EXPERIMENT I:

MEASUREMENTS AND ERRORS

PHYS 1815

Department of Physics

Seton Hall University

South Orange, New Jersey

Revised September 2014

1

EXPERIMENT I MEASUREMENTS AND ERRORS

I. PURPOSE

The objective of this experiment is for you to learn how to properly use the Vernier caliper and the

micrometer to measure length, and to study the methods of estimating and combining errors in

measurements and calculations.

II. APPARATUS

4 metal cylinders, 1 micrometer,

1 ruler, 1 pan balance,

1 Vernier caliper, and a laboratory PC with the LAB 1 icon on the Desktop.

III. THEORY

1. Types of Errors

Any measurement, no matter how carefully executed, contains a certain amount of error or uncertainty.

There are essentially two types of errors: systematic and random. The error to which we refer in this and most other experiments is probably better known to you as the plus or minus value or the uncertainty in a measurement. You should regard every number that you measure as having an uncertainty associated with it.

Systematic Errors

Systematic errors result from defects in the apparatus or in the experimental method used. For example,

if one end of a meter stick has been worn down, all measurements using that end would be too large by

a fixed amount. One cannot eliminate the systematic error present by averaging the lengths measured

using that meter stick. These errors may be minimized by a careful check of the instruments and

measurement procedures. For instance, the reading of a micrometer may not be zero when the jaws are

closed. This error must be corrected when using the instrument, otherwise, all measurements made with

the micrometer will be in error by the same amount.

Random Errors

If a given measurement is repeated several times, the results will generally not be the same, but will

tend to fluctuate somewhat. These fluctuations are due to uncontrollable conditions affecting the

observer, the measuring device, and the quantity being measured. Examples of such circumstances are

variations in room temperature and fluctuations in electrical-line voltage. These errors are presumably

random, i.e. they are just as likely to produce a reading that is too large as they are one that is too small.

Thus, if we take an average value of several measurements, these fluctuations should tend to partially

cancel out.

2. Accuracy of Measurement

A measurement is said to have good accuracy when its magnitude is close to the true value of the quantity measured. Good accuracy implies that the inherent systematic errors are small.

2

3. Precision of Measurement

A measurement is said to have good precision when its magnitude is close to the average value of the quantity measured. The precision with which a quantity is measured determines the range values about

their average value; the wider the spread, the lower the precision. The degree of precision is determined

by the fractional error (to be discussed shortly). Good precision implies that random errors are small.

For example, suppose that you are measuring the height of someone who is exactly 61 inches

tall; if you report the result as (59 4) inches, you are being accurate, but not precise. If you

say the result is 60.15 inches, the result is precise, but inaccurate.

4. Errors are not Mistakes

An error refers to the limitations on the accuracy or precision with which the value of some quantity can be measured. The use of the term error in no way implies that the experimenter has done anything incorrectly. A mistake, on the other hand, is due to a lapse on the part of the experimenter.

So, if you misread your instrument, you are making a mistake or a blunder, not an error as we use the term in Physics. Hence, you should never mention any human errors when discussing Error Analysis in your laboratory reports; human errors should be noted in your Data Analysis and Calculations and Conclusions sections.

5. Propagation of Error

In general, the quantity of interest is not directly measured. Instead, we often find that it is necessary to

measure several independent physical quantities such as length and mass, in order to compute the value

of the dependent quantity. But since the measured values contain errors that are propagated in the

computed value, this computed value will also have an error associated with it. In this section, we show

you how to account for the propagated error (or uncertainty).

There are two ways of presenting errors. We can present the absolute error, or, alternatively, the

percentage error. The absolute error is the actual amount of uncertainty in the quantity, whereas the

percentage error is the ratio of the absolute error to the accepted value multiplied by 100%, i.e.

100%absolute error

percentage erroraccepted value

(1.1)

We state a quantity by its value, X , and its absolute error X , as X X . This indicates that the true

(accepted) value of X is likely to lie within the range of X X and X X . Therefore, we have

100%X

percentage errorX

(1.2)

The ratio X X defines the fractional error, or the precision of a measurement.

Example: The width of a table is found to be (65.3 0.3) cm. What are the absolute and percentage

errors of the measurement?

The absolute error is 0.3 cm and the percentage error is

0.3 cm100% 0.5%

65.3 cm .

3

The following four rules must be applied to the propagation of error:

Rule 1: Addition or Subtraction

When adding (or subtracting), the absolute error of the sum (or difference) is simply obtained by taking the

square root of the sum of the squares of the absolute errors of the variables. If

R a X bY , (1.3)

where a and b are constants like 1, , , or 2 , et cetera, then

2 2R a X b Y ( ) ( ) , (1.4)

where X and Y are independent variables, R is the result, X and Y signify the error associated with

each variable, and R is the error of the overall result.

Example 1: If (10.5 1.2) cmX X and (12.5 1.2) cmY Y , find the average value of X and

Y , and the absolute uncertainty of the average.

The average value R is given by

1 12 2

11.5 cm2

X YR X Y

( )

,

and its absolute error is calculated using Eq. 1.4:

2 21 1

1.2 cm 1.2 cm 0.8 cm2 2

R

.

Therefore, the complete answer is (11.5 0.8) cmR R .

Example 2: Consider the case of a pan into which a sample is placed. The recorded data is

Mass of pan with sample pan with sample 1 1 (215.603 0.002) gM m m ;

Mass of pan without sample pan alone 2 2 (63.451 0.002) gM m m .

What is the mass of the sample?

Using Eq. 1.4, the mass of the sample, sampleM , is computed as

2 2

sample pan with sample pan alone 1 2 1 2 (152.152 0.003) gM M M m m m m ( ) ( ) ( ) .

Note: If R a X bY c Z , (1.5)

then the absolute error in R is found using both Rules 1 and 2, i.e.

2 2 2R a X b Y c Z ( ) ( ) ( ) , (1.6)

where a , b , and c are constants.

4

Rule 2: Multiplying a quantity by a constant

The absolute error in the product of a number, X , and a constant, C , is simply the absolute error in the

number multiplied by the constant. If

R C X , (1.7)

then

R C X . (1.8)

Example 3: The length of one side of a square and its uncertainty are given by (4.21 0.02) cmL L .

What is the perimeter, P, of the square?

Using Eq. 1.8 to determine the uncertainty, the perimeter of the square is

4 4 (16.84 0.08) cmP L L .

Rule 3: Multiplication or division

The percentage error in the product or quotient of two or more measured quantities is the square root of the

sum of the squares of the fractional errors of the variables.

If R C X Y or R C X Y , (1.9)

then the fractional error is

2 2R X Y

R X Y

(1.10)

(note that the coefficient C cancels), and the percentage error is

2 2

100% 100%R X Y

R X Y

. (1.11)

Example 4: A block of metal is weighed and its volume is determined. Given the measurements,

Mass of block (719.4 0.1) gM M ;

Volume of block 3(83.6 0.2) cmV V .

Find the density of the block and its percentage error:

Density of block 33

719.4 g8.61 g cm

83.6 cm

M

V .

To find the percentage error, we use Eq. 1.11:

2 2

100% 0.24%M V

percentage error inM V

Therefore, 38.61 g cm 100% 0.24% ( )( ), or equivalently, 3(8.61 0.02) g cm .

5

Rule 4: A quantity raised to the nth

power

The fractional error of a quantity raised to the nth power is n times the fractional error of the quantity. If

nR C X , (1.12)

then the fractional error is given by

R X

nR X

, (1.13)

where n is the absolute value of n.

Example 5: Given the diameter of a circle and its uncertainty as (3.10 0.10) cmD D , find the area

of the circle (note that the formula for this has the same form as Eq. 1.12).

The area of circle is

2 2 2 23.10 cm 9.61 cm 7.55 cm4 4 4

A D

( ) ( ) ,

while the fractional error in A is 0.10 cm

2 2 0.0653.10 cm

A D

A D

,

and the absolute error in A is 2 20.065 7.55 cm 0.45 cmA

A AA

( )( ) .

Thus, the area of the circle and its uncertainty are 2(7.55 0.45) cmA A .

Example 6: The density (mass / volume) of a cylinder in terms of its mass, m, its length, L, and its

diameter, D, is given by the equation

2

4m

L D

( )

. (1.14)

The fractional uncertainty in the density of a cylinder can be written in terms of the

uncertainties m , L , and D according to Eqs. 1.10 and 1.13 as:

2 2 2

2m D L

m D L

, (1.15a)

(1.15b)

Here, is the absolute error or absolute uncertainty. One calculates the fractional

uncertainty using Eq. 1.15a, then computes the absolute error utilizing Eq. 1.15b.

6

6. Derivation of Uncertainty Equations

The formulae used to calculate the uncertainty of a physical measurement in the foregoing section are

derived using the Calculus equation

R RR X Y

X Y

,

where the partial derivatives R

X

and

R

Y

imply differentiation with respect to the indicated variable

while keeping the other variables fixed. This is similar to the implicit differentiation of a function of two

variables, but here, one must take the absolute values of the partial derivatives, and the infinitesimals are

replaced by the differences (i.e. the uncertainties). This method usually leads to an over-estimation of the

total error in cases of two or more variables. Of course, it is always safer to over-estimate an error than

under-estimate it, but one wishes to obtain a formula that better expresses the true error inherent in such

situations. To evaluate the propagation of errors properly, one must use techniques from Statistical

Analysis to derive the formulae for the total uncertainty. Chief among these notions is the Method of

Least Squares, which will be described in detail in the manual for Experiment II Statistics of Measurement. In the present case, we do not need to know any of the particulars of this method other

than the fact that the square of the total uncertainty will be equal to a summation involving the squares of

the uncertainty of each variable. To calculate this, one squares the equation above and obtains

2 2

2 2 2R R R RR X Y X YX Y X Y

( ) ( ) ( ) ( )( ) ,

where the errors appear quadratically and the cross terms produce a modifying term related to the

Covariance (i.e. the joint Statistical Deviation, a subject explored in Experiment II ). Suffice it to say

that according to mathematical theory, the Covariance can dramatically increase the magnitude of the

errors, but in most Physics experiments, the measurements are independent of each other, so the

Covariance will be zero. Hence, the final term in this formula vanishes, yielding

2 2

2 2R RR X YX Y

( ) ( ) ( ) .

This is the same form as Eq. 1.4, and one can readily see that the result can be generalized to three or

more variables, e.g.

2 2 2

2 2 2R R RR X Y ZX Y Z

( ) ( ) ( ) ( ) .

The fractional uncertainty is then obtained by dividing both sides by R , which produces

2 2 22 2 2

2 2 2

R R X R Y R Z

R X R Y R Z R

( ) ( ) ( ).

With these results, one can derive all of the uncertainty relations presented in the previous section and

calculate new ones as the need arises.

7

7. Significant Figures

The significant figures in a number are the trustworthy figures that represent the degree of certainty of

the measurement. In recording data and results, it is incorrect to retain more figures than are significant.

To illustrate this point, let us assume that we wish to determine the area of a rectangle. We measure the

sides to be (472 1) cm and (186 1) cm. The result is 87,792 cm2. The measurement is accurate to three

significant figures and we express the answer as 4 28.78 10 cm . This means that the last two digits in the

number 87,792 are meaningless except for rounding (we round up in this example, but only quote three

significant figures in our answer).

The following rules help us to determine the correct number of significant figures in various

calculations.

Rule 1: If there is no decimal point, the leftmost non-zero digit and the rightmost non-zero digit are the

most and least significant figures, respectively. For example, there are four significant figures in

the number 3,501.

Rule 2: If there is no decimal point and the right-most digit is a zero, the zero may be significant or it

merely serves to locate the decimal point. For instance, the number of significant figures in 970 is

either two, or three, depending on the degree of certainty of the measurement.

To avoid this ambiguity, it is preferable to express such a number as its significant figures times

the appropriate power of ten. Thus, one can write

2970 9.7 10 (2 significant figures)

or

2970 9.70 10 (3 significant figures)

Rule 3: If there is a decimal point, the rightmost digit is significant, whether or not it is a zero. For

example, the number 9.7 has two significant figures, whereas 9.70 has three significant figures.

On the other hand, the zeros in a number like 0.0062 are not significant. This number has only

two significant figures.

Rule 4: Addition or subtraction

Arrange the numbers in tabular form. The number with the fewest significant figures after the

decimal point fixes the number of significant figures in the answer. When performing

calculations, we keep as many significant figures as the individual numbers possess, but our final

answer must be trimmed.

Example: Add the following numbers.

32.192

0.4687

+ 215.05

247.7107 (too many significant figures)

= 247.71 (two significant figures after the decimal point)

Note that the number with the least number of significant figures after the decimal is 215.05.

8

Rule 5: Multiplication or Division

The number of significant figures in the product or quotient is the same as that of the one with the

least number of significant figures, e.g.

2250.6 0.73 1.8 10 .

Rule 6: In recording measurements, it is always good practice to record the data together with its error, for

example, (7.09 0.16) cm. The number of significant figures used to express the error should

correspond to the number of significant figures in the answer itself, e.g. never express your result

as (7.09 0.2) cm or (7.09 0.158) cm. The value (7.09 0.2) cm is not a consistent expression

because the answer implies precision to the nearest 0.01 cm, which would be expressed as

(7.09 0.21) cm. Also, (7.1 0.2) cm would be proper if the data were only precise to 0.1 cm.

By applying the same logic, it should be clear that (7.09 0.158) cm is an incorrect expression.

Along the same lines (common sense, almost), try to remember the most obvious case(s):

If you are combining two numbers both with 5% uncertainty, your answer will be 5 2 %

or approximately 7% . If you are combining two numbers, one 10% and the other 1% ,

your answer is roughly 10% (i.e. 2 210% 1%( ) ( ) is nearly equal to 10%). These two

extremes (nearly equal errors or one dominant error) occur quite frequently.

Rule 7: When writing very large or very small numbers, it is customary to express them by a term

containing only the significant figures multiplied by a power of 10, e.g.

the average distance between the Earth and the Sun is 79.3 10 miles ,

the mass of the Proton is 271.67 10 kg .

8. Analysis of Random and Instrumental Errors

Random Errors

These are usually handled by analyzing a large number of measurements of a given quantity, since the

final error is reduced as the number of independent observations increases. If, however, only a few

measurements are available, it is still possible to estimate the limits of the (random) errors by inspecting

the measuring instruments.

Measuring Instruments

Meter Sticks. Measuring instruments such as meter sticks or plastic rulers are limited in their accuracy

by the spacing of the scale divisions. Although one can estimate the readings to a tenth of a division,

there is an inherent error approximately equal to one-half of the smallest division, i.e. 0.5 mm for a

ruler where the divisions are 1 mm apart. The same logic holds true for graduated vessels in Chemistry

experiments.

Vernier Calipers. The Vernier caliper uses the Vernier scale to estimate the distance between the main

scale divisions, providing an uncertainty of 0.01 cm . The scale line on the Vernier scale (i.e. the

smaller scale that is affixed to the movable jaw of the caliper) that aligns with one of main-scale

9

division lines indicates the fraction of the main-scale interval which the zero-point of the Vernier scale

has advanced past the nearest main-scale division line. Your instructor will demonstrate this using a

model.

Figure 1.1 A Typical Vernier Caliper. The

photograph illustrates the various features usually

found on a Vernier Caliper. These include the

Outer-Diameter Jaws for measuring the external

length of an object; the Inner-Diameter Jaws for

determining the internal width of an orifice; the

Depth Probe for measuring the depth of a cavity; the

Main Scale, which sports scales for measuring distances in both English (inches) and Metric (centimeters)

Units; the Sliding Vernier Caliper that enables a user to ascertain distances to one-tenth of the smallest

intervals on the Main Scale; a Fine Adjustment Knob for making critically small changes in the position of

the Sliding Vernier Scale; and a Lock to prevent the Sliding Vernier Scale from moving in order to retain the

measured distance, a useful feature when testing multiple objects to see if they satisfy a dimensional

requirement or not. No two models of Vernier Calipers are exactly alike, so they will not necessarily posses

all of these features. Additionally, some Vernier Calipers have dials or digital read-outs which enable users

to determine distances down to one-hundredth of the smallest interval on the Main Scale.

Inset: This shows how one uses the Sliding Vernier Scale to obtain a measurement with three significant

figures. First, the leftmost line on the Vernier Scale determines the first two digits of the measured quantity.

In the given example, this line on the Vernier Scale is between the lines for 2.4 cm and 2.5 cm on the Main

Scale, so we know that our distance is greater than 2.4 cm, but less than 2.5 cm. Next, we look for the line

on the Vernier Scale that is the closest to being aligned with one of the lines on the Main Scale. In this case,

it is the seventh line on the Vernier Scale that lines up with a line on the Main Scale, so 7 is our third significant figure, and our final measurement is 2.47 cm.

10

Micrometer Calipers. The micrometer caliper consists of a rotating scale called the micrometer head (also known as the thimble or the barrel), which subdivides the 0.5-mm interval of the main scale into fifty equal parts. The micrometer reading can be estimated to one-tenth of the smallest

division on the micrometer head, i.e. to 0.001 mm, so the inherent uncertainty in this case is

0.005 mm . Note that in any measurement, the last decimal place of the measured quantity must

coincide with that of the error, e.g. (12.637 0.005) mm .

Figure 1.2 The Anatomy of a Micrometer. The illustration shows

the parts found on most Micrometers. One places the object to be

measured in the gap of the Frame between the Anvil and the Spindle.

With the object flush against the Anvil, one rotates the Thimble until

the Spindle almost touches the surface of the object, and then uses

the Ratchet to decrease the gap distance to coincide with

with the diameter of the object (one must always use the Ratchet to make the final adjustment when closing the

gap in order to prevent damage to the Micrometer and the surface of object). One then utilizes the scales on the

Sleeve and Thimble to determine the measurement. On the Sleeve, one finds the horizontal Reading Line which

is flanked on the top and bottom by gradation lines. The upper gradation lines correspond to integer millimeter

values, while the ones on the lower side indicate half-millimeter intervals. The scale on the rotating Thimble is in

divisions of 0.01 mm, but its range is only between 0.00 mm and 0.50 mm. The Inset shows how these scales are

used to measure distances. Some Micrometers have a Lock Ring or Lever which prevents the Thimble from

rotating in order to preserve the current measurement, typically for comparison with other objects.

Inset: The scale on the Sleeve gives us the first two significant digits of our measurement, while the Thimble

yields the last two significant figures. The first two digits are determined from the exposed gradation lines to the

left of the leading edge of the Thimble. In this example, the mark corresponding to 5.5 mm is visible, so our first

two significant figures are 5.5 mm. Next, one sees where the scale on the Thimble aligns with the Reading Line,

which in this case coincides exactly with the 0.28-mm line. This value must be added to that from the Sleeve, so

we have 5.5 mm 0.28 mm 5.78 mm . Finally, one estimates a fourth significant figure by eyeballing the

approximate location where the Reading Line falls between two gradation marks on the Thimble scale. In this

case, since it exactly aligns with the 0.28-mm line, the estimated value is 0.000 mm, and we record 5.780 mm.

11

10n Number Prefix Symbol

010 One None

110 Ten deca-, deka- da, D

210 Hundred hecto-, hecta- h, H

310 Thousand kilo- k, K

610 Million mega- M

910 Billion giga- G

1210 Trillion tera- T

1510 Quadrillion peta- P

1810 Quintillion exa- E

2110 Sextillion zetta- Z

2410 Septillion yotta- Y

10n Number Prefix Symbol

110 Tenth deci- d

210 Hundredth centi- c

310 Thousandth milli- m

610 Millionth micro-

910 Billionth nano- n

1210 Trillionth pico- p

1510 Quadrillionth femto- f

1810 Quintillionth atto- a

2110 Sextillionth zepto- z

2410 Septillionth yocto- y

9. The Metric System of Units

The most widely used system of units in the World is the Metric System. This system is also known

by its proper name Systme International dUnits, which is French for International System of Units. In Physics textbooks, the Metric System can also be referred to by two abbreviations: SI, which comes from the French name, and MKS, which stands for meter, kilogram, second, the basic units of this system. Scientists and engineers prefer using the Metric System instead of the

English System because the various units in the SI are interrelated by powers of ten, which dovetails

nicely with Scientific Notation. Indeed, this structure of the SI is so important that a set of prefixes

have been developed for use with units in the Metric System, which are tabulated below. Please

note that these prefixes are never to be used with the English System of Units.

The prefixes deca- and hecto- have the alternative spellings of deka- and hecta- in some European countries.

The original abbreviations for kilo-, hecto-, and deca-, i.e. k, h, and da, respectively, are shown for reference; now, the standard custom is to capitalize all of the abbreviations for the prefixes denoting factors of ten

greater than one, while leaving all of the abbreviations of the prefixes for factors less than one in lowercase. Therefore,

one must always be careful to distinguish between uppercase and lowercase letters in their handwriting, especially in

those instances when the same letter is used for two different abbreviations, e.g. M for mega- and m for milli-.

The basic units of the Metric System are, for mass, the gram, abbreviated g; for length, the meter, abbreviated m; for time, the second, abbreviated in books as s., but it is standard practice to employ the longer abbreviation of sec. when handwriting values in order to avoid confusion; for volume, the liter, abbreviated l or L (note: a lowercase l might be mistaken for the number 1; so one should always use a capital L to stand for a liter). Also, only a single Metric System prefix should precede a unit abbreviation; they can never be combined together.

12

Here are a few examples of quantities expressed using the Metric System prefixes:

61 megagram 1 Mg 10 g ; 1 kilometer 1 Km 1,000 m ; 1 hectometer 1 Hm 100 m ;

1 decagram 1 Dg 10 g ; 1 deciliter 1 dL 0.1 L ; 1 centimeter 1 cm 0.01 m ;

1 milligram 1 mg 0.001 g ; 61 microliter 1 L 10 L ; 91 nanometer 1 nm 10 m ;

121 picogram 1 pg 10 g ; 211 zettawatt 1 ZW 10 W ; 181 attosecond 1 as 10 s .

Occasionally, one may encounter micro- abbreviated as mc, e.g. on vitamin bottles, where a microgram can be abbreviated as mcg. This is non-standard usage and should be avoided; always use the Greek letter Mu, , shown in the table on the previous page, as the abbreviation for micro-.

Note that in the older literature, one often finds obsolete terms for smaller quantities, e.g the length

unit known as an ngstrom, abbreviated and equal to 1010 m , was often used in Atomic

Physics; likewise, the word micron, an old term for a micrometer, was frequently employed in Surface Science, Microbiology, and various fields of Engineering.

10. The English System of Units

The English System of Units (also sometimes referred to as the British or Imperial System of Units) is an amalgamation of many different units taken from various cultures across the centuries.

This is why the relations between the various units denoting a given physical quantity appear to be

arbitrary, e.g. 12 inches in. 1 foot ft( ) ( ); 3 ft 1 yard yd ( ); and 5,280 ft 1 mile mi. 1,760 yd ( ) . Because the equivalences between different units are not based on powers of ten like the Metric

System, the use of the English System in Science and Engineering has waned in recent decades, but

there are still various industries where it is the standard. Despite their lack of use in the Sciences,

the English System is still just as accurate as the Metric System, and you should learn to perform

calculations using both systems.

Conversion factors in the English System are given in the table below (most of these conversion

factors should look familiar, since they are commonly used in everyday life; you must know all of

these relations, since other conversion factors can be derived from them).

Length Volume Weight (Force)

1 ft 12 in. 1 cup c. 8 fluid ounces fl. oz.( ) ( ) 1 pound lb 16 ounces oz.( ) ( )

1 yd 3 ft 1 pint (pt.) 2 cups 16 fl. oz. 1 ton 2, 000 lb

1 mi. 5, 280 ft 1,760 yd 1 quart qt. 2 pints pt.( ) ( ) 1 stone 14 lb

31 mil 1 10 in.

31 gallon gal. 4 quarts qt. 231 in. ( ) ( ) * 1 hundredweight cwt 100 lb( )

* The gallon used in the United States (internationally known as the U.S. gallon and abbreviated as U.S. gal.) is

defined as being exactly 3

231 in. . This should not be confused with the Imperial gallon, a volume still used in British

Commonwealth countries, which is defined to be exactly 4.54609 L or 3

277.41776827835377 in. .

13

The most important conversion factor between the Metric System and the English System is that for length:

1 m 39.37 in.

The Congress of the United States passed an act in 1866 which defined this fundamental relation

between linear measurement in the English and Metric Systems for economic purposes (i.e. trade

with other nations). This value is the exact relation, so only this conversion factor should be used

when converting length, area, volume, or quantities involving these when converting between

English and Metric units. There are many approximate conversion factors that one encounters in

conversion tables (often found in dictionaries), but these cannot be used in scientific or engineering calculations because the converted values would only be approximations as well.

11. Trigonometry

Trigonometry is the branch of Mathematics that deals with the relationships between the sides and the

angles of triangles, and the trigonometric functions, which ultimately describe those relationships.

Trigonometric functions are also found to correctly model various physical phenomena, such as

oscillatory motion and waves. Trigonometry is very important in every aspect of Science and

Engineering, e.g. it would be impossible to properly design a modern automobile or building without

using Trigonometry. Nearly all technologies that we take for granted today are researched, developed,

and manufactured utilizing trigonometric knowledge somewhere during the process. Many problems

encountered in the fields of Physics and Engineering could not be solved easily without employing

Trigonometry. Below is a chart of the most common trigonometric functions encountered in Physics

problems. It would behoove you to memorize this chart to the best of your ability.

0 30 45 60 90

sin ( ) 0 1

2

1

2 3

2 1

cos ( ) 1 3

2

1

2

1

2 0

Students should be alerted to a serious problem regarding the use of trigonometric functions on

calculators. Based on decades of teaching experience, a frequent mishap that befalls students on

homework assignments, examinations, laboratory reports, et cetera is having their calculator set to

the wrong angular mode. In many scientific applications, one employs radians instead of degrees to

express the value of an angle, so scientific calculators have an option for switching between the two

angular modes. Quite often, however, students will be working with degrees without realizing that

their calculator is in radians mode (or vice versa). Thus, when they input an angle into the

calculator, they believe that they are expressing the angle in degrees, but they are actually submitting

a value in radians, which leads to an erroneous numerical answer. Students can quickly determine

the angular mode to which their calculator is set by computing the value of the sin 30( ). If the

calculator returns a number other than 0.5, the calculator is in radians mode.

14

IV. PROCEDURE

P0. Your laboratory group will first participate in computer-based exercises that will familiarize you with

the use of significant figures (referred to as sig figs in the old computer program; do not use this colloquial phase in your reports). Double-click on the desktop icon named LAB1, then follow the step-by-step instructions. After that program concludes, it will automatically start the program

EXP1, which shows you how to use the measuring implements. At the conclusion of EXP1, you may need to type in the word exit followed by the Enter key in order to return to Windows.

P1. Now we will take actual experimental data. Measure the mass of the metal Cylinder Number 1 on

the pan balance and record its color appearance. We take the instrumental error to be 1 2 of the

smallest scale division when we estimate the last digit by eye.

P2. Measure the length of the cylinder using both the plastic ruler and the Vernier caliper. Estimate the

measurement to the nearest tenth of the scale division in the case of the plastic ruler. There is no need

for such an estimate with the Vernier caliper. The error associated with the Vernier caliper is

0.01 cm because we are not estimating a value between the scale lines.

P3. Measure the diameter of the cylinder utilizing both the Vernier caliper and the micrometer. Please be

gentlethe micrometer has a ratchet on the end of the barreluse this ratchet to close the gap until you hear a few clicks. Never use the barrel to do thisyou may apply too much force and damage the faces of the instrument and/or the surface of the object being measured. Take the zero reading into account for the micrometereither subtract or add it depending on the value read. The zero reading is the reading with the micrometer completely closed. Write this as the offset and include the sign.

P4. Repeat Steps P1 through P3 using Cylinders Number 2, Number 3, and Number 4.

You must complete the data sheets during the laboratory period. Calculate whatever values are

required to fill in the blanks on the data sheets before you leave the laboratory. Be certain you

understand that when you use the least-precise instruments (the ruler and the Vernier caliper) as

opposed to the most-precise (the Vernier caliper and the micrometer), you should obtain different

uncertainty values, and answers with different numbers of significant figures, but the values for the most- and least-precise densities for the same cylinder should be close. You shall see exactly what close means when you answer the report questions.

TABLE OF ACCEPTED DENSITY VALUES:

Material Density

Silver 310.49 g cm

Iron 37.87 g cm

Brass 38.44 g cm

Aluminum 32.70 g cm

Gold 319.32 g cm

Copper 38.93 g cm

15

V. PRE-LABORATORY QUESTIONS

All Pre-Laboratory Assignments in this course must be done individually. Calculations must be

in your own handwriting. Do not share any materials or collaborate with anyone on these

assignments. Plagiarism is a form of cheating and will be penalized as such. This assignment is

due at the beginning of your laboratory period.

Please note that calculator notation is unacceptable, i.e. never write 3.2141E + 9; instead, always

use proper scientific notation: 93.2141 10 .

Q1. Derive the equation for the volume of a cylinder:

2

4

L DV

,

where L is the lateral length, and D is the diameter. Make a sketch of a cylinder delineating the quantities

appearing in the equation. Begin your derivation with the equation for the area of the circular end caps,

and show all of your steps.

Q2. How many significant figures are there in the following values?

(a) 78.9 0.2

(b) 93.788 10

(c) 62.46 10

(d) 0.0032

Q3. The radius of a circle is measured to be 10.5 0.2 m( ) . Calculate the area of the circle and its circumference, and determine the uncertainty for each value.

Q4. A parking lot is 134.3 m long and 37.66 m wide. Compute the perimeter and area of the lot.

16

VI. REPORT QUESTIONS

All laboratory reports in this class must be done individually. Calculations must be in your own

handwriting. Do not share graphs or any materials that you have included in your report

sections with others. Reports containing shared parts or copied portions from other sources

that are not your own work will be penalized.

W1. (a) Calculate the volume of Cylinder Number 1 using both the most and then the least precise

combinations of the measurements (done with different instruments) for length, L, and diameter, D.

The volume of a cylinder is given by

2

4

L DV

(1.16)

(b) Repeat Part (a) for the other cylinders. Record these values in the tables on the data sheets.

W2. Calculate the density of each cylinder using the two values for the volume calculated in Report

Question W1. Label the computed densities as 1 and 2 . Based on your measured values,

determine the material composition of each cylinder using the accepted reference densities tabulated in

this manual by choosing the material with the density closest to your results. Compute the

percentage errors. Did your result indicate the material accurately? Is the finding what you expected?

Which density value, the one using the most-precise or the one utilizing the least-precise combination

of measurements, gave the smaller error? Logically, which density should have a smaller error and

why? Is that what you have found? Comment on your results.

W3. Check whether the accepted values lie within the range of your experimental uncertainty. To do

this, first calculate the fractional uncertainty using Eq 1.15a. The densities you calculated in Report

Question W2, 1 and 2 , should agree with each other within the limits of their errors. To examine

this, determine whether the ranges of 1 and 2 do in fact agree. In this experiment, the density 1

is the density that is calculated using the most-precise combination of instruments. The range of 1 ,

for example, is between 1 1 and 1 1 . If you know the percentage uncertainty in 1 , then

the absolute uncertainty is

11 1 1 1

1100%

percentage uncertaintyfractional uncertainty

( ) .

Comment on whether the range of 1 contains 2 . If they do not agree, then you either made a

mistake or under-estimated the errors associated with your readings.

W4. To improve your density measurement, which measured variable would you concentrate on the most?

Consider which variables contributed most heavily to the percentage uncertainty. Check the

uncertainty equation to answer this, describing each variable in the equation.

W5. Suppose that Cylinder Number 2 was instead a gold cylinder with the same length and mass. What

would its diameter be? Show all of the steps in your calculations.

17

Experiment I Measurements and Errors Name: ______________________________

DATA SHEET 1 Partners: _____________________________

Calculate the volumes, 1V and 2V , and densities, 1 and 2 required in the tables below. You need to

determine the correct uncertainty, e.g. L , for each instrument. Check with your instructor. Also, read

the Introduction in this experimental procedure.

Cylinder 1 (the reddish-orange-colored one)

Mass: _______ _______ (The uncertainty in the mass is the smallest division on the scale; see the text.)

Measurement Ruler Vernier Caliper Micrometer

Length, L (cm)

Uncertainty in L, L (cm)

Diameter, D (cm)

Uncertainty in D, D (cm)

Volume, 1V (caliper, micrometer) (cm3)

The density using 1V , 1 (g/cm3) ______________ _____________

Volume, 2V (ruler, caliper) (cm3)

The density using 2V , 2 (g/cm

3) ______________ _____________

Cylinder 2 (the silvery-colored one)

Mass: _______ _______ Micrometer Offset: _______

Measurement Ruler Vernier Caliper Micrometer

Length, L (cm)

Uncertainty in L, L (cm)

Diameter, D (cm)

Uncertainty in D, D (cm)

Volume, 1V (caliper, micrometer) (cm3)

The density using 1V , 1 (g/cm3) ______________ _____________

Volume, 2V (ruler, caliper) (cm3)

The density using 2V , 2 (g/cm

3) ______________ _____________

Note: measured known known100%percentage error R R R

You may argue about which term should be in the denominator of the percentage error (i.e. the known or

the measured result), but when the errors are small, it will not matter. If you want to be very precise, the

denominator should be the average value of the measured and the known values.

Instructors Signature: _____________________

18

Experiment I Measurements and Errors Name: ______________________________

DATA SHEET 2 Partners: _____________________________

Calculate the volumes, 1V and 2V , and densities, 1 and 2 required in the tables below. You need to

determine the correct uncertainty, e.g. L , for each instrument. Check with your instructor. Also, read

the Introduction in this experimental procedure.

Cylinder 3 (the dark-grayish-colored one)

Mass: _______ _______ (The uncertainty in the mass is the smallest division on the scale; see the text.)

Measurement Ruler Vernier Caliper Micrometer

Length, L (cm)

Uncertainty in L, L (cm)

Diameter, D (cm)

Uncertainty in D, D (cm)

Volume, 1V (caliper, micrometer) (cm3)

The density using 1V , 1 (g/cm3) ______________ _____________

Volume, 2V (ruler, caliper) (cm3)

The density using 2V , 2 (g/cm

3) ______________ _____________

Cylinder 4 (the yellowish-colored one)

Mass: _______ _______ Micrometer Offset: _______

Measurement Ruler Vernier Caliper Micrometer

Length, L (cm)

Uncertainty in L, L (cm)

Diameter, D (cm)

Uncertainty in D, D (cm)

Volume, 1V (caliper, micrometer) (cm3)

The density using 1V , 1 (g/cm3) ______________ _____________

Volume, 2V (ruler, caliper) (cm3)

The density using 2V , 2 (g/cm

3) ______________ _____________

Note: measured known known100%percentage error R R R

You may argue about which term should be in the denominator the percentage error (i.e. the known or

the measured result), but when the errors are small, it will not matter. If you want to be very precise, the

denominator should be the average value of the measured and the known values.

Instructors Signature: _____________________