses of the telescope.

Figure 5

25 A5

Spectrometer Table

Marked with a scale from 0 to 360 degrees.

Vernier Scale

Permits reading of the position of the telescope relative to the table to the nearest 0.1°.

Telescope Fine Position Adjustment

Permits extremely fine rotation of the telescope, when the telescope position lock is tightened. NOTE: travel is limited to a few degrees.

Telescope Position Lock

Permits coarse adjustment of the telescope when loose and must be tight for fine adjustment of the telescope. Be sure to unlock this clamp before rotating the telescope arm of the spectrometer by hand for coarse adjustments.

Table Lock

Allows the table to rotate when loose. This clamp should be kept locked during most of the experiment.

26 A6

BECKMAN 310 DIGITAL MULTIMETER

The Beckman 310 Multimeter can be used to measure DC and AC voltage, DC and AC current, and resistance.

DC Voltage: To measure DC voltage connect the positive lead to the leftmost input labelled V and the other lead to the COM input. The DCV section of the central dial has five divisions (200m, 2, 20, 200 and 1500) marked with the MAXIMUM voltage that can be handled without overloading the meter circuitry. The smaller numbered divisions have greater accuracy. To avoid overloading the circuitry, first turn the dial to the 1500 scale. If the voltage (or the maximum voltage expected) is less than the next lower division’s maximum (200 volts) you can safely rotate the dial counter-clockwise to the next division. You can repeat this procedure until you reach the division with greatest accuracy that does NOT overload the meter. If you reach the 200m scale the meter will read in millivolts.

AC Voltage: To measure AC voltage connect the leads to the V and COM inputs. Turn the dial clockwise into the ACV range. Follow a similar procedure to that described under DC Voltage to avoid overloading the meter.

DC Current: To measure DC current connect the positive circuit lead to the 10A input and the other lead into the COM input. Turn the dial to the 20m/10A position in the DCA range. The meter will now read in amps to a maximum of 10 amps.

If the meter reads less than 2 amps and you are sure that the maximum expected current will ALWAYS be under 2 amps, then you can safely obtain a more accurate reading by turning the meter off and reconnecting the positive lead to the A input. (NOTE: you will blow a fuse if the current ever exceeds 2 amps.) The dial can now be turned to the 2 division in the DCA range. If the current is less than 0.2 amps, turn the dial to the 200m division for greater accuracy. The meter will now read in milliamps. You can continue to switch to finer divisions providing you do NOT exceed the MAXIMUM current indicated on the dial. The finest division reads in microamps to a maximum of 200 microamps.

AC Current: Follow the same procedure as described under DC Current except that the dial is rotated into the ACA range.

Resistance: To measure the resistance of a circuit element, first disconnect the electrical connections of that element from any other part of the circuit. Then connect a lead from one end of the element into the ohm (Ω) input and a lead from the other end into the COM input. Rotate the dial into the ohm (Ω) range starting with the coarse scale (20M) where resistance will be read in Mega-ohms (MΩ). You can rotate the dial to finer divisions providing the resistance does NOT exceed the MAXIMUM RESISTANCE indicated. If you use the 200K, 20K or 2K divisions the resistance will read in kilo-ohms (kΩ). If you use the 200 division, the resistance will read in ohms (Ω) to a maximum of only 200 ohms.

27 A7

28 A8

SI UNITS

SI BASE UNITS

Quantity Name of Unit Symbol

length

mass

time

electric current

thermodynamic temperature

amount of substance

luminous intensity

metre

kilogram

second

Ampere

Kelvin

mole

candela

m

kg

s

A

K

mol

cd

SOME DERIVED UNITS

Quantity Name Symbol Equivalent to

force

pressure

work, energy, quantity of heat

power, rate of energy emission

quantity of electricity

electric potential

electric resistance

electric capacitance

magnetic flux

inductance

magnetic flux density

frequency

Newton

Pascal

Joule

Watt

Coulomb

Volt

ohm

Farad

Weber

Henry

Tesla

Hertz

N

Pa

J

W

C

V

Ω

F

Wb

H

T

Hz

Kg·m/s2

N/m2

N·m

J/s

A·s

J/C = A·Ω

V/A

C/V

V·s

Wb/A

Wb/m2

s–1

There are also two supplementary units in SI: the unit of plane angle, the radian, and the unit of solid angle, the steradian, which have the symbols rad and sr respectively. All of the other units used in SI are derived units and are expressed algebraically in terms of base units and/or supplementary units.

M39 STATIC EQUILIBRIUM (NON-CONCURRENT FORCES)

DATA & RESULTS Force and Torque Analysis for Object in Static Equilibrium

Force

Mass, m (kg)

Force, F (N)

Angle with x-axis,

(° ± ____°)

Fx (N)

Fy (N)

Lever arm, d(±______m)

Torque, (N·m)

1 ± ±

± ±

±

2 ± ±

± ±

±

3 ± ±

± ± ±

W (bar) ± ±

± ± 0.000 ±

± ±

±

M19 HOOKE'S LAW AND SIMPLE HARMONIC MOTION

DATA & RESULTS

mass of pan, mpan = _______________ ______________ kg

Force, Displacement, and Oscillation Data for Spring

Load Mass mload (kg)

Spring Force (mload g)

(N)

Scale Reading x (cm)

(±______cm)

Time for 10

oscillations (s ± ______ s)

Period, T (s)

(± ______ s)

Oscillating

Mass (mload +

mpan) (kg)

0 0 0.0

0.050

_______ ±

0.100

_______ ±

0.150

_______ ± ±

0.200

_______ ± ±

0.250

_______ ± ±

0.300

_______ ± ±

0.350

_______ ± ±

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

Spri

ng F

orce

(N)

Displacement from Equilibrium, x (cm)

Spring Force versus Displacement

0.500

0.550

0.600

0.650

0.700

0.750

0.800

0.400 0.450 0.500 0.550 0.600 0.650

Peri

od (s

)

__________________________________ (_________)

Measured Period versus ________________________

L11

DISPERSION OF LIGHT BY A PRISM

DATA & RESULTS

Source (Na or

Hg)

Colour Wavelength (nm)

1/2 ( 10–6 nm–2)

Deviation Angle, Dm

(° ± _____°)

Refractive Index, n (± 0.002)

1.610

1.620

1.630

1.640

1.650

1.660

1.670

1.680

1.690

1.700

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Ref

ract

ive

Inde

x, n

Inverse Square of Wavelength (x 10-6 nm-2)

Refractive Index versus Inverse Square of Wavelength to verify Cauchy's Dispersion Relation

MP6

HYDROGEN SPECTRUM

DATA & RESULTS

k = (______________ ____________ ) lines/mm

d = (_________________ ____________ ) nm

Pink line (m = 0): = (______________ ± ________)°

COLOUR m Left Side Angle, L (° ± 0.1°)

Right Side Angle,R (° ± 0.1°)

Difference, R –L(=2)

(° ± 0.2°)

(=22) (° ± 0.10°)

(nm)

±

±

±

±

±

±

±

±

MP9

RADIOACTIVE DECAY - MEASUREMENT OF HALF-LIFE

DATA & RESULTS

Indium-116 Half-life Data using 10 minute counts

Background Count (Cb) = _________________ _________________

Starting Time, t (min)

Count, C

C

Corrected Count Ccorr

(Ccorr)

ln(Ccorr)

(ln(Ccorr))

0.0

11.0

22.0

33.0

44.0

55.0

66.0

77.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

0 10 20 30 40 50 60 70 80

Nat

ural

Log

arith

m o

f Cor

rect

ed C

ount

s, ln

(Cco

rr)

Starting Time, t (min)

Determination of Half-life