Physics71.1 Activity Manual

2007 Edition

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Prepared byLeilani Torres

Elise Stacey AgraMaricor Soriano

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The 2007 Lab Manual AuthorsElise Stacey AgraJunius Andre F. BalistaMary Ann B. GoMargie OlbinadoAthena Evalour PazLeilani Torres

CoordinatorMaricor Soriano

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@2007 and2004 Lab Manual Authors

All rights reserved. No part of this publication maybe reproduced or transmitted or by any means,including photocopy, without written permissionfrom the 2007 and2004

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Published by the Philippine Foundation for Physics, Incorporated efpDfor the exclusive use of the National lnstitue of Physics, uP Diliman

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Table of Contents

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Preface

The 2007 Physics 7l . I Activity Manual is the 4'h outing of the Elementary Physics I Lab Manual series. This year,s volumehas l0 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 andMomentum) , (4) Torque (static Equilibrium), (5) Simple Harmonic Motion (Simple Harmonic Motioi: sprifi-MassSystem), and (6) Mechanical waves (Sound)

Of the l0 experiments in the current volume, 6 are new or revised. In Measurements, [Jncertainty and Deviation, rules forhandling significant figures and propagation of uncertainty are made more explicit. rn (Jsing Calipers, the use of the depthprobe of the Vernier caliper is explained and incorporated in the experiment. Instructioni oo Lo* to create plots usingMicrosoft Excel have been includedin Graphical Analysis while expiriments to demonstrate two conservation laws havebeen merged into one experiment in Conservation of Energlt and-Conservation of Momentum. T\e Simple HarmonicMotion experiment is totally new in that a spring-mass system replaces the simple pendulum which had been used in thepast 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 wasone experiment that req-uires a computer interface, in the 2007 ,rolu-" there are three. Besides Untfurmly AcceleratedLinear Motion (UALM, Simple Harmonic Motion (SHltl)and Sound requires the use of tre photogate and Vemiermicrophone respectively.

Because of the increased use of computers, we recommend the following flow of experimts fur parallel sections inPhysics 7l.l to avoid overlap in the use of interfaces.

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Section 1 Section 2

Experimental Skills in ElemPhysics 1

Expermental Skills in EbmPhysics 1

UALM (computer) Torque

Projectile Motion Sound(computer)

Conservation Laws SHM (computer)

Torque UALM (computer)

SHM (computer) Projectile Motion

Sound (computer) Conservation Laws

The lab and lecture topics of Physics 7l need not be synchroni zed. ln case a class follows thc Sctftn 2 plan, topics coveredin the lab

_may even be ahead of the lecture. This should not be a problem because fre iuo&ctory text 'su_f.flciently

discusses the necessary concepts for the experiments. Stadents are required ,o @ra b *.c ptryd by reading the rextcnd procedures prior to engagement in the lab. The prelab exercises have been dooc !r.y wift in this -.,oli,irn" butinstmctors 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 aregrateful to the students who participated in the pilot testing and to the instnrctors wto cilrH-odii.d lhe text during theGeneral Physics Committee workshop in June 2, ZOO7.

Itfuicor N. SorianoElise Stacey G. Agro

Ith- Leilani Y. Tones

Measu rement, U nceftai ntyand Deviation

Objectives

lntroduction

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

Report the best estimate of'observables and quantiff it.

Determine if a theoretical prediction is acceptable given the precision and

deviation of an experimental data.

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

Appreciate the role of measurements in scierrtific activity.

1.

2.

J.

4.

In a scientific endeavor, experiments involve collection of information or datathrough measarement.Datasets are presented to gain empirical knowledge abouta phenomenon, validate or invalidate an existing theoretical model and

demonstrate that a proposed method works. The measurement of certain variablescalled observables allows us to achieve this goal. Observables are also calledparameters. It is usually the quantity being controlled during the experiment.

Since measurement involves unknown quantities, there is always an uncertainty inthe measured values. This uncertainty is not always due to personal mistakes. Thedegree 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 oftheir magnitude, their conventional description and calculation of numericalvalues 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, ifan 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

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 farfrom the center target. Hence, this

case is high precision but lowaccuracy. Figure 1 (b) is low inprecision but the average of the

location of the stars is close to the

bullseye center, hence it has higher

accuracy compared to Figure l(a).Figure I (c) shows that most of the

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

(a)

(c) (d)

Figure l. Arrows on a bullseye. Four-point stars mark their landing. Arrows on(a) shows high'piecision but low accuracy,(b) low precision but high accuracy, (c)high precision and accuracy, (d) lowprecislon and accuracy.

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Physics 71.1 Measurement, Uncertainty and Deviation

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 ofthe instrument must also be checked. A body:-filerrnorneter cannot be used formeasirring 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 2shows 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.

15 16 t7 18 19 20 mmReporting and handling ofuncertainty can be categorized

into four approximations. The

use of each category depends on

the level of uncertainty the

experimenter requires.Figure 2. Measurement with parallax. What do

The first level of handling you expect the observer will read? What should

uncertainty is called zeroth ordi 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 jirstupproximation. 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 MagnitudeThe first order of approximation is done by estimating the measurement bypowers of 10. Fermi questions are answered by thinking of reasonable

assumptions followed by simple calculations that narrow down the range ofvalues where the answer lies. Hence, Fermi questions are answered in terms oforder 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 ofseconds in the year which is exactly 3 x107 s/yr, order of 106 to 107 is sufficient


Measurement, UncertainU and Deviation

an approximate.

Physics 71.1

is the least

Significant FiguresThe,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.

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 ofdigits that is significant

r. 1200-2sF2. 13.20-4SF3. 112000.-6SF4. 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, theanswer should be reported as 5.1. Anexperimental data cannot be made moresignilicant by' a mathematical operation.

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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 theaddends.

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 offNonsignificant 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 offare 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 obtainedthe following data: 215 rt" 222 m,219 m,231 m,224 m

The expectation value, <q>:

215+ 222+ 219+ 231+ 224

(q)= zzzm

To obtain the uncertainty, A q, subtract the <q> from themaximum value and the minimum value from A q. The larger ofthe 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 estimateofthe data is reported as

(9) = 222mt gm

3. In cases of multi-step mathematical operation, only the final result should

@ 2007 Lab ManualAuthors

Measu rement, Uncertainty and Deviation

be rounded off.

Physics 71.1

(1)

Absolute and Relative UncertaintyThe 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 q

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

be used for further calculations. For repeated measurements, the expectation value

is usually obtained by computing the mean value of the measurement trials.

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 ofthe measurement itself. Hence, it is desirable to compare an absolute uncertainty

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U ncertai nty Propagation

The rules for compounding uncertainty of measurements are still based onmaximum pessimism. For most laboratory work, the following rules aresufficient:

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

l. Addition and Subtraction. In addition or subtraction , the absoluteuncertainty of the sum or dffirence is the sum of the absoluteuncertainties of the terms.

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

Az=Ax*Ay2. Multiplication: '

uncertainty of the

factors.

Eg, z= xy

(z)=(x).(y)

Az (r/,1: Ax

Physics 71.1 .Measurement, Uncerlalnty and Deviation

with the acfual value of the measurement. For this purpose, we define a quantitycalled 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, therelative uncertainty is

h : 4.05 % . Therefore, the best estimate in terms

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

The relative uncertainty gives us a much better feeling for the quality of themeasureftrent, and we often refer it the precision of the .measurement. Theabsolute uncertainty has the same dimensions aqd units as the expectation valueof the measurement, whereas the relative uncertainty, being a ratio, has neitherdimensions nor units and is a pure number.

,:

If'"'twb numbers are"'being'multiplied, the relativeproduct is the sum of the relative uncertainties of the


(%)+ ^y

(%)

M6asu rement, Uncefiatnty a nd Deviation Physics 71.1

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.

DeviationIf a set of experimental data is compared to an aeceptable measurement of the

variable being measured to determine the accuracy of the measurement, it is

necessary to define a lew quanfity called d&iutior. '

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 = | ac c 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 x 100%

accepted value

Acceptability of Measurement Results \

'' To determine the ,acceptability of a measurement fesult, we follow the following

rules:

1. If the accepted value of measurement is given, a measurement is

acceptable if the absolute deviation is less than the absolute uncertainty.

2. If a maximum percent error is given, a measurement is acceptable if the


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.

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Referenceo 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 Physics 71.1

@ 2OO7 Lab Manual Authors10

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l{rm. DateSubmittod

Data'Pedomed

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Group Mombera

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Worksheet: Measurement, Unceftaintyand Deviation

A. Scientific notation and rounding off.Round off the following numbers up to three significant figures and express them in scientificnotation.

Table I

B. Rules on significant figures on operation.Perform the following operations. Write your answers in correct number of sisnificant figures.

Table 2

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

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

96.895 + 4.65 26.45312 x 6.500265.239 008 + 86 000 958 54.2 t26.5985.610 257 - 2.5 962.581t25

88.264 4 -15 26.53 x 12.5 + 6.98 -2.1 / 0.90513.265 x 4.1 53.24 + 15 x2.3615 -7.625 x26

@ 2007 Lab Manual Authors 11

Meaisu reiient; lln;iertaiity aid Deivtdtion Physics 71.1

Table 3. Best estimates of observables

obseruable trlal Eesfestimate

AcceptedvalueI 2 3 4 5

T( 3.1514 3.1421 3.1416 3.',|420 3.1501 3.1416

length (crh) 6.544 6.555 " 6.s23 6',.520 '', 6575- 6.61

volirme ( rn' ) 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

Table 4. Absolute and relative deviation

observable Absolute deviation Relative deviation

T(

length (cm)

volume ( 773 )

mass (g)

speed (m/s)

Questions1. 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?

12 @ 2OOl Lab Manual Authors

Physics71.1 M€abu rermeht; Uhceila inty ahd DCvi atian

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

o Square of time f if t: 50.00+ 2.0 s.

expectation value (t') : _; minimum t' : _; maximum i :

fr. Period of pendulum f =Zntl(L) if /: 100.00 + 2.OO cm and g : 9.81 + 0.10Igmlsz

E. Problems

o In measuring the volume:19.6+0i.2m3 and the mass

uncertainty qf the density ( I

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of a metal sample, the volume (n' -, '(m) obtained was 2.45 + 0.15 kg. What

) calculatbd usingithe eciuation, ,e=f

expectation value <T>:

Calculation:

Final Answer:

; minimum T: ; maximum T :

obtained was

is the absolute

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@2007 Lab Mddual Authors 13

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l@arrorrerf,' U n ee fiai n&t. an d, u9ev iall q O Physics 71.,.1

o A simple plndulum is used to measllr,g the"pqpelergiro3 due,to gtpvity.using , : ..1; , :

rTT =2n tl: . The period 7 was measured to be I .34 * .02 s and the length to be 0.58,1Ig

+ 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.m

d =- , where rn is''the niass, r is thb radius and / is the length of thb cylindricalTtr I i :".. ,r.. I

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

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

1. Appreciate the role of the available measurement precision to the practicalchoice of measuring device.

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

3. Identifu a metal sample based on its density.

lntroductionCalipers 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 andprecision in measuring devices.

Main and Fractional ScaleA 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 byreading the largest measurement the device can provide. On the other hand, thefractional 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 finalmeasurement is found by adding the niain scale reading and the fractional scale

reading, that is


x=xrr*xo, (1)

15

Using Calipers

estimated fraction : 0.2512: O.13 cmx = x,s+ xrJ estimated fraction

*: +.fS ".Figure 1. The length of an object is meqsured u$ng a ruler. The

estimated fraction is approximated afier visually dividing the ruler'sleast count.

Vernier Galiper

Physics 71.1

Figure 1 shows how the length of an object may be measured using a ruler withleast 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.I3cm is reported instead of 0.125 cm since adding all these make the 0.005

insignificant.

xrr: 4.00 cm 0.25 cm

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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.

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

16 O 2007 Lab ManualAuthors

Physics 71.1

How to

Using Calipers

The parts of the Vernier caliper are

' main'scale (4 and 5) - reads the main scale reading obtained by taking the lastmark 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 theorder 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) isused 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 afterapplying enough pressure (avoid squeezing the object) as the clamping mouthspans the diameter of the object. The zero reading of the Vernier scale is obtainedby closing the mouth completely and getting the reading. If the main scalereading is to the left of zero, the least count of the main scale should be subtractedfrom the fractional reading.

Before ,rirg measuring devices be sure that they are properly calibrated and arein good working condition. Calibration of instrumentsr,,imrolves ensuring theywork well within the range of values being measured and are properly zeroed. TheVernier caliper is properly zeroed if the zero mark of the rnain, scale coincideswith the Vernier scale when the clamping mouth is closed.

In using a vernier caliper the clampirrg *outh is,set after applying enoughpressure to keep the object in place but not enough to defonn or squeeze it. Thelock may be turned to ensure that the clamping mouth will not move even if themeasured object is removed.

Use a Vernier CaliperA Vernier caliper allows better estimation of the fractional part of a lengthmeasurement 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 isobtained by counting how rnany subdivisions the VS will make on the mainscale. 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 obtainedusing

tr=*=olo5cm


(2)

17

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 onthe main scale is I mm. Hence the LC of the caliper is

'!! :o.o2mm.50

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

determine the measurement of the Vernier caliper:

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

The uncertainty 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.

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

to

18 @ 2OO7 Lab Manual Authors

Physics 71.1

Micrometer Caliper

Using Calipers

Figure 4 shows the parts of a typical micrometer caliper.

Figure 4. Micrometer caliper showing its main (barrel) scale (M), thimble scale (T) for itsfractional 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 spans the diameter/length/width of the sample.

(B) barret - used to read out the main scale

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

(T) thimble - rotated to make the jaw clamp the object , this part is dividedequally along the edges so that the fraction of revolution can be obtained.

(R) rachet - this is a knob that is tightened or loosened to set the strength ofclamping to the object.

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

The micrometer screw must be turned at the rachet while closing the jaw toprevent the screw mechanism from wearing off and to avoid excessive clampingof the sample to be measured. One or two clicks from the rachet should indicate

enough tightness of the clamp.


0,1rPsnlu0+tJtP- 0

Using Calipers Physics 71.1

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

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

- rrr dependent) in the thimble reading can be done. See for example in Figure 6. The0 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 leastcount 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, wecan still make the reading finer by having fractional reading within the thimble'sleast count - the zero barrel mark is near the 22d notch, say, it may be around8/10 of 0.01mm or 0.008mm. The fine scale reading plus estimate will then be0.218mm. 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 a reading of 13.71Smm.


Physics 71.1

Materials

Using Calipers

:. ,. ,$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 digitalbalance 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 byusing the expression

a m(N1=4 ?-x I oo % (3)\m)

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

@ 2007 l-ab Manual Authors 21

Using Calipers Physics 71,1

Figure 8. Use the depth probe to measure the depth or the inner height of the metalsample.

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

Mp=T

Compute the relative and absolute uncertainty of the density values. Write

them down in Table 5.

Identifu what type of metal the samples are made of by comparing your

computed densities with densities of different metals.

4.

5.

6.

7.


Name DatsSubmitt d

DatePertomed

Scolt

Group ilembarg

lnstructor Slectlon

Worksheet: Using Calipers

l. Galibration of measuring devicesComplete the table below to determine the least count and estimated uncertainty of the verniercaliper and the weighing scale.Data Table I. Least count and estimated uncertainty of the measuring devices used.

Weighingscale

Ruler VernierCaliper

micrometercaliper

Main scale (least count)

Number of fractionaldivisions

Least count

Estimated uncertainty

. Based on the least count of ruler, Vernier caliper and micrometer caliper, which of thedevices 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 theestimated 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

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

Physics 71.1

@,2007 Lab ltrlanual Airthorsa

Measuring device Sample V (mm3) aY ("/;) AV (mms)

RulerA

B

Vernier caliperA

B

micrometercaliper

A

B

Physics 71.1 Using Calipers

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

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

Identifu what element comprised sample A and B.

B:

Measuring device Sample q (ilcm') Acp W Acp (g/cm')

RulerA

B

Vernier caliperA

B

micrometercaliper

A

B

@ 2OO7 Lab,Manual Authors 25

U@r:.Ga.fipers ,Physics'1,'t.1

ri

, 1, Shat assrrmption(p), if any, inthe shapeof-the samples is/are most likely 1o1_ry3li,ze$? ,

: !':i -r i--. -- :-1. ,-', ;.' :.i. l- ..-

,:,i

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 thisout by identiffingthe m.4jor element composition oJa 5 centayo cgin.

26 923;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 of data thatwill best showits purpose;

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

, 3. Leam how to use spreadsheets (Microsoft Excel) and some of its basicfunctions.

lntroduction

Theory

The most convenient way of presenting a dataset is through graphicalpresentation. A graph'is defined as the pictorial representation of a set of datawhich could be'in 2 or 3 dimensions. trt allows the experimenter to understand therelationship between 2 or'more parameters

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 twodimensional scatter graph which is most commonly used,as a way to present therelationship between two variables.


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 of the string. \ f-uur ur rrrs surrrs' 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 usuallyincluded, this part can be omitted due to redundancy i

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 ofdata concerned be shown. Oftentimes, the maximum and the minimumscale should also be adjusted to give the best display (the numbers are wellspaced and readable).

c) Labels - 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

28 @ 2007:Lab Mdnual Authors

Physics 71.1 Graphical Analysis

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

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 agraph by clearly pointing:out salient parts in the presentation (e.g. skewedpoints, alignment of points, trends, similarities). Important parameters notin any of the axes should be mentioned and described in this part. It is achallenge for the presentor to make captions as short as possible. Captionsmay include titles which may prove useful for quick glances.

e)

Graphing Procedures

Error Bars

Variables are commonly plotted in a rectangular coordinate system. Thedependent 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 ycoordinates, 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 tounderstand. With axis scales that are too small, the points will bunch together,making the plot incomprehensible.

Collection of data involves measurement; hence, this implies that uncertainties are

present. In plotting a set of data which includes the expectation value and itscorresponding uncertainties, the expectation value is plotted and the

corresponding uncertainty is presented as an error bar. Error bars show thepossible range of values of one or more variables in a data point. This is usefulsince it allows the experimenter to know the range of possible values under the

@ 2A07 Lab Manual Authors 29

Graphical Amlysis

influence of a certain variable.

Trendlines and linear fit

Physics 71.1

(3)

Hence, Equation (3)

Data points in an x-y scatter plot should not be individually connected by lines. Inthe event that the experimenter is certain about the relationship of the variablesbeing presented, a smooth line or curve called the best fit line can be drawn torepresent the known relationship. The word'osmooth" does not imply that the lineor 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 maincriterion for.this method is to minimize the distances of all data points from theline drawn.

Once linearized, the variables'can be represented:using the equation

y=mx'+b

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

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

AxAy

Any set of intervatr may be used to determine the:slope of a linear plot. But forbest results , points, showld be chosen within the best fit line. lf the data point is notincluded 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. Forexample, the equation

(l)

(2)

y- gx'+b

may be reduced to a linear equation if we let x =xreduces to the form

.r. y,=.gx'*b

which is just equivalent to Equation l.

30 @ 2007 Lab ManualAuthors

l

J,.

Physics 71.'1 Graphical Analysis

Graphing using a spreadsheetThe 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 thesample data from your worksheet.

2. Highlight these two columns and click "chart wtzard" icon on yourtoolbars.

3. Choose x, y scatter on your Chart type and click next.

4. 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. Youmay also click on the tabs to modifii the axes, gridlines, legend and datalabels. 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"

-lo - E / !t,E#gfig$iry#%i#, a : fl,al :ffijjom - r{S gr tdg ;1$ i AF EF, g: t$r - A -ffi

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

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

iI

6.

7.



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.

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'.

0650ffi,1_Dt'

!,n.15

8.

9.

11.57

i71q.si1.11

1 :lE

'.r -,i.*.rd.&-=;=d wt"

32 A 2007 Lab Manual Authors

Xama . Date$ubmittod

Dat6Perioamed

Scorc

Group tlemberc

lnstructor Section

Worksheet: Graphical Analysis

A. PreSenting data'set graphicallyDuring 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 Ploty as a function of x, 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 fromthere.

o Ploty as a'function of x2 , . Canyou conclude with certainty that the plot is linear?

@ 2047' Lab Manual Atrthors 33

Graphical"Analysis Physics 71.1

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

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 fromthere.

II

I

i

I

I

iI

I

I

I

I

I

I

t

N

i

:

IL

From your answers in items 1-3, determine the degree (in x) of the equation relatingy andx. (Recall that the equation y=ax2 *bx-lc has a degree of 2 in 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 findthe 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 toand at a distance d from the center-of-mass axis has a moment of inertia I pabout that axis written as

I ,= I "^*Md2

where I "^ 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

34 @ 20Ol Lab Manual Authors


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:

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

(1,)(s'cm') (d)(cm)

(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 2007 Lab Manual Authors 35

Gruphical Analysis Physics 71.1

B. Plot I ovs. d2 and paste it' on the space below. Calculate the best estimate of the mass of the

mysterious heirloom.

Solution:

36

Final Answer: M:

@ 2007' Lab Manual Authors


2. Off to the moon!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

I 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 initialheight h. He used a timer with 0-001 s precision (least count) and a meterstickwith a least count of I mm. His estimated fraction for the meterstick is 0.5 mm.Heobtained the following data below. However, he has no Physics 71.1 trainingwhen 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

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

@ 2OO7 Lab.Manual Authors 37


B. If / and hare related by n=)S t' , obtain the best estimate for gmoon.

Solution:

38

Final Answer: I *oon :


Vectors a'nd Force Table

ObjectivesAt the end of this activity you shoutrd be able to:

L Show that the sum of forces acting on a system inzero.

equilibrium is

2. Obtain the equilibrant of two or more forces using the concept ofequilibrant.

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.

A vector 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 vectoris represented by an arrow with three-dimensional direction.

Oftentimes it is difficult to imagine the graphical representation of vectors makinggraphical approach impractical and analytic representation comes handy.

Analytically vectors can be decomposed into its orthogonal (graphicallyperpendicular; physically' independent) components. Since vectors are

mathematical entities, they follow certain rules of combinations. The simplest

static

@ 2AO7 Lab Manual Authors 39

Vectors and Force Table Physics 71.1

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

ileil<l7tt+ltEll (2)

Geometrically, there are two ways vector addition is viewed: head-to-tail method

and parallelogram method, each consistent to the other. The magnitudes can also

be , obtained by measuring the lengths and scaling or by calculations using

trigonometry. Furthermore, thp magnitudes can also be obtained from the values

of the vectors'known comPonents:

Hoad-to-tail method

Equation 1 can be analyzed graphically by forming a triangle with the sides as the

vectors as shown in Figure 1. The length of the sides corresponds to the

magnitude of the vectors. It should be obvious that the magnitude of C can

be less than the sum of the magnitudes of and with maximum equal to the sum

of the magnitudes of i and B : Adding two vectors does not necessarily

result to a vectof with larger magnitude than that of either term!

To draw vectors, their magnitudes should be converted to a length unit. For

example, a force vector with magnitude 500N can be scaled fu t.-* With this

scaling, we see that a lcm vector has actual magnitude of 100N and so on. Thus

40 @ 2007 Lab Manual Authors

Physies 71.1 V*tors and F.orce Table

every time a physical vector is drawn either a scale (say,"lcm 's"ro 100N'?) isindicated or the vector is labEled with its corresponding magnitude. Taking thesum of two vectors would then involve drawing them as in Figure I andmeasuring.the length of and scaling it back to the actual magnitude value.

Mathematically, Equation 1 can be rearralged. tq become a subtraction:B=t-i , just as Figure I can be rearranged into a similar figure shown in

Figure 2 via the concept of translation . It.,should be obvious that the sum of avector and its negative is zero (null vector, 0 ) with the negative of a vectorrepresented by the same vector but pointing towards the opposite direction.

Figure 1. Head-to-tail method of how two vectors Z and B add up to1aC : C = A+ B . The same figure represents the difference of two vectors:

b =t -2 . The vectors 2 , b , and e represent the same rype

of physical quantities. Note that the "head" ofb

A is placed onto the "tail" of

tu "-*l

Figure 2. Head-to-tail method addition of the negative of a vector, - 2 , to a

vector e is considered the subtraction process that yields b . Again, the

vectors must represent the same type of physical quantities.

4

A

@ 2007 Lab Manual AUthors 41

Vectors and ForceTablePhysics 71.1

,4

Figure 3. Parallelogram formed by the vectors

can be obtained using trigonometric concepts'+

and B

and B .Themagnitude

is the angle between 72e

The same relation can be obtained trigonometrically from Figure

Equation 2. Note that the angle O between e and icosine law:

c2+ A2-E i

cosO=-- ,;a-

1 consistent withis related by the

Parallelogram method

Figure 1 can atso be viewe{lvi.o.iTrr,*: :ti,t1^::"-1. ;Jl, ,f. ,.Til:t;parallelogram as shown in Figure 3' In the same mannel

method, the magnitude of e can also be obtained by measuring the length

and converting it into the actual magnitude. Trigonometrically, with the angles

O and e as ,shown, the magnitudes of the vectors obey the relation

(obtained from the cosihe law ):

cz = Az + 82 + 2ABcos o (3)

Resultant and equilibrant

A set of forces (or other physical quantities that can be represented by vectors)

can be replaced by a single force - the net force, which renders the same effect.

In terms of vectors, a set of vectors can be replaced by one vector called the

resultant. The resultant is merely th9 sum of all the vectors it will replace. In the

.*u*pi.'uUor., d is the resultant of the vectors 7 and b The

concept of resultant is commonly used in engineering where a set of forces acting

on an object can be replaced by a resultant without changing the overall effect.

Lr*.

42@ 2A07 Lab Manual Authors

Physics 71.1 Vectors and Force Table

The action of a set of forces (again, force is just an example) can be

countered/nullified. Indeed, a particular single force introduced into this system

can produce a zerolntll overall effect. In terms of vectors, this particular vector isa vector that will cancel the resultant of the set and is called equilibrant. The sum

of the resultant and equilibrant is therefore zerolnull vector making equilibrantand resultant a negative of each other.Symbolically ,

E=-h. (s)

with E as the equilibrant and fr as the resultant. Figure 4 shows the

graphical relation of E and fr

' Figure 4. Graphical representation of the equilibrant E . Note that the dashed

vector (tfanslated ) placed beside fr shows that it can cancel fr and thus

fr 's effect.

Unit vectors

Vectors with unit length or magnitude are called unit vectors. Unit vectors are

used to indicate direction and are represented by symbols with a hat, e.g. 2Thus the direction of 2 is along 2 The unit vector can be obtained byrescaling a vector into one unit length/magnitude. This involves multiplying a

vector with a (unitless, dimensionless) scalar, say s, changing only its magnitude

not its direction . Thus to get the unit vector of i , we scale it with a scalar

equal to the reciprocal of its magnitude:

r.)

Ft

2=tltz (6)

Orthogonal vector components

Vice versa to the problem of finding the resultant, a set of vectors can be sought

so that the given vector will be their resultant. The vectors belonging to this set is

called the components of the vector. This is the same as asking what forces

should be combined to yield an effect equal to the single given force. Additional


Vee&rs atld FordeTable Physics 71,1

Condition however; isiimposed for these contponents: they should be orthogonal

to eaohiother. ,This physically, moalls'tfiat thebe compondrts have to be directed

along a fixed set,of directions.

FE

Figure 5. The component F " of F along the direction A

The unit vectors corresponding to orthogonal vectors are called orthonormal

vectors or basis vectors. Finding the components of a vector along a given a set

of unit direetions (frame of referehce) inv-olves'finding the oomponent of a vector

along a gi$en direction. See Figure 5 for'an example' The direction of F "

extends along the direction of o and ends at the point where a perpendicular

line dropped from the "head" of crosses the direction of F ' If the angle

between O and F' is e , then the magnitude of F " is'grVe,lr by.l i ,'

! "=l c9se (7)

Thus, if 'we take 1 and ' i' as the urtit airecti<m1 , then' F can be

decOmpos6$ to the respective comportents' F, and F y as shown in Figure

6. Furthermore, the magnitudes of the iomponents are given by

iil{i,,ft'I#rI-

and

F ,: Fcos,0

F r=Fsine

(8)

(e)

M @ 2007 Lab Manual Authors


(10)

(12)

lih.#*

rnr- rr

Figure 6. The components of F along F,between F and, i is e

Fy .Theangle

Using trigonometry, we immediately See in Figure 6 that the magnitude of Fis related to the magnitude of its components by the Pythagorean relation:

p+= fi+ F2,

The vector F canalso be written as

F=F,i+rri (ll)The angle between F and x-axis, e is related to the components by theequation:

tane=L-F,

Equation 10 can also be obtained from Equation 3 by replacing the angle betweenthe components with gO' A vector with known components can now be

normalized by scaling it with the reciprocal of its magnitude derived fromEquation 10.

The axis directions are arbitrarily chosen and each chosen set of axes results to adifferent set of component vectors . These components however, still add up tothe same vector (the resultant of the components). This is advantageous in cases

when the axes have to be oriented so that most components of the vectors liealong one direction only, making trigonometric analysis (as well as othermathematical arguments) straightforward.

@ 2AOV Lab Manual Authors 45

Vectors and.Fsrce Table Physics 71:1

Adding vectors using its orthogonal components becomes straightforward. Each

of the components of the sum of two vectors, say i and h , are simply the

sum of the corresponding components of the vectors . Symbolically,

C,=A*+8,

C ,--'Ar+ B,

As an example, consftler i=5i+6i and h=i-7 ystraightforward to see that their sum is t=i+B=6i-i

Referenceo D.Halliday, R. Rgsnick, and J. Walker, Fundament4ls of Physics 6th Ed.

(John Wiley & Sons, !nc: Singapore,200l). i

MaterialsForce table and the accompanying weights and a ring, level, digital balance (for

the total mass), graphing papers, rulei, protractor, pencil, calculator or equivalent.

ProcedureThe experiment utilizes a force table to examine the effect of forces acting on a

ring. The forces are supplied by hanger with weights pulling towards directions

controlled by the position of thg pulleys as indicatedby q large 350' protractor

printed on the foroe table. The pulleys are much lighter,than.the loads and can be

assrrmed to have insignificant effects compaled to forces. The magnitude of the

force applied to the string (and therefore to the ring) is equallo the weight of the

hanging mass (the container included). Since the weight is the product of the

corresponding total mass M and the acceleration due to gravity g (considered

constant all around the experirnental proa) M,may be, considered to be the force

magnitude. To reeover the, actual force strength, we just have to multiply it with

and

(13)

(14)

Then it should be

I

46 @ 2007 LabManual Authors


Figure 7. The force table and its accessories. Shown are the weights (W) with

hooked hanger, pulley (P) and its locking screw (L), the ring (R), string (S), and the

balancing screws (B). The force direction is read from the angular scale (C) marked

along the perimeterof the table like a big 360-degree protractor.

A complete setup of the force table is shown in Figure 7. The ring serves as the

object at which the forces act together. The sum of these forces becomes the net

force acting on the ring. Once the net effect to the ring is null, it is expected to

stay on the center. The aim of adjusting the masses and their directions is to place

the ring at the center indicative that the effects of the forces (provided by the

strings) on it have been canceled out.

The hooked hangers afid a set of masses are shown in Figure 8. These may be

replaced by other unconventional weights like water bottles, sand and cups. The

actual weights just have to be weighed using a (digital) balance.

Figure 8. Hooked hangers (H) and a set of masses (M). These may be replaced by

other weights like water in bags or sand in cups, etc.

@ 2OO7 Lab Manual Authors 47


A. Setting up the force table

To avoid systematic errors introduced by the weight of the ring, the entire set-up

must be leveled. A level is a device that uses water (or other liquid) to indicate

leveled surface. A bubble resting in the center of the markings indicate level

surface only along the direction of the level, so it will be advantageous to take two

level readings - the second reading perpendicular to the first. The force table can

be balanced by adjusting its three balancing screws. There are other things to be

kept in mind to avoid erroneous readings. The strings have to virtually pass the

center ofthe force table so that all forces (vectors) intersect through one point at

the center of the table. The pulleys have to be made sure to rotate freely about

their axles so that they offer insignificant added tension to the strings. The

orientation of the pulleys should also be aligned with its strings' direction of pull.

B. Reading the angular position

The angular position is read from the mark on the force table (C in Figure 7) that

is aligned with the string. Make sure that you read directly above the string,

pelpendicular to the table otherwise, parallax error will be committed (see Figure

e).

Consider 0o as the positive x-axis direction and 90o as the positive y-axis

direction. All calculations for the angle should be measured or determined relative

to the 0' mark.

Three basic cases will be studied, each case trying to show that the vector

representation of forces is valid.

All experimental value should have uncertainties given by the experimenter.

C. (Case l) Resultant and equilibrant

r

I

iI

l

t

x

I

I

t

Assume that there is already a given resultant force with magnitude of about 2009

directed towards the 2lO' direction. Locate the equilibrant (the force that will

nulliff the effect of this resultant) for this force.

@ ,tttu"f, masses (total of about 2009 including its hanger) on one string, tie

it to the ring, and pull it over a pulley. The pin placed in the center of the



force table should pass trough the ring holding it in place while there is anonzero net force.

@/eaiust the position of the pulley by loosening its lock and sliding it alongthe circumference of the table until the string aligns with the 2lo" ;mark. once in place, lock the pulley again. This pull serves.as the given ':iresultant.3) i i ,.,,.,i"""i',

@ril-i.rthe entries for F, inTable I oftheActivitySheet. I ";'t ;

@) r"another string on the ring, pass it oytranother pulley, and then o';t ,*i" i.

matching combination of mass and,ldnger to be attached to the end of tliisstringtocounterorba1ancethepullofthefirststring-f,.'.,:Thepositionof this pull may also have to be adjnsted by movin!-ihi: pulley lik.'th." ' '"Jfirst. This pull has completely countered F t if the ring is at the center

of the table (indicated as the pin passes though center of not touching, thering as shown in Figure 10). The total mass and,position of this pullcorresponds to the magnitude and direction of the equilibrant of the givenresultant.

@ or.. the ring is at the center, record the experimentally obtainedv equilibrant as F, in Table I of the Activity Sheet.

6. Record the expected (theoretical) magnitude and direction of theequilibrant in Table II as E along Case I row.

Determine the angle 0o, between the predicted and experimentalresults of the equilibrant and fill in the corresponding column (case I).This angle is simply the absolute of the difference between the angles0t and e2

Complete Case I row by computing for the percent deviation A F (%) .

The percent deviation is computed using the following formula:

7.

8.

AF=@and

AFAF (%): ;t7

This formula is derived from the difference:

(l s)


AF=E-Ft

49

F,


D. (Gase 1l) Equilibrant of two forees

Two known forces are given: one pull, F, , clirected towards the

100' direction and the other, F, , towardd 2oO;';. The

equilibrant; assigned as F, ', willbe sought. Clear the fotie tablb of the\previous pulls and ieplace them with new pulls as follows.

than 2009. in the directions

in CaseJ in setting up th$eand F2 in Jaflsl.t.;I

6 ) 0,u.. two forces with net masses greater

l0cf and 200' Follow the directions

pulls. Fill in the column for F,correspondingly.

td, r * rer ) 2a9o ;H:fn",:,?_TJ,T;,ff1;11',"f,,};;X :;::J* #Itr*r,, :(given) resultant in the previous case: Find the equilibranti;ofkhese pul'lq,

just as what is done in Case I.

6) On.. the ring is placed at the center, record the obtained magnitude,a!rd\/ direction of the e[uilibrant as F, in the Activity Sheet's Table I.

4. Graphically determine the theoretically predicted restrltant in the space

provided in Question la of the Activity Sheet. ..,Label the vectors

correspondingly as well as the angle eR it makes with the x-axis.

Indicate the scale used. A scale of 40g to 1cm is rdcommended.

5. Based on the results of the resultant, recordlhe expected magnitude and

direction of the equilibrant in Table II as F r along the row of Case II

(graphical). The equitribrant can be determined using the concept of Case I

(Equation 5).

6. Compute for 0 oo as in Case I and complete the row for Case II on

, Graphical method (also filI in its A F (%) oolumn, use Equation 15)..

7 . Label the angle between F, and F 2 as in the drawing in Question

la. Use Equation 3 to compute for the magnitude of the resultant.Utilize

the space provided in Question lb for the computation from rt to Eusing trigonometry. Fill in the row for Case II (Trigonometric).

8. Determine the angular position e E of the solved b (equal to

180'-0, ) using Equation 4. Complete the trigonometric method row

by computing for 0 o, ahd A F (d/o). :

9. For the prediction using component method, compute for the component

of the vectors in Table III along the x-axis and the y-axis. The components

of F are just the sum of the corresponding cornponents of F 1 and

50 @ 2A07 Lab Manual Authors

Physics 71.1

@g

6vFr

Veetors and Force Table

F, . Use Equation 5 to get the components of h

10. Using Equations 10 and 12, calculate the magnitude and angular positionof and complete Table II. Use the space below Table III for calculations.

IV.

E. (Gase lll) Orthogonal components of a forceThe equilibrant of a given force F , can be decomposed into two other forces

that are perpendicular (orthogonal forces) to each other. The magnitude and

direction of the equilibrant of a single force is already established from Case I so

now, the case deals with decomposing an unknown equilibrant into twoorthogonal components by showing that a given resultant can be countered by twoorthogonal pulls. Again, clear the force table of the previous pulls and replace

them with new pulls as follows.

1. Similar to Case I, place a pull with magnitude about 2009, directedtowards 200'

2. Record the magnitude and direction of this pull as F, in Table I.

3. Now place two other pulls directed towards 0' and 90' so that they

are perpendicular to each other.

4. Adjust the magnitudes of these two forces such that they counter the pullof the first. Note that these two forces should cancel their componentsalong the direction perpendicular to the given first pull.

5. Once the ring is placedron the center, record the magnitudes and directionsof the two pulls as F 2 and F 3 in Table I. Note that these vectors

are actually the experimentally determined components of h=-Falong the i and i directions, E, and E y respectively.

( 0. yCgmnute for the theoretically predicted magnitude of the components of\-/ R=F, using Equations 8 and 9. Record R, and Ry in Table

G, "ur"d

on Equation 5, determineV rubl" IV.

E * and E y Record your results in

(QCo*plete the table by calculating the percent deviation of the predictedV value from the experimentalvalue.

@ 2007 .Lab Manual Authors 51

Vec{ors a nd Farce, Tdhle Physics 71.1

'' ,.t'' a- .---"'

&,*

52 @2O07 Lab Manual Authors

Nams DatoSubmitted

DatePedomed

Scorc

Group llemb€E

lnalluclor Sectlon

WorkSheet: Vectors and Force Table

Data Summary

Data Table I. Experimentally obtained equilibrants (Cases I ond II) and orthogonalcomponents (Case III).

Uncgrtainty in magnitude : Uncertainty in position:

Comparison of theoretical predictions and the experimentally obtained,vectors, .1

Data Table 2. Theoretically predicted equilibrant in comparison to the experimental result.

Case

Fl F, F3

Magnitude(g)

Positione1

Magnitude(s)

Position02

Magnitude(g)

Positione3

I 210. N/A N/AII 100" 240',

m 200o' 00 900

case methodi

oo, * aF (o/o)Magnftude,E

Direction,eE

I expected

II

graphical

trigonome'tric

component

* 0 o, is the angle between the theoretically predicted and the experimentalty obnined equitibraiil

@ 2OO7 Lab,Manual Authors 53

vectors' aid,Folee rloit Physics 7'1,1

Data Analysis*

) When can we say that the predicted

: ' '

tt i !

E is close enough to its experimentally determinedi.:.' ,"' '

:,i

i

,,!i

A. Graphl66l p6ffigsJ.:. 'ri

. i..3. Show youlcosputatrons necessary for determining the theoretical prediction ofequilibranf E'

- ln tho spacls provided below usinglgraphical, trigonometric and

component method ; Indicate if-a separate sheetis attached' -

Oo- :'j

2700

5/+:

scale: _ cm: _g

@,20O7 Lab.Mdriural Authtks''

u

Physics 71.1 Veclotsand Force TaNe

B. Trigonometric methodComputation of fr ( is just equal to,this):

Computation of e R (relative to the Oaposition, can be sho$m'graphically):'

Computation of 0 E (still relative to the 0" position):

C. Gomponent method

Data Table 3; Coi,mponents of theforces invblved in obiaining the components of

magnitude s (component method, able 3):

E

vector x-component (g) y-component (g)

F,

F2

n

ECalculations for the masnitude E and the ansle e I)afe T

@ 2007lab Manual Authors

Veator componen{ Prdtctton (g) ',t, oawas*6h.,W).

R,,3F1* N/A N/A

'R'-F"' N/A N/A

Er=- R,

E ,=- R,

:,lffiorsafr&Fo*pe.:GWeI F.hlsbs:7tll

Dare Tshle.A Cowparisons of the components of F, alo@rlrall+? t arl:d ,9*drped&ais.':i

,l l;. tl

i

2: Summarize your"eonclusiofls iii llne-with'the obje0tive -clf the Sctivify.-l '.j ..:

-..- .-:',,*.., .-, .. , -.

;I ,'illl: !- i:i.r'.i ..r,r;-ri:';.i:i ;!:J,i{..:{iLir}.;} i: ;;1'1;9i1i 1"; ' i'r l ' '} :'ti'':rt;ill1g r; :iii:

s56,@ffiS7,E&,.fit'anu&liAilth6rs

U niformly Accelerated Li nea rMotion (Ball)


1. Determine experimentally the niagnitude of acceleration of an objectundergoing un i form ly accel erated I inear m oti on.

2. Plot experimentally the graphs illustrating the position and velocity as afunction of time for an object undergoing uniformly accelerated linearmotion

lntroduction ,

An object moving in one dimension with a constant acceleration is said to be

undergoing uniformly accelerated linear motion. One example of type of motionis an object which is dropped from an initial height h which is allowed to fallfreely to the ground. At all times, its acceleration is constant (with a magnitude of

9.8mls2 ) and is directed downward. Based on this knowledge, we will observe

a freely falling object and, with the help of computer interfaces, determine thegraphs that illustrate the position, velocity and acceleration of this object as

functions of time. It turns out that the graph of the position is quadratic and for thevelocity, it is a straight line with a negative slope, having a value very close to thepredicted value of the acceleration due to granity g which is 9.8mlsz

TheoryConsider an object undergoing free fall. This object may either be dropped from a

57@ 2047 Lab Manual Authors

Uniformly Accelerated Linear Motion (Ball) Physics 71.1

height above the ground or tossed into the air and allowed to fall back down

again. Assuming that the object is moving in a uniform gravitational field and that

there are no other forces present, the only force acting on the object is the

gravitational,forc,e, whigh impartS.,an acceleration gfmaSmtude 9.8ru1s' to the

objebt. Th'b magnitude of this acoeleration is cbnsttint'''and'' is always directed

downward; hence, since the object is traveling in one dimension only, it is a

perfect exarrtpfe' pf uniformly accelerated linear motion.

Since the magnitude and direction of the acceleration of the object (hereby

represented as g) is constant, from the definition of acceleration (which is the first

derivative with respect to time of the object's speed and the second derivative

with respect to time of the object's position), we find that:

a) the object's speed is expected to be a linear function of time (speed is

directly proportional to the time), and

b) the object's position is expected to be a quadratic function of 'time

(position is directly proportional to the square of time).

Specifically, the equations describing the object's velocity v(t) and position y(t)

with respect to time are

v(t)=vo- 8t1y(t)=y"-v,rt-)Bt2

where vo is the object's initial velocity and !o is the object's initial position.

Note that the acceleration here is denoted by g. Hence, from the form of equations

(1) and (2), the graph of the velocity is a straight line slanting downward with a

slope of g : 9.8mls2 and the graph of the position is a parabola opening

downward.

MaterialsVernier LabPro@ computer interface, motion detector, large round ball

(basketball, soccer ball, volleybalt)

Procedure1. Connect the Vernier LabPfo@ interface to the computer. Follow the

instructions in Appendix A illustrating how to carry out this procedure.

Connect the Vemier Motion Detector to DIG/SONIC 2 of the LabPro or

PORT 2 of the Universal Lab Interface. Place the Motion Detector on the

(1)

(2)

il

2.

I58 @ 2007 Lab Manual Authors

Physics 71.1 Uniformly Accelerated, Linear Motion (Ball)

table.

Open the file in the Experiment 6 folder of Physics with Computers. Threegraphs will be displayed: distance vs. time, velocity vs. time, and

acceleration vs. time.

Toss the ball straight upward above the Motion Detector and let it fallback toward the Motion Detector. This step may require some practice.

Hold the ball directly above and about 0.5 m from the Motion Detector.

Click "collect" to begin data collection. You will notice a clicking sound

from the Motion Detector. Wait one second and then toss the ball straight

upward. Be sure to move your hands out of the way after you release it. Atoss of 0.5 to 1.0 m above the Motion Detector works well. This can be

achieved by tossing the ball in such away that it will reach the tip of yournose. You will get best results if you catch and hold the ball when it isabout 0.5 m above the Motion Detector.

Examine the distance vs. time graph. Repeat Step 4 if your distance vs.

time graph does not show an area of smoothly changing distance. Check

with your teacher if you are not sure whether you need to repeat the data

collection

NOTE: The COM, USB and other cables are indicated on the box containing the

Vernier LabPro@ interface.

l. Connect the COM or USB cable to the COM or USB ports on theLabPro@ interface and the computer.

2. Connect one end of the power supply to the corresponding outlet on the

LabPro@ interface. The other (socket) end will be connected to the plug ofthe power supply.

3. If the LabPro@ device is connected properly, after a few seconds a tone

will be heard and the lights in front of the device will blink.

4. Double click the Vernier LabPro@ icon on the desktop. If the device isproperly connected, you should see the WELCOME screen immediately.If the SCAN screen is seen, click the SCA.N button to giVe the computer

aJ.

4.

5.

APPENDIX A. CONNECTING THE VERNIER LABPRO@INTERFACE TO THE COMPUTER


APPENDIX.B. TROUBLESHOOTING GUIDE FOR THELABPRO@ INTERFACE

Unifomrly Accelerated. Linear Motion (Ball) Physics 71.1

more time to connect. If the connection still fails, consult the

troubleshooting guide.

PROBLEM:, The SCAN, not the YruLCOME, screen is seen after double

clicking the LabPro@ icon

'C1ick'the SCAN button. If, after a few minutes, the WELCOME screen is

seen,' proceed with the'experiment.

If after clicking the SCAN button, the SCAN screen is still seen, close the

LabPro@ window and remove the COMruSB cable attached to the

computer. Reconnect the cable to another COMruSB port on the computer

and repeat steps 1- 4 in Appendix A.

3. If, after the previous step, the SCAN screen is still seen, replace the

COMruSB cable with another COMruSB cable, and repeat steps 1 4 in

Appendix A.

I.

2.

4; If ,after thg,,previous step,.the SCAN, sprgen is,still seen,.replace

COMruSB'cablb with-a USB/COM bable, and repeat steps 1 4Appendix A.

the

in

5. If, after the previous step, the SCAN screen is still seen, check the

- connection of the power supply. If it is not connected properly, reconnect

it and repeat steps 1 4 in Appendix A.

6. If, after the previous step, the SCAN screen is still seen, the CPU may

have a problem interfacing with the unit. Replace the CPU or, if the CPUs

are not enough for the class, merge with another group whose unit is

functioning properly.

PROBLEM: No tone is heard andlor the lights on the LabPro@ unit do not lightup after beiong connected to the power supply

1. Re check the connection of the power supply with the unit. If the

connection is faul.ty, recoqnect it aqd repeat step 2 in Appendix A.

2. If, after'the previous step, no tone is heard and/or the lights do not blink,

60 @ 2007 Lab Mdnual Authors

Physics 71.1 Uniforqly Accelerated Linear Mation (Batt)

the power supply may be defective. Replace it with a new one or a fullyfunctional one, and repeat step 2 in Appendix A.

If after the previous step, no tone is heard and/or the lights do not blink,electrical outlet may be defective. Move to another table with functionalelectrical outlets, and repeat step 2 in Appendix A.

If after the previous step, no tone is heard and/or the lights do not blink,the CPU or the unit itself is defective. Replace it with a new one or a fullyfunctional one.

PROBLEM: No data is being collected by the motion sensor/photogate after the

COLLECT button is clicked.

1. The device may be connected to the wrong port on the LabPro@ unit. Re

check the port where it is supposed to be connected, and reconnect the

device.

2. If, after the previous step, the problem still persists, close the LabPro@

window then, after a few minutes, double click the LabPro@ icon on the

computer and repeat the experiment.

3. If, after the previous step, the problem still persists, the device may be

faulty. Replace it with a new or fully functional one.

PROBLEM: The graph produced by the motion sensor/photogate is truncated

1. Check the calibration of the motion sensor/photogate and the settings on

the graph. Adjust them in such a way as to produce an untruncated graph.

If, after the previous step, the problem still persists, the motion sensor orthe CPU is faulty. Either replace the CPU or refer to the previous section

and carry out steps I 3 there.

aJ.

4.

2.

61@ 2007 Lab Manural Authors

Lhtitutnly'Ais,el€rated l[,i'nwi *5666,,6Bla t0 Physios'7f .1

@2OA7 Lab Mdnual Authors

Name DateSubmitted

DatePerfomad

Scotr

Grcup MembeB

ln3tructor S6ction

Worksheet: Uniformly Accelerated LinearMotion (Ball)

For this portion, sketch or attach the printouts of the graphs produced by the interface on thespilces provided and answer the questions listed below each graph.

Graph 1. Distance vs. time

On the graph above, identiff and mark the region where the ball is being tossed but stillin your hands.On ttre graph above, identiff and mark the region where the ball is in free fall.From the graph, what is the maximum height that the ball reaches?

a

a


,,, IrdLlairr.tu*'-.,^.u . *^i!di

lJniformly Accelerated Linear Mofron (Ball) Physics 71.1

. Click and drag the mouse across the portion of the distance vs. time grapn'that is -

parabolic, highlighting the.free-fall portion. Click the Curve Fit button, select Quadraticfit f.o* the list of models and click utry frt". Examine the fit of the curve to your data

and click "ok" to return to the main graph. Now consider the value of your "a" tetrn on

the. graph (as co_mputed by the interface). Compute for the percent difference with respect

Percent Difference:

Graph 2. Velocity vs. time

mark the region where the ball is being tossed but still

in your hands.

o On the graph above, identifu and mark the region where the ball is in free fall.

. From the graph, what is the maximum velocity that the ball reaches?

@ 2007 LabrManual Authors

Physics 71.1 Uniformly Accelerated Linear Motion (Ball)

. From the graph, what is the velocity of the ball at the highest point of its motion?

o Click and drag the mouse across the free-fall region of the motion. Click the Regressionbutton. Now consider the value of the slope (as computed by the interface). Compute forthe percent difference of the slope with respect to the theoretical value ofg, which is 9.8

m/s2.

Percent Difference:

Graph i. Accelerationvs. time

. Is your graph for the acceleration as a function of time perfectly straight? If not, whatcould be the reasons why it is not perfectly straight?

@ 2007 Lab ManualAuthors 65

Ua ffid Phyeice.71,1

o Click and*ry6o:BQ*rQ9rasress fre ftee:fall.sporion o[thernption,qs,clic{c rhe Statbticsbutton. How olossly does the mean acceleration value compare to the values ofg found inthe previous,steps?

I

ll

1ll

- ,: -_ iitl

.r *i..

j

i

!

I

i1.

.,1

ri

:

I

{.

.i;ii , t .

,1it ., .,. i,1;,, , i... ii.

U n iform ly Accelerated Li nea rMotion (Picket fence)


1. Determine experimentally the magnitude of acceleration of an objectundergoing uniformly accelerated linear motion.

2. Plot experimentally the graphs illustrating the position and velocity as afunction of time for an object undergoing uniformly accelerated linearmotion. motion

Introduction

This experiment extends the previous experiment (Unifomly Accelerated LinearMotion: Ball) by considering a picket fence released fiom rest. The same theoryapplies and the main difference is found in the procedure. The student is alsoadvised to consult the appendices of the previous experiment which detail how toconnect the sensors and how to troubleshoot the setup.

In this experiment, the ball is replaced by a plastic bar (the picket fence) withequally-spaced black-painted strips. The motion detector is replaced by a

Photogate which is an infrared light source coupled to a detector. When infraredlight to the detector is blocked, the detector records the time. In this manner,dropping the'plastic picket fence through the photogate allows us to measure thetime between dark bands as the picket fence accelerates,


l! niformly Accelerated Li near Motion (Picket fence)

Materials

Proced0ie

vernier LabPro@ computer interface, photogate, picket fence

1.,.

1

Attach the vernicr LabPro@ interface. .to . tht computer. Follow the

instructions in the APPendtx.

fu.t", the Photogate rigidly to a ring stand so the arms are extended

horizontally. The entire length of the Picket Fence must be able to fall

free'ly through the Photogate. To avoid damaging the Picket Fence, make

sure it has a soft surface (such as a carpet) to land on'

Connect the Photogate to the DIG/SONIC 1 input of the LabPro or the DG

Physics 71.1

a-r.

1 input on the ULI. . :

4. Open the file in the Experiment 5 folder of Physics with computers'Two

graphs will appear on the screen., The top graph displays distance vs' time,

and the lower graph, velocity vs. time'

5. Observe the reading in the status bar of Logger Pro at the bottom of the

screen. Block the Photogate with your hand; note that the Photogate is

shown as blbcked. Remove your hand and the display should change to

unblocked. This means that the photogate detector is ready.

6. Click "collect"to prepare the Photogate. Hold the top of the Picket Fence

and drop it through the Photogate 5 to 8 seconds after the "collect" butlon

is clicked, releasing it from your grasp completely before it enters the

Photogate. Be careful when releasing the Picket Fence. It must not lyltthe sides of the Photogate as it falls and it needs to remain vertical. click

"stop" to end data collection'

7. , ixamina your graphs. The slope of a velocity vs. time graph is a measure

of acceleratioo. ti the vplocity graph is approximately a straight line of

constant slope, the acceleration is constant. If the acceleration of your

Picket Fence appeafs constant, fit a straiglrt line to your data. To do this,

click on the velociry graph once to select it, then fit the line y : mx * b to

the data. Record the.slope in the data table'

8. To determine the shape of the distance vs time curve, click and drag the

mouse across the graph. Click the Curve Fit button, select Quadratic fit

from the 1ist of models. Examine the fit of the curve to your data and

return to the main graPh.

68@ 2OO7 Lab Manual Authors

Physics 71.1 Uniformly Accelerated Linear Motion (Picket fence)

If you are not satisfied with just one trial, you may repeat steps 5 and 6 as

many times as you want to obtain an average value of the slope. Do notuse drops in which the Picket Fence hits or misses the Photogate. Recordthe slope values in the data table.

9.

@ 20,07 Lab Manual Authors 69

.t

.iltl

I

Uniform ly Accdl6raled Llnea r M'iltio n;(Picket fe nce) Physics'71.1

@ 2007 Lbb Manual Authors,l

t

L

l{tme DreSubmftf.d

o#F.rfomld

Scdr

GEup ltembeE

lnstruc{or Sacdon

Worksheet: Uniformly Accelerated Linear

For this portion, sketch or attach the printouts of the graphs produced by the interface on thespaces provided and answer the questions listed below each graph.

Graph 1. Distance vs. time

o What is the shape of your distance vs. plot? Using the curve fitting tool, vnite out theequation describing your graph. What is the value of the acceleration due to gravity ?

Motion (Picket Fence)


lJniformly Accelerated Linear Mofrfli (Plcket fence) Physics 71.1

o What is the shape of your velocity vs. plot? Using the curve fitting tool, write out the

pquation describing your graph. What is the value of the acceleration due to gravity?

Obtain several measurementsof the acceleration due to gravity using this setup.

Determine the best estimate and use this as your experimental value. Calculate your

percent deviation using 9.81 m/szas your theoretical value, which is the accelaration due

to gravity at the Earth's surface.

72 @ 2A07 Lab Manual Authors

Kinematics of ProjectileMotion


Veriff that in a projectile motion, the horizontal and vertical motions are

independent with each other.

Determine the trajectory of a projectile motion.

1.

2.

lntroduction

Theory

The most common example of two dimensional motion is projectile motion.Consider for example a tall thrown at an angle less th*' 90' rror" irr.horizontal. Assuming that only the gravitational force significantly acts on the

ball, the trajectory or path observed is parabolic. Many bodies in motion exhibitsprojectile motion. Some examples are) a cannon shot,a ball thrown upwards and

an affow shot by a bow.

Projectile is the motion of an object that has an initial velocity vo movingundgr the influence of graviff. In the absence of air resistance, gravity is the onlyforce that acts on the object which acts only along the vertical rnotion. Since thereis an absence of horizontal force to affect the horizontal motion of the object, themagnitude of the component of velocity along the horizontal does not change.Hence the acceleration alongx ( ox )and,y ( an )are givenby equation l.


Knematics of Proiectile Motion

ar=0 and or=-B

Figure 1. An object in a projectile motion has an initial velocify v o

Physics 71.1

(1)

of initial velocity is given by the

(2)

. . ,. (,:)

(4)

The path followed by the motion of an object is called a trajectory. To derive this

mathematically, thg trajectory of a body in projectile motiorr, we consider an

$qjeet.rtrlthaainitidt veloc ; 'vr1;tnqwa;at an auglq '0. .iias shown in Figure

:i

tilllI

lu

The horizontal and vertical

expresslons

component

y= yorcos0

v=vorsin?

Since the acceleration of the object along the x-axis is zero, at every point in the

path of the objeg! the horizontal component of yelo-city i.s alyays equal to the

ro, . The horizontal'dis'tance x, traveled by the objeit is given by

While the vertical position, y canbe described by the equationl

I '.y=yo+v,sin0t-;gt'

I(s)

:

it willGenerally, the projectile may not be released at the same height at which

land. The initial height of felease'is expressedtas:' !o in expression 5.

By obtaining the expression of timb, / from equation 4 and substituting it to 5, we

derive the form

12!=lo* *tanT-!-L-

" v|"cos2 e(6)


Physics 71.1 Kinefiatics otf"Projectite Motion

The curve ex'hibited by equation 6 is an inverted parabola. In this experiment, we' will experimefltally.obtain the trajectory of a body under a projectile motion.

ReferenceTipler, Paul A., Physics for Scientists and Engineers, Fourth Edition, W.H.Freeman and Company, USA, 1999.

Materialsinclined plane, protractor, ruler, metal ramp, carbon paper, marble

, : : , ,, : ; :

Procedure .

A. Galculation of the initial velocity of the projectile.

)

Figure 2. The experimental setup.

2. Set the inclined plane to a specific angle. Use angles less than 20'

@ 2OO7 Lab Manual Authors 7i

K nematics of Prsjestile MOlisn

Drop dhe marble at the high end of the

B, the marble will,undergo projectile

marble land (point C).

Physics 71.1

metal ramp (at point A). After pointrnotion. Mark the range where the

,.J.

4.

Figure 3. The schematic diagram in obtaining theinitial velocity of the projectile

Choose four (4) angles and write ouf your ddta in Table 1.

Do not forget to obtain several trials for the measured range and present

your data in terms of the best estimate.

B. Determination of the trajectory of the projectilel. Set up the inclined plane,

4.

Carbonppu

(-r*-: ',: l]l

Figure 4. The schernatic diagram in obtaining thetrajectory of the projectile

metal ramp and ca-r,bon paper as shown in Figure


Physics,71.1 Kl n em atles of. Proj eati I e M oti o n

using the same angles in Table 1, set the horizontal distance x from thecarbon paper. Drop the marble at point A and, obtain the correspondingvertical position y. Vary the horizontal distance and obtain thecorresponding vertical position. You should have a minimum of five (5)data points to observe the trajectory of the marble.

Again, do not forget to obtain several trials for the measured verticalposition. Write out your best estimate of y in Table 2.

compute the theoretical value of vertical position y by substituting thevariable.r on expression (6) and write it out on Table 2.

Plot the vertical position as a function of the horizontal distance usingExcel and superimpose the theoretical y'on the graph. Make sure that thesize of your graph is not too small. Use one sheet of paper for every graph.

Compare the theoretical and experimental hajectory. What is the generalshape ofthe curves for each angle?

2.

3.

4.

5.

6.

@ 2007 Lab,Manual AUthors Tf

K n e ma ti c s of . P rol ec ti ! p,l/. I ati sn Physies 71.1

@ 2007 Lab Manual Authors'78

Name D.teSubmitted

DatePodomed

Scor?

Grcup MembeB

lnstructot Section

Worksheet: Kinematics of Projectile

Data Summary

A. Galculation of initial velocity of projectile

Data Table I. Range and initial velocity of a projectile at dffirent angles of release

Angte of release (d"S) Range (cm) lnitial velocity (cm/s)

. ril/hat do you observe about the projectile's range

Motion

@ 2007 Lab Manual Atdthors

Kinematics of Proj*tile Motion Physics 71.1

B. Projectilels trajectory

Dato Table 2. Horizontal and vertical components of the trajectory of a projectile.

Angle of release (deg) Angle of release (deg)

Initial height (cm) .:

Hortzontal'Illstancex (cm)

Horizontal Distancex (cm)

Vertical Distance Vertical Distance

Angle of release (deg)

lnitial height (cm)

! theo

Veftical Distance

Angle of release (deg)

lnitial height (cm)

! theo

Veftical Distance

lupr I *eo / theo

. On a separate sheet of paper, paste your graphs of the superimposed theoretical andexperimental plot of the trajectory of the marble. Use one sheet of paper for every plot.

@ 2OO7 Lab Manual Authors

Conseruation of Energy andMomentum


o Illustrate the law of conservation of energy and momentum using a

pendulum-proj ectile system.

lntroductionOne of the basic laws of Physics is the law of conservation of mechanical energy(COME). It states that f,or a physical system where the only internal forces actingon it are conservative, the total mechanical energy, i.e., the sum of the kinetic andpotential energies is constant. In the presence of external and dissipative forces,the law becomes more general. In this system, some mechanical energy may belost and transformed into another form of energy, ensuring that the total energy isconstant.

The law of conservation of momentum (COM) is important in situations wherewe have two or more interacting bodies. This conservation law is valid when thevector sum of all extemal forces acting on the system is zero. In these type ofsystems, the momentum before and after collision of two objects is constant.

The pendulum-projectile experiment allows us to'simultaneously veriff theenergy and momenfum conservation laws. The collision of the pendulum bob and


Conseruation of Energy and Momentum Physics 71.1

the marble can be analyzedusing the momentum conservation. On the other hand,

the dependence of the angle of release of the bob with the range of the marble can

be understood by means of energy conservation.

Consider the system in Figure l. Alength / is raised at some angle 0

pendulum bob ( m B ) attached to a string of

.+x +Figure 1. A pendulum-projectile setup

After releasing the;bob, it hits a marble ( m* ) which is initially at rest.

The velocity of the bob just before'it collides with the marble ( vrl ) is obtained

by'applying'COME from point Ato B. We find that the expression for the velocity

of the bob before colliding with the marble is given by:' i I i .

,ur=J2g( 1- coso) (1)

Upon collision of the bob with the marble and noting that the marble is initially at

rest and hence v*t : 0, we apply COM to obtain the equation

mBv Bt=-mov szlm*y*z (2)

where v az and 'v -z are the velocities of the bob and the marble just after the

collision. We may express equation (2) in terms of v*2 and vsr since for

elastic collision, the speed of approach is just equal in magnitude but opposite in

direction with the speed of recession. Hpnce, we may vsz is given by the

following expression

82 @' 2007 Lab:Mdnual Authors

Physics 71.1

Reference

V sy=V 12- | Bt

Substituting equation (3) to (2), we obtain an expression of inmm and vat :

Conseruatlon of Energy and Momerilum ,

, (3)

termsof mB ,

'"'(4)

After collision, the marble then undergoes projectile motion and will land at some

distance x from its initial position. By applying conservation of energy from pointB to point C, we obtain the expression

,, -2mou u,

Y m2- lTlst lll-

1p2 2 LLoth,i

V ^r=V^rl-- Zgnmru

The expression above maybe fttoie concretely shown in termsthe projectile by substitutiirg'expression (4) and (6):

(5)

where E o,0", ,7s the energy diqsipate&in the systern.,The velocity v*3 in the

right-hand side of the equation is the velocity of the marble as it hits the ground,

v ^3=x !tr (6)

By manipulating equatiori ($), wre obrtainp final form for v *2 :

(7)

ofx, the range of

2vnt

g(m r+ *-)' *' (mu+m^)2(E *o*-m^gh) (8)

8m2rh Zm'rm-

Therefore, from the law of COE and COM, we were able to derive an equation forthe velocity of the bob ( vat ) just before it hits the rnarble as a function of the

range of the projectile (r). A plot of ,'r, vs. *' will give us a slope of

s(rypl?^)' and a y-inrercept of ,?\*^Y (Eo,n",_m,gh)8m"uh zm Bmm

Tipler, Paul A., Physics for Scientists and Engineers, Fourth Edition, W.H.


Conseruation of Energy and Momentum

Freeman and Company, USA, 1999.

pendulum setup (bob, string, tripod

digital balance, carbon paper

Physics 71,.1

stand), protractor, ramp, meterstick, marble,

Materials

Procedure

Figure 2. The experimental setuP.

Measure the mass of the marble and the bob using the digital balance.

Setup the pendulum such that the length of the string is just right for the

bob to hit the r,narble at different set angles.

Attach a sheet of paper under a carbon paper where the marble will most

likely land.

Place the marble on top of the ramp and displace the pendulum bob at

some angle. Measure the eorresponding range (x) traveled by the'marble.

Carefully take note of the uncertainty in your measurements.

1.

2.

3.

4.

84 @ 2OA7 Lab Manual Authors

Physics 71.1 Conservation of Energy and Momenturn

Figure 3. To determine the angle of release, place a protractor with the 90-degree angle

aligned to the string. The angle is then measured from the vertical angle.

Repeat step 4 for five (5) different angles.

Plot 4, vs. * and answer the questions provided in your answer

sheet.


ff

Conservation of Energy and Momentum Physics 71 .1

I

I

I

r

l.

86@ 2OO7 Lab Manual Authors

Name DateSubmitted

DatePerfomed

Score

Group Members

lnstruct6r Section

Worksheet: Cohservation of Energy and

Data Summary

l. Measurement of Constants

Tabulate below the results you obtained from steps 4 and 5 of the experiment.

Table l. Range and velocity of the marble corresponding to the angles of release.

Table 2. Square of the range andvelocity of the marble corresponding to the angles of release.

Angle of Release ( 0 ) Square of the range of themarble ( *' )

Sguare of the velocity of thebob ( v2u, )

Momentum

Range of the Marble (x)Angle of Release ( 0 ) Velocity of the bob ( v r, 1

A 2OA7 Lab.Manual Authors 87

Conservation of Energy and Moryq1tytm Physies 71.1

Questionsl. On i graphing paper, plot a graph of v'r, vi. *' fiom the values recorded in Table

2. The y-axis of your plot should correspond to the square of the initial velocity of the

bop ( ,tr, ), yhil-eu tle x_, q.Iil lltogtq cg{rg.:pgn.$ lp ttrp sggale qf.lhg -rar1gg .9f,the,rnarble- ' ,lY .x

)' ;rlomiute th6 flop6 bna thb f-inierdept ortnr plot.and rvli'te dbwn"tH v'dhres in,:.;: , ,; the;paQe nroVided below. These will correspond to the experimental slope and y-' intercept.

Slope:y-intercept :

,i

To obtain the theoretical value for the slope, compute the value of the quantity. / , .. r2 using the measured value of the height of the ramp. Write down yourg\mB+ mn)-@;

calculation and the final value in the space provided below.

Theoretieal slope:

Calculate the relative deviation between the theoretical and experimental values of the

slope. Write down your calculations below.Relative Deviufion:

88 O 2007 LtbManual Authors

Physics 71.1 C,onliCruafrdn' of [email protected] Momentu m

o ' What does this deviation s'ignifu? Can fiisib€ aprodfdf the'conservatibn of eirergy in thesystem? Why or why not?

To obtairt'the valiie'of the dissipated energy' E o*n , follow the step:6r-.r* calculation:1. Using the expression of your slope from your v'r, vs. x' plot, obtain the value for

the h of the ramp corrosponding to the experimental slope

2. Using this value of h,wite the expression for the y-intercept of the vs. x2' plot

3. Equate the expression obtained above to the experimental y-intercept

4. Compute for the value of Eotnn

Amount of energt dissipated:

.2,vst

@ 2WT Lab,Manuhi Alttltors 89

Consgrvation of Energy and.Mo" mefitu m Physics.71.1

o . Cornpare thE arnoqnt of enorgy dissipate.{go the perqent diff,erence of your slope. How

can you relate these two quantities?

Is the rrrechanioal-energy of the system conserved? E5plain'by using data obtained in the

experiment

o Is the total energy of the system conserved? Explain by using data obtained in the

experiment.

90 O 2007 Lab,MaQual A't4hors

Static Equilibrium


2.

lntroduction

Determine experimentally where an object must be suspended (center ofgravity) and the conditions which it must satisff (conditions ofequilibrium) for it to be in static equilibrium.

Apply the conditions of equilibrium in finding the mass of an object.

Theory

t.

Whenever we see an object perched or mounted on any surface and is perfectlystill, we say that the object is "balanced". Examples of these are a bird perched ona wire or a person standing on a ledge. How do these objects maintain theirbalance or state of equilibrium? In this experiment, we aim to answer thisquestion by observing an object which is suspended at a point. We find that these

objects must satis$r certain conditions to remain balanced, or what is known inphysics as in a state of static equilibrium. First, nothing must be causing the objectto move (the net force acting on the object must be equal to zero). Second, the

object must not rotate or tip over (the net torque due to the forces acting on the

object must be equal to zerc). To achieve the second condition, the object must besupported or suspended at apoint which we call the object's center of gravity.

An object must satis$r two conditions for it to be in static equilibrium. The firstcondition is based on Newton's law and the second condition on the dynamics ofrotation of rigid body. A body"iatisfiing the first and the second conditions of


Static Equilibrium Physics 71.1

equilibrium is said to be in static equilibnum.

When a rigid body is in equilibrium, it does not accelerate. This is often called the

first condition of equilibrium. That is, the vector sum of all the (external) forces

acting:ron thg bPdY:is zero, or ' ,l' 'l

) r,=o

When the vector sum of all the torques acting on a rigid body is zero, it does not

rotate. The sum of the torques due to all the external forces acting on the body,

with respect to any specified point, must be zero. This is the second condition of

equilibrium, or in equation form,

I r:owhere the torque T (which is a vector quantity) is defined as

i:7 xF

where 7 is the radius vector pointing from any axis point to the point at which

the force vector F acts on the object. The magnitude of torque is given by

r=rF sin?

(1)

(2)

(3)

(4)

Materials

where e is the angle between the vectors F and iThere is a particular point in a rigid body where the sum of the torques. due to its

weight elements is zero. This point is called the center of gravity of the object.

We can think of the center of gravity as the point where the weight effectively

acts. An object suspended along a line through its center of gravity will not rotate.

plastic beam, ruler or tape measure, metal pans, a set of standard masses, digital

balance, hanger or beam holder

92 @ 2007-Lab Manual Authors

Physics 71.1

Procedure

Static Equilibrium

Figure 1. Some of the equipment for this activity. The beam suspended bythe hanger is the setup for Part I in the Procedure.

Part l. Determining the center of gravity of a uniform object.Insert'the beam into the holder then slide the holder along the beam until itreaches an arbitrary point on the beam. Tighten the screw in the holderthen suspend the beam from this point. Observe how the beam moves as

you release it after suspending it. Determine the forces acting on the beam

in this case. Note the distance of the point where the beam was suspended

from the right side of the beam.

Locate the center of gravity of the beam by' first moving the beam alongthe holder to another point on the beam then releasing it until, upon

release, the beam is no longer moving and is almost parallel to the

horizontal. Suspend the beam at this point and determine the forces acting

on.the beam in this case. Again, note the distance of the point where the

beam was suspended from the right side of the beam in this case.

Part ll. Determining the mass of a'uniform object using theconditions of static equilibrium

1. Support the beam at a point 30 cm from its left end. Put a 1009 mass onthe shorter end of the beam and restore equilibrium by putting masses on

the other side. Take note of their position with respect to the point ofsuspension of the beam. Using the values of the masses and their position,

' calculate the mass of the beam by means of the conditions of static

equilibrium.

l.

2.

@ 2007 Lab Manual Authors g3


2. Determine the mass of the beam using the electronic balance. Use this as

the reference value for the mass of the beam. Calculate the percent

difference with respect to the value calculated in the preceding part.

Part lll. Finding the center of gravity and mass of anonuniform object using the conditions of static equilibrium.

1. Attach an arbitrary mass to any point of the beam. Once the mass is

attached, consider it to be part of the beam. :

Locate the center of gravity of the beam masq system using the same

procedure in number 2,Pafi I of this experiment, then suspend the beam

from this point. Note the distance of the point of suspension from the right

side.

Suspend the nonunifofm beam from a point not at the center of gravity,

then restore equilibrium by adding masses to the left,and right sides of the

beam. Note their positions with respect to the point of suspension. Using

these values, apply the conditions of static equilibrium and calculate for

the mass of the beam.

Using the electronic balance, obtain the mass of the beam and use this as

your reference value for the beam's mass. Calculate the percent difference

of the reference mass against the mass calculated in the previous number.

2.

aJ.

4.

Figure 2. The experimental setup for Part II.

94 @ 2007 Lab, Manual Authors

Name DateSubmitted

DatePerfomed

;Score

Group Members

lnstructor Section

Worksheet: Static Equilibrium

I Center of gravity of a uniform object. Initially, suspend the beam at some arbitrary point. Write it down as your initial point of

suspension. Observe the direction of rotation. Based on this, locate the final position ofsuspension of the beam. At this position, the beam is not rotating.

Initial point of suspension (from the right of the beam):

Direction of rotation: (clockwise/counterclockwise)

Final point of suspension (from the right of the beam): cm

. Draw a free body diagram of the beam when it is supported at the final point ofsuspension.

. Is the net force and the net torque on the beam equal to zero? Why or why not?

. From your observations in Part 1, would it be better tb simply weigh the beamdigital balance than to obtain its mass indirectly using the principles of staticequilibrium? Why or why not?

using the


,,


ll. Determining the mass of a uniform obiect using theconditions of static equilibrium

Left of pivot Right of pivot

Location of the added massfrom the point of support

Mass

Draw the schematic diagram of the setup with corresponding measurements.Write out your

solution in determining the mass of the beam. . :'

100 g

Analytical mass of the beam:

Measured mass of the beam :

Percent deviation :

(}E

g (measured using digital balance)

%

and2,what are the conditions that a body must satisfu

o List down all the forces acting on the beam in this case:

. Are the net torque and net force acting on the beam both equal to zero? Why or why not?

From your observations in parts Ifor it to be in static equilibrium?


Physics 71.1 Static Equilibrium

lll. Finding the center of gravity and mass of a nonuniformobject using the conditions of static equilibrium

*Indicate wether CoG is to the left or to the right of the point of support.

Draw the schematic diagram of the setup with corresponding measurements.Write out yoursolution in determining the mass of the beam. Write out your solution in determining the mass ofthe non-uniform beam.

Suppose now that the mass which is part of the non-uniform beam be moved to anotherportion of the beam. Would the center of gravity of the mass-beam system change?

Location of center of gravity of the non-unfformbeam, measured from the right end (cm)

Added arbitrary mass,right of support (g)

Added arbitrary mass, left of support @)

Distance of the point of support to the center ofgravity of the non-uniform beam (cm)*

Distance of the point of support to the arbitraryadded mass, right of support(cm)

Distance of the point of support to the arbitraryadded mass, left of support(cm)

Measured mass of the non-uniform beam (g)

Analytical mass of the non-uniform heam (g)

Percent devi atio n (o/o)



lr(

;I,

!rr

n,,

r,ix


Simple Harmonic Motion:Spring Mass System


1. Determine the dependence of the period of a simple harmonic motion onthe amount of displacetnent and mass of the object.

2. Obtain the best estimate of the elastic (spring) constant for the verticalspring-mass system.

3. Determine the mass of an object using the concept of simple harmonicmotion using a spring-mass system.

Many types of motion are repetitive. A ship bobbing up and down in water, aswinging pendulum of a clock and the vibrations of guitar strings - these types ofmotions are called periodic or oscillatory. The simplest form of periodic motionsis called simple harmonic motion. This occurs when the restoring force is directlyproportional to the displacement from the equilibrium. The classic examples ofthis type are the simple pendulum and the spring-mass system.

In this activity,we shall study the motion of the spring-mass system, and examinethe parameters that affect its motion.

lntroduction


Theory

Simple llarmonic Motion: Spring Mass System Physics 71.1

(1)

Newton's 2dlaw states that F, is also relatedto the mass of the object and its

acceleration such that

Consider a system consisting of a spring with spring constant k also termed as the

stiffness constant, and an object with mass m attaghed to the qnd of the spring

FigqG 1). Whgrr rhe, Qbjeet iso disp'lacel Of s'bla? fistance

.r, a force, F* is

exerted by the spring on the object, given by Hooke's law:

F "=ffia,

where ax is the acceleration of the object with mass la

displacement of the object.r given by

d2xa*=m7/

Substituting equations I and 3 into 2,we obtain"

: d2xr , _t*=mlt

And finaily an expression for a " 'the' in terms of the spring constant k and the

displacement of the object from the equilibrium position

d'x -kn :-=-vxdim

The acceleration of the object is proportional to and opposite in direction from its

displacement. This is a characteristic of an object in simple harmonic motion.

The period T of a simple harmonic motion is the amount of time required for the

object to completely oscillate back and forth about its equilibrium position (

(2)

and is related to the

(3)

(4)

(s)

Figure 1. A spring-mass sYstem


Object on a Vertical SpringWhen an object hangs vertically fiom a gpring, in addition to the restoring force F: - k, exerted by the spring on the object, there is a force equivalent to mg

directed downward., Chooslng the downward.direction to be positive, Newton's

Phlsics 71.1 Simple Harm on lg, M oti on /$gri ng':Mass,sysfem

x= xo ). It is related to the niass of the blopk aud the spring constant, and the

relationship is given by

r=2nE (6)

(7)

How do we

handle this extra term? ; ; I

If y,=T is the distance the spring is stretched when the object is added and

the system'is in equilibrium, then making'a change of varidble in the form

second law reads I

,d'vm':=-krl mg

This differs from eqtibltiofl (2)'o*V the addition of the constant mg.

which reduces equation 3,into;12, aytn*=-lsY'

dt'

which is now similar to equation (2).

.1 *l .i

(8)

(e)

@ 2047 Lab, Manual Authors 1.B{

Simple Harmonic Motion: Sprrhg,613s5 Sysfem Physics 71.1

Figure 2. An object suspended from a verticalspring. (A) Equilibriumposition of the springwhen the object is not yet attached. (B)Equilibrium position of the ryatem when theobject is attached. The spring is sffetched by anamount of ye:mgllc(C) The object oscillates about the equilibriumposition with a displacement of y':y-yo.

The effect of the gravitational force mg is simply to shift the equilibrium positionby an amount lo :mglk, fromy:0 to y':9. When the object is then displaced

by an amount y', the spring exerts a restoring force of -lry'on the object. The

object oscillates about this equilibrium position with a period equal to equation 6

the same as that for an object on a horizontal spring. Hence, even in the presence

of gravitational force, the spring-mass system also undergoes simple harmonic

motion.

MaterialsVernier LabPro@ computer interface,Photogate, A set of similar springs,

Pendulum setup, Set of masses, Object of unknown mass, Digital balance

Procedure

Determination of the spring constant of a single spring1. Connect the Vernier LabPro@ computer interface to the computer.

2. Fasten the Photogate rigidly to a ring stand (using the pendulum set-up)

such that the arms are suspended horizontally. Make sure that the masses

are able to pass freely through the Photogate as shown in Figure 3

142 @ 20i07 Lab Manual Authors

Physics 71.1 Simple Harmonic Motion: Spring lUass Sysfem

Connect"the Photogate to DIG/SONIC 1 input of the computer interface.In

the ffi icon, double click Physics with Computers, open the folder

marked Experiment 14: Pendulum Periods then the file Photogate. A plotof the Period as a function of the Trial Number will appear on the screen.

Attach the spring-mass system to the ring stand. Make sure that inequilibrium position, the mass is blocking the Photogate (as in Figure 3).

This can be seen in the status bar of the Logger Pro at the bottom of the

screen - if the mass is blocking the Photogate, the status is noted as

blocked, otherwise it is unblocked.

5. Displace the spring-mass system from its equilibrium position by a

distance y'.

6. Click the ffi button, and release the mass. The mass then oscillates

about the equilibrium position, and the period of oscillation is shown inthe graph. Note that you may have to wait for a -few seconds before the

period of oscillation is registered by the Logger Pro.

7. After 10 trials, click the Stop button. Highlight the plot, then click the

3.

4.

.Figure 3. Simple Harmonic Motion setup.


$irnpl€ Harmonic Motion: Spring /lfass Sysfem Physics 71.1

button to obtain the mean period of oscillation of the system.

8. Obtairr tho'period for varying values of the mass displaced and the amount

of displacement.

9. From the period of oscillation obtained, calculate the experimental spring

constant. This value will be used to obtain the mass of the unknown

object.

Determination of the mass of an unknown object

Using a set of springs of the same spring constant, create five setups withdifferent resultant spring constants.

1. Obtain the period of oscillation for each of the setups created.

2. From the,period of oscillation obtained, calculate the mass of the unknown

object, ., ':"

Reference. Tipler, Paul A., Physics for Scientists and Engineers, Fourth Edition, W.H.

Freeman and Company, USA, 1999.

104 @ 2007 l-ab Manual Authors

Name DateSubmitted

Datd.Performed

Score

Group Members

lnsiructor Section

Worksheet: Simple Harmonic Motion

A. Period dependence on the angle 6f releaseObject's mass :

Table 1. Period dependence on the amount of displacement

. How does the period depend on the amount of displacement of the object?

B. Period dependence on mass of the objectAmount of displacement :

Table 2. Period dependence on mass of the object

Mass (g) Period T(s)

@)

Amount ofdisplacement (cm)

Period T(s) ol5 Difference

Experimental Theoretical

@ 2007 Lab Mdnual Authors

Srmple Harmonic Motion: Spring /l{ass System Physics 71.1

. How does the period depend on the mass of the object?

lll. Spring constant calculationPlot the square of the experimental period ( T' ) as a function of the mass (z) obtained fromTable 2,then answer the following questions:

o What is the slope of the plot of f vs. m?

o From this slope, calculate the experimental value of the spring constant.

{,06 @ 2407 Lab Mdnual Authois

Physics 7't.1 Sr:mple Harunonia Motion * Spring Mass Sysfem

lV. Period dependence on the spring constantAmount of displacement : km)

setups and calculation.

Galculations

o How does the period depend on the spring constant?

Table 1. Period dependence on the amount of displitcement

Spring constant(dyne/cm)

Period T(s) %o Difference

Experlmental Theoretical

. How did you obtain the different values for the resultant spring constant? Show all


Simple Harfiionic Motlon : Spiing ltfassisyb(em Physics 71.1

V. Unknown mass calculationPlot the square of the experimental period ( r ) as a function of the inverse of the spring

constant(i/k)obtained from Table 3, then answer the following questions:

. What is the slope of the plot of I vs' 1/k?

. From this slope, calculate the experimental value of the unknown mass'

r08 @ 2007' Lab Mdnual Authors

Physics',7{.1 s t m pb : narhiarffEllt'dfioif r spriiigr uessqrc'rerri:

. Measure the mass of the object using a digital balance. What is the percent deviation ofthe experimental mass to the acbloilmass of the object? What are the possible sources oferror in the experiment?

@ 2'007' Lab,Ma n ual Atrttlorb {oe

gimile,HarmonicMation:$-pr-pgMass,Sys{efi :. Physics 71.1

O 2007 Lab ManualAuthors

Sound

At the end of this activity you should Ue:iUle to:

Measure the speed of sound.

understand and observe interference and beats using sound waves.

Measure the beat frequency of two tuning forks.

1.

2.

aJ.

lntrsduction

Sound waves are longitudinal wavds passing through any medium such as air,solid or liquid that have frequencies within the range of human hearing. Soundwaves may alsq be in the form not audible gnopgh to be perceived by humans. Forexample, medical practitioners usq ultrasound waves to form an image of a fetusinside a pregnant woman's womb. Sound waves have also been used to detect oilin the earth's crust. Ships cany with them sound emiuing equipments calledSONARS to detect underwater objects. -

In this experiment, we will measure the speed of sound by detecting the echo orieflected sound of a finger snap. Also, we will study the interference of two soundwaves with slightly different frequencies called beais using two tuning fo.ks and aVemier microphone.

Sound is a form of mechanical wave that isunderstand how sound waves are produced,

produced by a vibrating object. Toconsider a loudspeaker. When its

Theory

@ 2007 Lab'Manual Authors

SoundPhYsics 71.1

diaphragm moves outward, the air in front of it is compressed and will cause an

increase in air pressure. This region with increased pressure is called a

condensation. After producing a condensation, the diaphragm immediatety

reverses its motion and moves inward. The inward motion produces a region

knows as rarefaction,withpressure less than the ambient surrounding air'

These oscillatory changes in pressure propagate and arrive at the ear' It forces the

eardrum to vibrate with the same frequency as the loudspeaker' The vibration of

the eardrum is sent to the brain as sound. Keep in mind that the change in pressure

is the one propagating. The air molecules are disturbed, moving back and forth

parallel to the disturbance.

The sinusoidal behavior of the pressure shown in Figure 1 can be measured by the

Vernier microphone. The microphone converts the pressure signal to an electric

signal that is recorded by the interface'

Figure 1. The oscillatory motion of pressure amplitude as a function of time' The

points of condensation (C) and rarefraction (R) are labeled'

Hence, the Vernier microphone can provide us some measurements in order to

calculate the speed of sound. Theoretically, the speed of sound (v) in air is related

to the tempelature of air which can be approximated by the equation

vo33l.4+0.67 "mls (1)

where T " is the temperature of the air in celsius. Remember that the speed of

sound is dependent on the properties of the medium and not on the properties of

the wave. From Equation 1, the speed of sound increases with air's temperature'

When two or more sound waves are present, the resulting sound is due to the

summation of the waves. This is called interference.In a special case where in

you have two tuni4g forks with slightly different frequencies -f , and 'f 'such that the oscillatory equations of pressure for both tuning forks are given by

Pr=Acos2nf ,t und Pz=Acos2n.f zt , where A is the amplitude of the

sound waves. If the two sounds reached your ear at the same time, the resulting

wave willbe a sum of the waves such that

P = A(cosrr f ,t +cosn f ,t)

Using triginometric identities

cosa*cos fr=2cosf,'"-P)cos lO- A

(2)

(3)

1',12@ 2007 Lab Manual Authors

Physics 71.1 Sound

This will allow us to write the resultant wave in the form of

P=2Acos(wt+oo't)

t,EoE

o

ooE0-

where the angular frequencies ur and or' are given by the expressions

*=){zn fi2'n .f ,\ and, *,=}yzn f r2n.f ,) (s)

2.5

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5 Loud

TimG (s)

Figure 2. A plot of two interfering waves that form beats. Notice that resultingwave have periodic loud and faint sound with variation in intensity.

This phenomenon is what we call as beats. The frequency that reaches our ears isthe average of the two frequencies. Physically this is manifested as an alternateloud and faint sound that repeat at a certain beat frequency, .f uot given by

.fru,=ft-f,

(4)

(6)

Musicians often used beats to tune their instruments. They tune their instrumentsin comparison to a certain reference tone. once the beat disappears, theinstrument can be said to be in tuned with the standard.


Physics 71.1

Speed of Sound

Materials

Procedure

computer, vernier LabPro, Logger Pro,Vernier microphone, meterstick, PVC

pipe, thick hardbound books

Measure the room temPerature.

I 1.' Connect the VemierlahPrd@ computer interface to the computer'l'r .

' . Z. Connect the Vernierimicrophone to the Vernjer LabPro@ interface. In the

' i"or, double click Physics with Computers, open the fiilder marked

Setup by the materials by covering one end of a PVC pipe with a thick

(hardbound) book to avoid great loss of sound. Place the microphone near

the entrance of the PiPe.

click the "collect" ,button to begin the data collection, and snap your

finger near the opening of the tube. This will trigger the interface to start

collecting the data.

3.

4.

++iiffil*#j.*,+;fut,;::,r.;,.:,r'ir.i=I li,:,.i

114@ 2007 Ldb ManualAuthot's

Physics 71.1 Sosnd

5. Ybu should'be able to distihguish the incident wave and the reflected(echo)" Determine the time interval between the lst wave and the 2ndwave. You may use the examine button on your toolbar.

6. Repeat the measurements aa.d'obtain several trials and calculate the bestestimate of the speed of sound.

Sound waves and beats

Materials

Procedure

computer, vernier LabPro, Logger pro,vernier microphone, two (2)tuning forks

Using the tuning'fork produce a sound and hold it close to the microphoneand click "collect" . The data plot should be a sinusoidal curve.

In your data plot, count and record the number of complete cycles shownafter the first peak in your data. . .

click the "Examine" button. Drag the mouse across the graph and recordthe times for the first and last peaks of the waveform. Divide the timedifference by'the number of cycles to deterihine the period of the tuning

l.

2.

J.


Sound Physics 71.'t

fork.

Calculate the frequency of the tuning fork in Hz and record it in your data

table.

Drag the mouse across the graph and record the maximum and minimumy values for an adjacent peak and trough.

Calculate'fie amplitude of the waVe. Record the values in your data table.

Plot the data using excel. Calculate the wavelength of this sound. Record

on the graph the information rbgarding the sound such as wavelength,

amplitude, period, and frequency.

Repeat Steps 3 - 9 for the second frequency.

To observe beats, the tuning forks must be struck at the same time. Listen

for the combined sound on the tuning fork. Beats is observed when there is

a variation of intensity or an emergence of a third pitch.

Figure 3. Equiptment for investigating sound beats andwaves. A rubber mallet, not pictured, is used to strike the

tuning forlcs.

10. Collect data plot of this waveform. Strike the tuning forks equally hard

and hold them the same distance from the Microphone.

ll.Count the number of amplitude maxima after the first maximum and

record it in a data table.

12. Click the "Examine" button. Drag the mouse across the graph and record

the times for the first and last amplitude maxima. Divide the time

difference by the number of cycles to determine the period of beats (in s).

Calculate the beat frequency. in Hz from the beat period. Record these

4.

5.

6.

7.

8.

9.


FhysieT4rtl' &rmd.

values in your data table.

ReferenceTipler, Paul A., Physics fgr,scjegJislg an{ Engineers, Fourlfu Edition, W.H.FreemanandCom$airy,uSAj,rgg9..'...:]1':::]ii..)

@ 2007...L4b i Mdri uhl AUft rors. 1flI

Physics 71.'1


lfeim DateSubmitted

DatePetbmed

ScoE

Grcup temb€E

lnatruc{or Setion

Worksheeil Sound waves

Data Table 1. Temperature of the room and length of PVC pipe

Room tgmperature.f,C)

Length of PVe p:ipe (m)

Graph 1. Amplitude of soundwaves as afunction of time

A. Speed of sound

@20OT Lab Marual Authors'1 19

Physics 71.1

Duta Table 2. Tirne measurement of.sound waves

. Using the room temperature measured what should be the theoretical speed of sotrnd?

Calculate the percent deviation between the experimental and theorctical speed of sound,

Calculations

Data Table 3. sound

Experimental speed of sound (m/s)

Theoretical speed of sound (m/s)

Percent devlation (/o)


Physics 71.1 Sound

Questions1. If you use a longer or shorter pipe will the calculated speed of sound change?

2. What happens to the plot when you gradually move the book away from the end of thepipe? You could try this. What is the difference between the open ended pipe and closedend pipe?

ll. Sound waves and beatsObserve the markings on the tuning forks. Usually, the frequency of sound emitted by thetuning forks is specified and carved on the side of the tuqing forks.

Data Table 7. Frequency of tuningforl<s

Frequency of tuning fork A (Hz)

Frequency of tuning fork B (Hz)


Physics,71.1

sound waves as a

Graph 2. Amplitude of sound wavrjs as afuntction of time (tuntingfork B)

O 2007 Lab ManualAuthors

Physics 71.1

Data Table 2. Tuningforl<s

Parameters Tuning fork A Tuning fork Bno. of complete cycles

Time interval between 7"t and 2dpeak (s)

Period (s)

Frequency (Hz)

Max y-axis value

Min y-axis value

Amplitude

Wavelength

Strike both tuning forks equally hard. Paste the plot on thebu bebw. On your graph, mat thebeats period and the wave period. At which points is the bea md loud and faint?

Graph 3. Beats

l

@ 2OO7 Lab Manual Authors ia!

Souad Physics 71,1.:

Data Table 3. Results

F,rqwtffbf tuning fork A (Hz)

Frequency of tqning fork B (Hz)

Experlmental wave frequbncy (Hz)

Tieoretical wave frequency (Hz)

Experimental beat frequency (Hz)

Theorctical beat frequency (Hz)

Percent devlatlon f/o)

. Explain the beat pattern by noting at which points is the beats loud antlwliete it is faint?What happens to the sum of the trigonometric functions at these points?

!r.4

124 @ mOTrLab. Manual Authors

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