Physics71.1 Activity Manual

2007 Edition

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Prepared by Leilani Torres Elise Stacey Agra Maricor Soriano .:t

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The 2007 Lab Manual Authors

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Elise Stacey Agra Junius Andre F. Balista Mary Ann B. Go Margie Olbinado Athena Evalour Paz Leilani Torres

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Coordinator Maricor Soriano

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@2007 and2004 Lab Manual Authors

All rights reserved. No part of this publication may be reproduced or transmitted or by any means,

including photocopy, without written permission from the 2007 and2004

*ilHttlijS,'i -iiil,.r,1111]/i?;5 l0 ii Y'I11, i4tji;i ri{rj&t^[0 &. Jll,lAi.l- 0:1Ii.{ Published by the Philippine Foundation for Physics, Incorporated efpD for the exclusive use of the National lnstitue of Physics, uP Diliman

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Table of Contents

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Preface . I Activity Manual is the 4'h outing of the Elementary Physics I Lab Manual series. This year,s volume experiments. The concepts covered by theselxperiments ur. (t1 Experimental skills in FundamenLt rhysics I (Measurement, (Jncertainty and Deviation,Graphical Analysis, Using Calipers, Vectors) , p) Motion in 2D or 3D (Untformly Accelerated Linear Motion, Projectile Motion) , (3) Conservation Laws (Conservation of Energt and Momentum) , (4) Torque (static Equilibrium), (5) Simple Harmonic Motion (Simple Harmonic Motioi: sprifi-Mass System), and (6) Mechanical waves (Sound)

The 2007 Physics 7l

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Of the l0 experiments in the current volume, 6 are new or revised. In Measurements, [Jncertainty and Deviation, rules for handling significant figures and propagation of uncertainty are made more explicit. rn (Jsing Calipers, the use of the depth probe of the Vernier caliper is explained and incorporated in the experiment. Instructioni oo Lo* to create plots using Microsoft Excel have been includedin Graphical Analysis while expiriments to demonstrate two conservation laws have been merged into one experiment in Conservation of Energlt and-Conservation of Momentum. T\e Simple Harmonic Motion experiment is totally new in that a spring-mass system replaces the simple pendulum which had been used in the

past 3 volumes. Finally, sound explores the properties mechanical waves.

Tlte 2007 version makes increased usage of the Vernier LabPro computer interface system. If in the 2004 volume, there was one experiment that req-uires a computer interface, in the 2007 ,rolu-" there are three. Besides Untfurmly Accelerated Linear Motion (UALM, Simple Harmonic Motion (SHltl)and Sound requires the use of tre photogate and Vemier microphone respectively.

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Because of the increased use of computers, we recommend the following Physics 7l.l to avoid overlap in the use of interfaces.

flow of experimts fur parallel sections in

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Experimental Skills Physics

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Section 2

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Expermental Skills Physics

UALM (computer)

Torque

Projectile Motion

Sound(computer)

Conservation Laws

SHM (computer)

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Torque

UALM (computer)

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SHM (computer)

Projectile Motion

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Sound (computer)

Conservation Laws

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The lab and lecture topics of Physics 7l need not be synchroni zed. ln case a class follows thc Sctftn 2 plan, topics covered in the lab even be ahead of the lecture. This should not be a problem because fre iuo&ctory text 'su_f.flciently _may discusses the necessary concepts for the experiments. Stadents are required ,o @ra b *.c ptryd by reading the rext

cnd procedures prior to engagement in the lab. The prelab exercises have been dooc instmctors may give a quiz before the experiment to check on the student's

readiness.

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The 2007 Physics 7l.l Activity Manual was pilot tested in the second semest€r and slrllrg.of AY Z(/)f.-2007. We are grateful to the students who participated in the pilot testing and to the instnrctors wto cilrH-odii.d lhe text during the General Physics Committee workshop in June 2, ZOO7.

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Itfuicor N. Soriano Elise Stacey G. Agro Ith- Leilani Y. Tones

Measu rement,

nceftai nty and Deviation U

Objectives At the end of this activity you should be able to: 1.

Report the best estimate of'observables and quantiff it.

2.

Determine if a theoretical prediction is acceptable given the precision and deviation of an experimental data.

J.

Report the final data in terms of the proper degree of precision.

4. Appreciate the role of measurements in scierrtific activity.

lntroduction In a scientific endeavor, experiments involve collection of information or data through measarement.Datasets are presented to gain empirical knowledge about a phenomenon, validate or invalidate an existing theoretical model and demonstrate that a proposed method works. The measurement of certain variables called observables allows us to achieve this goal. Observables are also called parameters. It is usually the quantity being controlled during the experiment. Since measurement involves unknown quantities, there is always an uncertainty in the measured values. This uncertainty is not always due to personal mistakes. The degree of uncertainty is mainly due to the precision of the measuring device used

and the quantity'that is measured. These uncertainties determine the signif,rcance of the measurement. Hence, proper handling of uncertainties must be known.

@ 2007 Lab Manual Authors

Measurement, Uncertai nty and Deviation

Physics 71.1

This activity deals with the analysis of uncertainties; that is, proper judgment of their magnitude, their conventional description and calculation of numerical values based on individual measurements.

Precisiofi'and'AcCUracy : -

,,1

. Individual measurefllents

'

do not yield the same result. Hence,

measurements

become uncertain and deviate from true value. The agreement among repeated measurements or the closeness of these measurements with each other is defined as precision. The measuae of precision is called uncertaingt On the other hand, if an accepted value is present, the closeness of the measured value to the accepted

one

is

termed as accuracy and is

presented in terms of deviation.

To

understand more clearly the difference between precision and accuracy,let us consider arrows shot (a) (b) into a bull's eye. Precision and accuracy are two independent terms. Figure 1 (a) shows that most of the stars,are on one location only but far from the center target. Hence, this case is high precision but low accuracy. Figure 1 (b) is low in (d) (c) precision but the average of the location of the stars is close to the Figure l. Arrows on a bullseye. Fourbullseye center, hence it has higher point stars mark their landing. Arrows on (a) shows high'piecision but low accuracy, accuracy compared to Figure l(a). (b) low precision but high accuracy, (c) Figure I (c) shows that most of the high precision and accuracy, (d) low precislon and accuracy. stars are on one location only and is at the center target and is the ideal case. While Figure 1 (d) shows the worst case scenario where the marks are both low in accuracy and precision. Uncertainty is not only due to mistake or sloppiness. It is brought upon by the ambiguity of the real value of the quantity being measured. The variation in each measurement may be due to the fluctuations in the quantities measured such as temperature, current or light intensity. It is also dictated by the qualrty of the measuring device or the fineness of its scale. For example, one digital balance may have a reading of 2.13 kg while another reading is 2.134 kg. The latter

@ 2007 Lab Manual

Authors

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Measurement, Uncertainty and Deviation

Physics 71.1

measurement has more certainty.

Deviation maybe minimized by properly calibrating the measuring device. For example, a weighing scale should read zero if there is nothing on it. The limits of the instrument must also be checked. A body:-filerrnorneter cannot be used for measirring the temperature of a,,boiling water while a l2-inch ruler cannot be directly used to measure the Oarth.moon distance.

During the measurement proOess, deviation may also occur due to mistakes, improper use of devices, and, most commonly, due to parallax. Parallax can be removed by ensuring that the eyesight is perpendicular to the scales. Figure 2 shows a reading with parallax. In manual time measurements, the finite human rbaction time (in the order of milliseconds) may greatly affect the accuracy of the result. Hence, it is not advisable to have manual timers for highly precise time measurements.

Reporting and handling of

15 16 t7 18 19 20

mm

uncertainty can be categorized into four approximations. The use of each category depends on the level of uncertainty the experimenter requires.

The first level of handling uncertainty is called zeroth ordi

Figure 2. Measurement with parallax. What do you expect the observer will read? What should the readins be?

approximation which deals with the order of magnitude of the value. The next level involves the use of the significant figures (SF) which is the jirst upproximation. The second approximation deals with the maximum and minimum range of measured quantities. The third approximation involves the rules of probability and statistics which will not be discussed here.

Order of Magnitude The first order of approximation is done by estimating the measurement by powers of 10. Fermi questions are answered by thinking of reasonable assumptions followed by simple calculations that narrow down the range of values where the answer lies. Hence, Fermi questions are answered in terms of order of magnitude. The order of magnitude is the power of ten at which a quantity is expected to fall in. For example, in calculating for the number of seconds in the year which is exactly 3 x107 s/yr, order of 106 to 107 is sufficient

@ 2007 Lab Manual

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Measurement, UncertainU and Deviation

Physics 71.1

an approximate.

Significant Figures The,significant figures in an experimental measurement include the numbers that can be directly read from the instrument scale plus an additional estimated number. Some of the rules in counting the number of SF are listed below.

1.

The leftmost nonzero digit is the most significant.

2. If

there is no decimal point, the rightmost nonzero significant.

is the least

3. If there is a decimal point, the rightmost digit even if it is zero is the least significant.

4. All digits between the least and the most significant digits are considered to be significant.

Example 1: Numbers and the number of digits that is significant

r. 2. 3. 4.

1200-2sF 13.20-4SF 112000.-6SF 0.003456 -4 SF

Problem may arise if the decimal point is omitted and the rightmost digit is zero. This maybe solved by presenting the data in scientific notation. For example, 3560 has 3 SF but the zero may be significant. Thus, the number may be wriffen in the powers of ten, that is, 3.560 x 103 which shows that the last zero digit is

significant.

Multiplication and Division In multiplication and division of trro or more measurements, the number of SF in the final answer is equal to the least number of SF in the measurements. Example 2: Multiplying two measurements: 2.34

x2.2: 5.148: 5.1

Since, the least number of SF is two, the answer should be reported as 5.1. An experimental data cannot be made more signilicant by' a mathematical operation.

@ 2OO7 Lab Manual Authors

Physics

71.1

Measurement,llncertainty and Devialion

Addition and Subtraction In addition or subffaction, the sum or difference has SF only in the decimal places where the digits of the measurement are both signifrcant. Hence, we report the sum or difference which corresponds to the least number of decimal plaoe,of the addends.

Example 3: Adding two measurements: 6.56 + 3.1

:9.66 = 9.7

Since, the least number of decimal place is one, the answer should be reported as 9.7

.

Rounding off Nonsignificant digits are removed if they are at the right of the dpcimal point. The rightmost significant digit is retained and rounded off. The rules for rounding off are as follows:

l.

If the fraction is less than 0.5, the last SF is leftunchanged.

2. If the fraction is greater than or equal to 0.5, the SF is increased

by

l.

Example 4: Best estimate of repeated measurenients.

A student did a repetitive measurement of length

and obtained

the following data: 215 rt" 222 m,219 m,231 m,224 m The expectation value, :

215+ 222+ 219+ 231+ 224

(q)= zzzm

To obtain the uncertainty, A q, subtract the from the maximum value and the minimum value from A q. The larger of the two differences is the uncertainty of the data.

t ,

231 m_ 222m:9 m 222 m-215 m: 7 m The difference 9 m is greater than 7 m, hence the best estimate ofthe data is reported as (9) = 222mt gm

3. @

In

cases

2007 Lab ManualAuthors

of multi-step mathematical operation, only the final result should

Physics 71.1

Measu rement, Uncertainty and Deviation

be rounded off.

Absolute and Relative Uncertainty The second approximation to uncertainty analysis is based on maximum pessimism. This implies that a measurement cannot be expressed as a single, exact value but is a range of values wherein the true measurement lies, called the best estimate of the measurement. The best estimate of an experimental data set is usually presented as

(q)*.A

(1)

q

where (q) is termed as the expectation value or the central value, which

can

for further calculations. For repeated measurements, the expectation value is usually obtained by computing the mean value of the measurement trials. be used

The ubsolute uncefiainty of the measurement is denoted by lq. The absolute uncertainty gives us the quality of the measurement process, and its value can be used in continued calculations on uncertainties. Note that, as the name implies, the absolute uncertainty represents the actual amount, or range by which the

expectations value is uncertain. For single measurements, the absolute uncertainty is defined as the least count of the measuring device divided by two. The least count of a measuring device is the smallest division in that device. For example, in Figure 2, the least count of the device is 1 mm, because that is the smallest division in the device that you can obtain. To calculate the absolute uncertainty of repeated measurements, refer to Example 4. For example, in measuring the length of a table, a best estimate of 35 cm t 2 cm implies that the true length lies within the range of 33 cm to 37 cm. Example 4 shows how to obtain the absolute uncertainty of a data set. To determine the signilicance of the uncerlainLy, we have to extend its definition. For example, if you obtained an absolute uncertainty of +0.1 cm, how do you explain its significance? When we measure the length of a book, or perhaps a table, the value of this absolute uncertainty is significant to some extent. However, if we are to measure the distance between two provinces, or interplanetary distance, an absolute uncertainty of +0.1 cm is highly insignificant. On the other hand, an absolute uncertainty of +0.1 cm becomes meaningless if we are to measure the size of microscopic organisms such as viruses'

Obviously, the significance of an uncertainty value depends on the magnitude of the measurement itself. Hence, it is desirable to compare an absolute uncertainty @ 2007 Lab Manual Authors L

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Physics

71.1

.Measurement, Uncerlalnty and Deviation

with the acfual value of the measurement. For this purpose, we define a quantity called the relatiue uncertainty, /q(o/o), of the measurement. It is defined by

Aq%_#

(2)

The relative uncertainty is often quoted as a percentage so that in Example 4, the : 4.05 % . Therefore, the best estimate in terms relative uncertainty is

h

of relative uncertainty may be reported in the form 222 m + 4%o. Note that number of SF in the absolute uncertainty is equal to the number of SF in

the the

relative uncertainty

The relative uncertainty gives us a much better feeling for the quality of the measureftrent, and we often refer it the precision of the .measurement. The absolute uncertainty has the same dimensions aqd units as the expectation value of the measurement, whereas the relative uncertainty, being a ratio, has neither dimensions nor units and is a pure U

number.

,:

ncertai nty Propagation The rules for compounding uncertainty of measurements are still based on maximum pessimism. For most laboratory work, the following rules are sufficient:

Let v:(v)+Ax , y=(y)tAy , z=(z')*A,

l. Addition and Subtraction. In addition or subtraction , the absolute uncertainty of the sum or dffirence is the sum of the absolute uncertainties of the terms.

Eg. z=x+ ! (z)=(x)+ (y)

Az=Ax*Ay 2. Multiplication: If'"'twb numbers are"'being'multiplied, '

the relative uncertainty of the product is the sum of the relative uncertainties of the factors.

Eg,

z= xy

(z)=(x).(y)

Az (r/,1: Ax (%)+ @ 2007 Lab Manual

Authors

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(%)

M6asu rement, Uncefiatnty

a

Physics 71.1

nd Deviation

3. Power. If a number is raised to a power, the relative uncertainty of the result is the product of the relative uncertainty of the number and the absolute value of the power to which the number is raised. Eg. t ,

=x"

where a is any number

L\-1.\o \-/\""/

Az (%): lalAx (%) From rules 2 and 3, it is obvious that for division, the relatiue uncertainty of the quotient is the sum of the relative uncertainties of the numbers being divided, as in multiplication. The assignment of uncertainff bounds depend on the judgment of the experimenter based on many factors such as fte measuring device, the quantity to be measured and the precision needed.

Deviation data is compared to an aeceptable measurement of the variable being measured to determine the accuracy of the measurement, it is quanfity called d&iutior. ' necessary to define a

If a set of experimental

lew

The absolute deviation

of a measarement is the absolute difference between the

accepted value and the experimental value of the measurement. ab s o lu t e dev i at ion

=

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ac

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ep t e d v alu e

-

exp er iment al v aluel

To determine the significance of the absolute deviation, we define the relative deviation of a measuri:ment as the'ratio betweerf the absolute deviation and the accepted value:

relative deviation=

absolute deviation accepted value

x

100%

Acceptability of Measurement Results ''

To determine

the

\

,acceptability of a measurement fesult, we follow the following

rules:

measurement is given, a measurement is the absolute deviation is less than the absolute uncertainty.

1. If the accepted value of acceptable

if

2. If a maximum

percent error is given, a measurement is acceptable

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if

the

2007 Lab ManualAuthors

rg-lqtiue uncertainty is less tl4an the maximum peycent

eff9r given.

Note that if both the accepted -value.of the medsurement and a maximum percent error are given, then a measurement is acceptable only if both the above conditions are satisfied. I

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Reference

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D.C. Baird, Experimentation:

en tntroOubiion'tot'Meai*eil.nt'Theory

and Experiment Design, 3rd Edition, Prentice-Hall,Inc., USA, 1995.

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Measurement, Uncertainty and Deviation

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Physics 71.1

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Lab Manual Authors

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Date

Data'

Submittod

Pedomed

Scorc

Group Mombera

lnalructor

Sec{lon

Worksheet: Measurement, Unceftainty and Deviation A. Scientific notation and rounding off. Round off the following numbers up to three significant figures and express them in scientific notation. Table

I

0.000 856 400

10.562 3

3 26 500

8.595 00

56.450 001

96.442 s

90 523.5

4 646.56

146 500 000 001

10.050 000

B. Rules on significant figures on operation. Perform the following operations. Write your answers in correct number of sisnificant figures. Table 2 96.895 + 4.65

26.45312 x 6.500

265.239 008 + 86 000 958

54.2

5.610 257 - 2.5

962.581t25

88.264 4 -15

26.53

13.265

x

4.1

t26.598

x

12.5 + 6.98 -2.1 / 0.905

53.24 + 15 x2.3615 -7.625 x26

C. Acceptability of measurement results. Compute for the best estimate of the observables presented in Table 3, given a number of its corresponding estimates. Write your best estimate in the form qlAq Complete Table 4 based on data from Table 3. Briefly answer the questions that follow. Use proper units. Indicate if additional sheet/s is/are used.

@ 2007 Lab Manual

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Meaisu

Physics 71.1

reiient; lln;iertaiity aid Deivtdtion

Table 3. Best estimates of observables

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obseruable

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4

3

2

5

Eesf estimate

Accepted value

3.1514

3.1421

3.1416

3.',|420

3.1501

3.1416

6.544

6.555 "

6.s23

6',.520

6575-

6.61

i.045

'1.203

1.158

1.009

1.001

1.100

mass (g)

5.5

5.3

5.1

5.3

5.5

5.2

speed (m/s)

1.507

1.601

1.512

1.514

1.500

1.6

T(

length (crh) volirme

(

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)

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Table 4. Absolute and relative deviation Absolute deviation

observable

Relative deviation

T(

length (cm) volume

(

773

)

mass (g) speed (m/s)

Questions 1.

How did you estimate the value of the uncertainty for the best estimate? Explain why this is valid.

Based on Table 3 and Table 4, which observable has absolute deviation greater than the uncertainty obtained?

3.

Which of the observables can be considered to have an acceptable experimental proofl

whv?

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Physics71.1

M€abu rermeht; Uhceila

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D. Uncerta'inty' of caleulated' values.,' Compute for the minimurn possible value for each of the quantities given below. Given are the best estimates of the yariables needed. Olserve proper units.

o

Square of

expectation

.

time

f

if

t:

50.00+ 2.0 s.

value (t') : _;

Period of

minimum

fr

pendulum f =Zntl(L) Ig

t' : _;

if /:

maximum

100.00

+

2.OO

i

cm and g

:

:

9.81 + 0.10

mlsz

expectation value :

;

minimum T:

E. Problems

T

; maximum

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o In measuring : the volume of a metal sample,' the volume -, (n ' 19.6+0i.2m3 and the

uncertainty qf the density

Calculation:

mass

( I

(m) obtained was 2.45 + 0.15

kg.

) calculatbd usingithe eciuation,

obtained was What is the absolute

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Final Answer:

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Physics

A simple plndulum is used to measllr,g the"pqpelergiro3 due,to gtpvity.using rT T

. =2n tl: Ig

The period 7 was measured to be I .34

*

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71.,.1

,

:

.02 s and the length to be 0.58,1

+ 0.002 m. Whatis the resulting value for g with its absolute and relative uncertainty? Calculation:

Final Answer:

o

An experiment to measure the density , d, of acylindrical object uses the e{uation d

=-.m Ttr I

and / is the length of thb cylindrical , where rn is''the niass, r is thb radius :".. i ,r.. I

object. The dimsnsions of the object is listed below. m 0.033 + 0.005 r: 8.0 + 0.1

:

kg,

mm l:

14.6

+b.t

mm.

What is the absolute uncertainty of the calculated value of the density? Calculation:

Final Answer:

14

@

2007 Lab,Manual Authors

Using Calipers

Objectives At the end of this activity you should be able to:

1.

Appreciate the role of the available measurement precision to the practical choice of measuring device.

2.

Measure the dimensions of an object using a ruler, aVernier caliper and micrometer caliper.

3.

Identifu a metal sample based on its density.

a

lntroduction '

Calipers are devices that can measure dimensions of small objects and hard to measure observables. The main advantage of usingrone is it allows user to find the very small fractional measurelnents (up to micrometer scale).

This activity teaches the use bf calipers and the application of uncertainty and precision in measuring devices.

Main and Fractional Scale A measurement of a specific device consists of two parts (a) main scale reading ( x us ) and (b) fractional scale ( xrs ). The main scale reading is determined by reading the largest measurement the device can provide. On the other hand, the fractional scale is the fraction of the least count (smallest possible measurement) of the device or may be estimated by the experimonter. In the end, the final measurement is found by adding the niain scale reading and the fractional scale reading, that is

x=xrr*xo, @ 2007 Lab Manual

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(1)

15

Physics 71.1

Using Calipers

Figure 1 shows how the length of an object may be measured using a ruler with least count of 0.25 cm and an estimated fraction part of 0.13cm. The experiment may report 4.38 + 0.07cm or 4.4 + 0.1 cm as his or her best estmate as long as the pnge of lhe reportiqg is" practical and consistent with maximum pessimism or ,.iounding'off prineiples. Also, the reporting of uncertainty should also be consistent. In adding the main scale, fractional scale and estimated fraction 0.I3 cm is reported instead of 0.125 cm since adding all these make the 0.005 insignificant.

xrr:

0.25 cm

4.00 cm

estimated fraction :

0.2512: O.13 cm

x = x,s+

xrJ

estimated fraction

*: +.fS ".

i I I

Figure 1. The length of an object is meqsured u$ng a ruler. The estimated fraction is approximated afier visually dividing the ruler's

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ir

least count.

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Vernier Galiper The French mathematician Piorre Vernier (1580-1637) invented the Vernter caliper in 1631, a device that can measure outer and inner diameters or lengths as well as depths. Figure 2 shows the parts of a Vernier caliper.

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Fignre 2. A;picture of a typical Vernier c'aliper showing'the main scale (4 for metric and 5 for English systeru), Vernier scale (6 for metric.ani 7 for Englisk system), clamping mouth (l for outer diameters and 3 for outer), locking screw (8 ) and depth probe (3).. (graphics by Joaquim Alves Gaspar)

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Physics 71.1

Using Calipers

'

The parts of the Vernier caliper are

main'scale (4 and 5) - reads the main scale reading obtained by taking the last mark of the main scale before the zero of vemier scale (edge of zero mark).

Vernier scale (6 and 7) - estimates the fractional scale reading by taking

the

order of the Vernier scale mark that literally aligns with the main scale mark.

clamping mouth (1)

- used to measure

diameters, opposite to this mouth (2) is

used to measure inner diameters of pipes.

depth probe (3)

- used to measure depths.

locking screw (8)- used to lock the caliper after sbtting it. The caliper is set after applying enough pressure (avoid squeezing the object) as the clamping mouth spans the diameter of the object. The zero reading of the Vernier scale is obtained by closing the mouth completely and getting the reading. If the main scale reading is to the left of zero, the least count of the main scale should be subtracted from the fractional reading. Before ,rirg measuring devices be sure that they are properly calibrated and are in good working condition. Calibration of instrumentsr,,imrolves ensuring they work well within the range of values being measured and are properly zeroed. The Vernier caliper is properly zeroed if the zero mark of the rnain, scale coincides with the Vernier scale when the clamping mouth is closed.

In using a vernier caliper the clampirrg *outh is,set after applying

enough pressure to keep the object in place but not enough to defonn or squeeze it. The lock may be turned to ensure that the clamping mouth will not move even if the measured object is removed.

How to Use a Vernier Caliper A

Vernier caliper allows better estimation of the fractional part of a length measurement by the use of its VERNIER SCALE (VS). To read the vernier scale, the LEAST COLTNT (LC) or the precision of the caliper must be known. This is obtained by counting how rnany subdivisions the VS will make on the main scale. The caliper in Figure 2 has the smallest reading on the main scale at 0.1 cm.

Meanwhile, the Vernier scale can create 20 subdivisions. Hence LC is obtained using

tr=*=olo5cm @ 2007 Lab Manual Authors

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Using

Calipers

Physics 71.'1

Figure 3 shows an example of Vernier caliper reading. The caliper has 50 (including the smaller tick marks) Vernier divisions and its smallest reading on the main scale is I mm. Hence the LC of the caliper is

'!!50

:o.o2mm.

In reading a Vernier scale measurement, take the main scale reading at the left of the zero mark of the VS, not the edge. In Figure 3, the main scale reading is 26 mm. Next, take the VS scale line which is coincient with the Vernier scale. Note that in Figure 3, the VS mark coincides at the 17'h line. From these values we can determine the measurement of the Vernier caliper:

::t?il {i?1rl 4i

#{ *rt,

ittl

Figure 3. A close up view of the Vemier caliper. What is the least count of the caliper? What is the reading of the caliper?

in readings is

subjective. Its value must be given by the experimenter. As a rule of thumb, the uncertainty should be half of the least count as long as no other technical reason interferes with the measurement process.

The uncertainty

Try out the simulation in http://www. physics. smu.ed u/-scal ise/apparatus/caliper/tutorial/ practice reading a Vernier caliper.

18

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Physics 71.1

Using Calipers

Micrometer Caliper Figure 4 shows the parts of a typical micrometer caliper.

Figure 4. Micrometer caliper showing its main (barrel) scale (M), thimble scale (T) for its fractional scale, lock (L),jaw (J), and rachet (R).

A micrometer caliper estimates the fractional scale using a screw mechgpism. The displacement of the barrel is proportional to the number of turns of the thitnBf# For example, if the thimble moves at a distance of 0.5mm per rotation, then dividing the thimble into 50 equal parts would make the least count to be 0.01mm.

Figure 4 shows the parts of a typical micrometer caliper: (J)

jaw -partthat actually

(B) barret

-

spans the diameter/length/width of the sample.

used to read out the main scale

(M) reading (the last mark the edge of the thimble has passed), in case of ambiguity, look at the value of the thimble reading (if less than half a revolution it means the thimble has just passed the mark).

rotated to make the jaw clamp the object , this part is divided equally along the edges so that the fraction of revolution can be obtained.

(T) thimble

(R) rachet

-

-

this is a knob that is tightened or loosened to set the strength of

clamping to the object.

(L) lock - this is used to keep the setting of the instrument for reading (used if the sample is hard-to-reach and the micrometer need to be removed from site to read the measurement).

The micrometer screw must be turned at the rachet while closing the jaw to prevent the screw mechanism from wearing off and to avoid excessive clamping of the sample to be measured. One or two clicks from the rachet should indicate enough tightness of the clamp. @ 2007 Lab Manual

Authors

19

Using Calipers

Physics 71.1

ilf#.1",i;,,'"?,T"""iJ:'",i"i1.il:H:J:ii;1,:'Ji"1#:[.Tiff:l'i;:

0, 1r

-- turning

the

Each complete rotation is divided and marked into equal subdivisions which make reading of the fractional part straightforward. Arbitrary further division (user

Psnl

u0+

dependent) in the thimble reading can be done. See for example in Figure 6. The - rrr tJtP- 0 0 6 2 6W main scale reading is x^ : l3.5mm. since the upper marks correspond to lmm ' and the lower to 0.5mm marks. This particular micrometer caliper has 50 divisions in the circular scale. One fullturn moves it 0.5 mm. Therefore, the least count of the fine scale is 0.50mm/50 : 0.01mm. The fine scale has passed the 21"

notch therefore xr"=0.01

mmx2l:0.21mm

However, as can be noticed, we can still make the reading finer by having fractional reading within the thimble's least count - the zero barrel mark is near the 22d notch, say, it may be around 8/10 of 0.01mm or 0.008mm. The fine scale reading plus estimate will then be 0.218mm. So the final reading would be: x : t3.5mm + 0.2l8mm or x :

l3.7l8mm.

Figure 6. An example of micrometer reading. The marks show

20

a reading

of 13.71Smm.

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Lab Manual Authors

Physics 71.1

Using Calipers

Materials :. ,. ,$uler, Vernier caliper, micrometer screw, dlgital balance€nd metal samples .

' :.

i i I l: :'

,. r.

Proc6dUfe,' 1.

Calibrate the ruler, Vemier caliper, micrometerj caliper and the digital balance by noting the least count of these equipment, the least count of bo. Input your data in Table 1 of the worksheet.

2.

Measure the mass of the metal samples using the digital balance. Use Table 2 to record the masses. Compute for the relative uncertainty by using the expression

a m(N1=4 ?-x \m)

3.

@ 2007 l-ab Manual

I oo

%

(3)

Measure the dimensions of the sample and tabulate in Table 3.

Authors

21

Physics 71,1

Using Calipers

Figure

8.

Use the depth probe to measure the depth or the inner height of the metal

sample.

4.

5.

Compute for the volume of the samples. Assume a specific shape for each sample. Write out your computed volumes Table 4.

Finally, compute for the density o/orho of the samples using

M p=T 6.

Compute the relative and absolute uncertainty of the density values. Write them down in Table 5.

7.

22

Identifu what type of metal the samples are made of by comparing your computed densities with densities of different metals.

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Lab Manual Authors

Name

Dats

Date

Submitt d

Pertomed

Scolt

Group ilembarg

lnstructor

Slectlon

Worksheet: Using Calipers l. Galibration of measuring devices Complete the table below to determine the least count and estimated uncertainty of the vernier caliper and the weighing scale. Data Table I. Least count and estimated uncertainty of the measuring devices used. Weighing scale

Ruler

Vernier Caliper

micrometer caliper

Main scale (least count) Number of fractional

divisions Least count Estimated uncertainty

.

Based on the least count of ruler, Vernier caliper and micrometer caliper, which of the devices is most precise?

ll. Calculation of the density of the sample

A. Mass measurement Data Table 2. Masses of the metal samples. The relative uncertainties are based on the estimated uncertainty in Data Table l. Itetal sample

Mass (g)

Re I ati ve U n ce rta i n

ty (/o)

A B

A 20AT Lab Manual Authors

23

.

.i.

11

Using Calipers

Physics 71.1

B. Volume measurement Data Table 3. Measured dimensions of the metal samples using the ruler (R),Vernier caliper (yC) and micrometer caliper (MC). The relative uncertainties A x are based on the relative

a

@,2007 Lab ltrlanual Airthors

Physics 71.1

Using Calipers

Data Table 4. Yolume of the samples. The uncertainties are calculatedfrom the absolute uncertainties in Data Table 3.Write out your solution in a separate sheet of paper. Measuring device

Sample

V (mm3)

aY

("/;)

AV

(mms)

A

Ruler

B

A

Vernier caliper

B

A

micrometer caliper

B

Data Table 5. Sample identification. From the values of the mass and volume found in Data Tables 2 and 4, calculate the best estimate of the density of the samples. Reseorchfor the densities

of

these samples.

Measuring device

Sample

q

(ilcm')

Acp W

Acp (g/cm')

A

Ruler

B Vernier caliper

A B

micrometer caliper

A B

Identifu what element comprised sample A and B.

B:

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Lab,Manual Authors

25

,Physics'1,'t.1

U@r:.Ga.fipers

ri

, 1, Shat :

assrrmption(p), !':i

if

any, inthe shapeof-the samples is/are most likely 1o1_ry3li,ze$? -r

i--.

--

:-1. ,-', ;.'

:.i.

l-

,

..-

,:,i

26

2.

Would the use of a more precise length measuring device improve the performance of the method used to determine the density of the sample metals?

3.

Can this method accurately identiff the major percent composition of analloy? Try this out by identiffingthe m.4jor element composition oJa 5 centayo cgin.

923;gr Lab Maaual Authors

G,raphical Analysis

At the end of this activity you should be able to:

1.

Create a graphical representation of a given set its purpose;

2.

Formulate a theory or a model based on the parameters from a graph experimental data using linear fit and trendlines.

, 3.

of

data

thatwill best show

of

Leam how to use spreadsheets (Microsoft Excel) and some of its basic functions.

lntroduction The most convenient way of presenting a dataset is through graphical presentation. A graph'is defined as the pictorial representation of a set of data which could be'in 2 or 3 dimensions. trt allows the experimenter to understand the relationship between 2 or'more parameters

Theory Graphs may involve shapes, curves and symbols. Some types of graphs are pie,

bubble, scatter, bar and line graphs. Figure I shows an example of two dimensional scatter graph which is most commonly used,as a way to present the relationship between two variables.

@ 2007 Lab Manual

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27

Graphical Analysis

Physics 71.1

\

Figure {. The plot shows a linear relationship between the squad of the period of a simple pendulum and the length ur the rrrs string. surrrs' \ fuur of

L

| ""eil;l

Graphs have basic parts that need some attention before they could express

their purpose well.

"

Shown in 'Figure

I is a Sample graph with parts

described below.

a) Title -

This part is usually placed at the top of each graph. It tells a specific thought about what the graph shows. Since a caption is usually i included, this part can be omitted due to redundancy

b) Axes -

This is ,the part that shows the values of the variables involved. The x-axis usually contains the parameter values (independent variable) and y-axis contains the observable in ques.tion (dependent variable). The range of values in the'axes should be reasonably enough for the range of data concerned be shown. Oftentimes, the maximum and the minimum scale should also be adjusted to give the best display (the numbers are well spaced and readable).

28

c)

Labels are words or phrases that best describe Jhe quantity being represented by 1n axiq, Thus there are two labels for a 2- dimensional graph since there are two variables (thus two axes) involved. It should be noted that a label includes the unit used in measurement.

d)

Symbols - These could be filled circles, squares, triangles, and other shapes that represents a point or a thought about a datapoint. These

Labels

-

@

2007:Lab Mdnual Authors

Physics 71.1

Graphical Analysis

symbols should be clear enough (not too big but not too small) so that other datasets plotted in the same set of axes can be easily differentiated. Color atdlor shading should be utilized to maximize this effect. Shadows and other o'special" or "aesthetic" effects should be avoided specially for graphs with technical or formal purposes. e)

Legend - This describes each of the dataset used in a graph. Using a word or a short phrase, the legend differentiates different symbols used. This is not necessary for graphs that shows only one dataset.

Caption - This is used to briefly describe the idea being presented by a graph by clearly pointing:out salient parts in the presentation (e.g. skewed points, alignment of points, trends, similarities). Important parameters not in any of the axes should be mentioned and described in this part. It is a challenge for the presentor to make captions as short as possible. Captions may include titles which may prove useful for quick glances.

Graphing Procedures Variables are commonly plotted

in a

rectangular coordinate system. The dependent variable is placed on the y-axis and the independent variable is placed on the x-axis.. The location of a point on a graph is defined by its x and y coordinates, written (x,y), with respect to a specific origin.

In plotting a dataset, the axis scales should be chosen such that the plot is easy to understand. With axis scales that are too small, the points will bunch together, making the plot incomprehensible.

Error Bars Collection of data involves measurement; hence, this implies that uncertainties are present. In plotting a set of data which includes the expectation value and its

corresponding uncertainties, the expectation value is plotted and the corresponding uncertainty is presented as an error bar. Error bars show the possible range of values of one or more variables in a data point. This is useful since it allows the experimenter to know the range of possible values under the @ 2A07 Lab Manual

Authors

29

Graphical Amlysis

Physics 71.1

influence of a certain variable.

Trendlines and linear fit Data points in an x-y scatter plot should not be individually connected by lines. In the event that the experimenter is certain about the relationship of the variables being presented, a smooth line or curve called the best fit line can be drawn to represent the known relationship. The word'osmooth" does not imply that the line

or curve must pass exactly through'each point. But the best fit line should best represent the data set. This type of plotting is called eyeball method. The main criterion for.this method is to minimize the distances of all data points from the line drawn. Once linearized, the variables'can be represented:using the equation

y=mx'+b

(l)

where m and b are constants that represent the slope and the y-intercept of the plot respectively.

Slope is an algebraic relationship of the line and is given by the equation

Ax Ay

(2)

Any set of intervatr may be used to determine the:slope of a linear plot. But for best results , points, showld be chosen within the best fit line. lf the data point is not included in the best fit line, it should not be used to calculate the slope of the graph.

Other forms of nonlinear functions may also be represented as a linear plot. For example, the equation

may be reduced

y- gx'+b to a linear equation if we let

(3)

Hence, Equation (3)

x =x

reduces to the form

.r.

y,=.gx'*b

which is just equivalent to Equation

30

l

J,.

l.

@

2007 Lab ManualAuthors

Physics 71.'1

Graphical Analysis

Graphing using a spreadsheet The following is a step by step way to plot data using Microsoft Excel.

1.

Input your x and y data in two separate columns. Try this out using the sample data from your worksheet.

2. Highlight these two

columns and click "chart wtzard" icon on your

toolbars.

3. 4.

Choose x,

y scatter on your Chart type and click next.

You will see a preview of your plot. Ensure that the option you choose is series in 'column'.

5. click 'next' and enter the chart title, x -axis label and y-axis label. You may also click on the tabs to modifii the axes, gridlines, legend and data labels. Just continue clicking next and you have your plot. g& 4Bw r]hHt

FE@ Idc

[*6

ffitu

EeIF

li.l i.]S$t L} ElidJ irh,1F'Ht, l( *:e,J&: {fl,'rr " ,r" i#, a : fl,al :ffijjom - r{S g -lo - E / !t,E#gfig$iry#% r tdg ;1$ i AF EF, g: t$r - A -ffi

i I

6.

To include error bars on your plot, just type half of the magnitude of your error bar on a column beside your y-data points.

7.

Right-click your data points on the plot and choose 'Format data series'.

@ 2007 Lab Manual Authors

31

Graphical Analysis

Physics 71.1

8.

After choosing 'Format data series', click the y-error bar tab. Choose 'Custom'. You may opt to type the series on the + and - space or you may click the icon beside the space and highlight the corresponding series.

9.

To add a trendline, right-click again the data points and choose the option 'Add trendline'. Choose the corresponding best fit curve for your plot. To insert the equation of your trendline, click on the 'Options' tab and check the box on ' Show equation'.

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A 2007 Lab Manual Authors

Xama .

Date

Dat6

$ubmittod

Perioamed

Scorc

Group tlemberc

lnstructor

Section

Worksheet: Graphical Analysis A. PreSenting data'set graphically During an experiment, a physics student obtained the following data: x + 0.10

v

-5

384.5

-4

208

-3

96.5

-2

35

-l

8.5

I

0.5

2

-l I

3

47.5

4

-124

5

-255.5

The variables x and y are the independent and dependent variables of the experiment, respectively.

o

of x,

Ploty

as a function Can you conclude with certainty lhat.the plot is linear? Explain your answer. You may try to fit a line using the eyeball method and argue from

there.

o @

Ploty

as a'function

2047' Lab Manual Atrthors

of

x2 ,

.

Canyou conclude with certainty that the plot is linear?

33

Graphical"Analysis

Physics 71.1

Explain your answer. You may try to fit a line using the eyeball method and argue from there.

Ploty as a function of *' . Can you conclude with cgrtainty that the plot is linear? Explain your answer. You may try to fit a line using the eyeball method and argue from there.

From your answers in items 1-3, determine the degree (in x) of the equation relatingy and (Recall that the equation y=ax2 *bx-lc has a degree of 2 in x.)

x.

B. Problem solving using graphical analysis 1. The

Chronicles of Narnia: The King, the Prince and the Heirloom. 'On his King-father's deathbed, Prince Caspian of Narnia was mandated to find the mass (M of the royal family's heirloom. After days of sleepless nights, he was reminded of a very important lesson from the great Professor Digory: The Parallel-Axis Theorem. This states that a body rotating about an axis parallel to and at a distance d from the center-of-mass axis has a moment of inertia I p about that axis written as

I I I

I

i I

where

I

,= I "^*Md2

is the moment of inertia about the center of mass. By the Prince's command Regpicheep, {he ,commander, of the Army, conducted a series of

I

i I I

"^

I I I I I

t N

i :

I

34

@

20Ol Lab Manual Authors

Physics 71.1

Graphical Analysis

experiments using Vernier LabPro@ that eould determine d and I p at precisions (least counts) of 0.10 cm and 0.50 g.cm2 respectively.Reepicheep was V great wa.r.rior, but so poor physicist, that he tabulated his data so horrendously:

(I o)G'cm')

(d)(cm)

IJ

2.51

2.s2

3.6

4.010

7

8.1200

8.667

11.010

9.41

A. Re-tabulate Reepicheep's data correctly by writing the expectation value of the moment of inertia and the distance from the center of mass based on the given precisions.

(1,)(s'cm')

A 2007 Lab Manual Authors

(d)(cm)

35

Physics 71.1

Gruphical Analysis

B.

Plot I

ovs.

d2

and paste it' on the space below. Calculate the best estimate of the mass of the

mysterious heirloom.

Solution:

Final Answer:

36

M:

@

2007' Lab Manual Authors

Physics 71.1

Graphical Analysis

2. Off to the moon!

I

The accepted value for the acceleration due to gravity of the lunar surface gmoon, is 1/6 that of the earth, gno,,^=9.8m1s2 You decided to go to the moon and

conduct experiments

to verify this value. However, because of your busy

schedule, you have no time to go to the moon and decided to send your younger brother instead. He conducted free fall experiments, measuring the time it takes for a freely-falling ball to ,reach the lirnar surface upon release from an initial

height h. He used a timer with 0-001 s precision (least count) and a meterstick with a least count of I mm. His estimated fraction for the meterstick is 0.5 mm.He obtained the following data below. However, he has no Physics 71.1 training when it comes to reporting measured data. t(s)

h(m)

0.34

0.1

0.58

0.27

I

0.85

1.3410

1.6s

r.604

2.5

A.

Retabulate your younger brother's data'correctly by writing the expectation value of the time and the initial heisht based on the si

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37

Physics 71.1

Graphical Analysis

B. If

/

and hare related

by n=)S t'

, obtain the best estimate for gmoon.

Solution:

Final

38

Answer: I

*oon

:

@ 2007 Lab Manual Authors

Vectors a'nd Force Table

Objectives At the end of this activity you

L

shoutrd be able to:

Show that the sum of forces acting on a system in static equilibrium is zero.

2.

Obtain the equilibrant

of two or

more forces using the concept of

equilibrant.

3.

Obtain the orthogonal components of a force.

lntroduction Vectors are mathematical representation of physical quantities that involve a rnagnitude and a sense of 'direction. Examples of physical quantities that can be represented by vectors are: position, velocity, force, and electric fields. These quantities follow rules of addition and multiplication just as vectors do . The magnitude and direction of vectors do not necessarily need to be real. can be represented by an affow in space. A two-dimensional vector needs an arrow in a planar surface. On the other hand, a three-dimensional vector is represented by an arrow with three-dimensional direction.

A vector

Oftentimes it is difficult to imagine the graphical representation of vectors making graphical approach impractical and analytic representation comes handy. Analytically vectors can be decomposed into its orthogonal (graphically

perpendicular; physically' independent) components. Since vectors are mathematical entities, they follow certain rules of combinations. The simplest

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Lab Manual Authors

39

Physics 71.1

Vectors and Force Table

means

of combination are addition (and subtraction) and multiplication (division

is not possible for vectors).

This activity.deals with comparing theoretical (graphical and analytic) approaches in dealinglwith combining physical vectors, force in particular, including about the concept of resultant and equilibrant.

Theory Vector addition (and subtraction)

'

Just like the physical quantities vectors represent, they can be added (or subtracted) to (or from) each other. It should be emphasizedthat only vectors that represent the same physical quantity. can be added or'subtracted. This translates to the idea that only vectors with same units can be addgd together or subtracted from each other. Thus the vectors i , E , and e should have the same uriit so that

t=Z+B

(1)

has a physical meaning. The magnitude of the vectors follows the inequality below (2)

ileil