Experiment 2: Vector Addition

Experiment 2: Vector Addition Addition Laboratory Report

Kamylle Consebido, Hazel Dacuycuy, Jose Gerardo Del Rosario, Ira Gabrielli Delos Reyes, Ancilla Diamante Department of Occupational Terapy Colle!e of Reabilitation "ciences, #ni$ersity of "anto Tomas %spa&a, 'anila (ilippines

Abstract

2. Theory

This experiment aims to determine the the resul esulta tant nt disp displa lace cemen mentt usin using g thre threee variou variouss method methodss namely namely,, the compone component, nt,  parallelogram, and the polygon method and   prove that vector addition is both comm commut utat ativ ivee and and asso associ ciat ativ ive. e. For For the the  polygon method, a group member’s initial   position and displacements of 1 m E, 2.5 m  , and ! m !"# of $ %ere mar&ed then, a resultant displacement from his initial to his  final position %as measured. For the  parallelogram method, the same displa displaceme cements nts %ere %ere used used and a resul resultan tant  t  displacement %as determined using a scale of 1m'1cm. For the component method, the  x and y components of the resultant resultant vector  %ere derived. The results sho%ed that the magnitude of ( is ).!1 m, *+.2#  of $. n conclus conclusion ion,, vector vector additio addition n is prove proven n to  sho% commutative and associative  properties.

-ector ctor addit additio ion n is addi addin! n! to to or  more $ectors to!eter into a $ector sum and is acie$ed usin! tree different metods. component, parallelo!ram, and poly!on+ A $ector, defined as a measurement it bot ma!nitude and direction, is caracterized by tree tree compone components nts namely, namely, its its ), y, and z components+ Te addition of tese components respecti$ely ould result to te $ect $ector or sum sum or resu result ltant ant $ecto $ector, r, and and tis tis metod is called te component metod+ On te oter and, te poly!on metod in$ol$es drain! scaled $ector dia!rams and usin! te ead/to ead/to/t /tail ail metod metod erei erein n diffe different rent disp displa lacem cement entss ill ill be under underta ta0e 0en n by te te sub1ect and a resultant $ector deri$ed from te tail of te first displacement to te ead of te last displacement, closin! te fi!ure dran+ Te parallelo!ram metod in$ol$es  pro1ectin! parallel $ectors, ic ould a$e a$e e2ua e2uall comp compon onen ents ts as te te ori! ori!in inal al $ect $ector or,, and form formin in! ! a four four/s /sid ided ed fi!u fi!ure re++ 3astly, te parallelo!ram4s dia!onal ould  become te $ectors4 resultant $ector, due to te e2uality of te $ectors pro1ected+

1. Introduction

Te Te prac practi tica call appl applic icat atio ion n of tis tis e)peri e)perimen mentt is manife manifeste sted d in aircra aircraft* ft*si sip p na$i na$i!a !ati tion on++ It is also also bein bein! ! used used in te te mode modern rn day G("+ G("+ Tat Tat e)per e)perim imen entt as as cond conduc ucte ted d to det determ ermine ine te res result ultant ant disp displa lacem cement ent by te te compon componen entt met metod od,,  parallelo!ram metod and poly!on metod+ It as also conducted to so tat $ector  addition is commutati$e and associati$e+

5ormulas used. 2  R= √ ( Σ x )+( )+ ( Σ y 2 ) −1

θ= tan  I 

y I   x

 I S − E I   Error = x 100 S

3. Methodoo!y

5or te poly!on metod, one of te members in te !roup as tas0ed to stand at an initial position ic ser$ed as te ori!in+ Displacements ere made it te folloin! ma!nitude and direction. 6 m e, 7+8 m n, and 9 m 9:; n of + Te final  position as recorded and te resultant displacement, ma!nitude and direction ere obtained+ Te member made anoter set of  displacements startin! from te same initial  position+ Te displacements ere. 7+8 m n, 9:; n of  and 6 m e+Te last set of  displacements made ere 9:; n of , 7+8 m n and 6 m e+  In te second acti$ity, te resultant displacement as obtained usin! a scale and te parallelo!ram metod+ Tere ere to trials made to obtain te result+ 5irst, $ector  a <6 m e= and $ector b <7+8 m n= ere dran and te resultant of a and b as obtained+ After obtainin! te resultant of a and b, $ector c <9:; n of = as dran and added to te resultant+ "econd, $ector b and $ector  c ere added first ten teir resultant as added to $ector a+ Te tird acti$ity included te use of  te component metod+ Te components of  te $ectors ere obtained and added to !et te summation of ) and y+ Te ma!nitude and direction of te resultant displacement as obtained, and tus used as te accepted $alue+ %)perimental "etup.

". Resuts and #iscussion Tabe 1. $oy!on Method Magnitude of R % error for magnitude Direction of R % error for direction

 Trial 1

Trial 2

Trial 3

4.5

4

4.3

4.41%

7.19 %

0.23%

60⁰  of !

60⁰  of !

60⁰  of !

11.76%

11.76%

11.76%

Tabe 2. $araeo!ram Method %& 'cae o( 1m:1cm "engt# of arro$ rere&enting R Magnitude of R % error for magnitude Direction of R % error for direction

 Trial 1 1cm

Trial 2 1cm

4m 7.19%

4m 7.19%

60⁰  of ! 11.76%

60⁰  of ! 11.76%

Tabe 3. )omponent Method Di&lacement

'(comonent

)(comonent

*

1

0

+

0

2.5

,

(3 co& -32

3&in12

/' (1.59 Magnitude of R  4.31 m

/)4 Direction of R  6.2

°

>it te use of te component metod, eac of te displacements ere assessed to determine te $alues of teir ) and y components+ Te results from te ma!nitude and te direction are ten used as te standard or accepted $alues to compute te ? errors of te ma!nitude and direction for te resultant displacement obtained in te acti$ities usin! te poly!on and  parallelo!ram metod+ In te poly!on metod, tree trials ere performed in order to !et te

  of !

ma!nitude and direction of te resultant $ector+ Hoe$er, te order or se2uence of  te $alues of eac trials ere can!ed ic soed te commutati$e property of $ector  addition+ Despite te fact tat teir orders ere can!ed, te results ended up as  precise as possible it te $alues @+8 m+, @ m+, and @+9 m+ respecti$ely+ >en computin! for te ?error for ma!nitude, it is e$ident tat Trial 9 as te least $alue of  ?error, manifestin! tat it obtained te most accurate measurements compared to te to trials before+ Te oter trials may a$e attained lar!er ?errors mainly because of  te occurrence of paralla) errors+ "ince te e)perimenters can $ie te measurement in different an!les, eac of us reported different $alues dependin! on o e con$eniently seen te measurement from our perspecti$e+ On te oter and, a random error as present due to te fact tat a meter stic0 as a restricted precision tat limits its ability to measure distances smaller  tan its smallest scale di$ision+ 5or te direction of te resultant $ectors, all of te trials obtained te same ?errors since tey also ad te same $alues for te direction of  teir resultant $ectors+ In te parallelo!ram metod, to trials ere re2uired and bot of te trials  performed obtained te same results in te ma!nitude and direction, tus te ?errors ac2uired ere also te same+ Tis metod also pro$ed te associati$e property in $ector addition, since te results are te same despite te can!es in te manner of  !roupin! te displacements+ Te presence of  errors may be attributed to uman errors and  paralla) errors+ Tere flas in measurin! may come from basic incompetence and bias of te e)perimenter, meanin!, mista0es from ali!nin! te ruler and protractor properly, and incorrectly readin! te accurate measurement trou!out te trials, respecti$ely+

*. )oncusion Te resultant displacement can be obtained usin! tree different metods. te  poly!on, parallelo!ram and component metod+ Te most accurate $alue as obtained usin! te component metod tus  becomin! te accepted $alue to compute for  te ?error of ma!nitude and direction+ Te commutati$e property of addition as son in te poly!on metod ile te associati$e property of addition as son in te parallelo!ram metod+ %rrors may a$e been made due to o te displacements ere dran+ "mall spaces  beteen lines could lead to a lar!er $alue tus te component metod is used as te most accurate and accepted $alue for it does not re2uire te displacements to be dran+ Instead, fi)ed formulas are used to determine accurate results+ +. Appications 6+ Ran!e of ma!nitude of te resultant Te smallest $alue is possible en te to $ectors point in te opposite direction+ Te $alue is 6 unit+ <@ units B 9 units= Te lar!est $alue is possible en te $ectors  point in te same direction+ Te $alue is  units+ <@ units  9 units=

a+ 'a)imum resultant.  b+ 'inimum resultant. c+ 'a!nitude of 8 units.

0° 180 °

90 °

d+ 'a!nitude of E units. Appro)+ E9 °

7+ Distance is simply te amount of space from one point to anoter+ It simply refers to o muc !round an ob1ect as co$ered up durin! its motion and bein! a scalar 2uantity, it is dependent only to its ma!nitude+ On te oter  and, displacement is te o$erall can!e in  position of an ob1ect and measures o far out of place an ob1ect is relati$e to its point of  ori!in+ It is possible for one to a$e no displacement tou! tra$elin! !reat distances+ A !ood e)ample ould be en a 1o!!er from startin! point, runs alon! an o$al trac0 o$er 8 times, co$erin! $ast amounts of distances+ Fut as te 1o!!er returns to te startin! point, te

distance co$ered may be !reat, but te displacement of er position relati$e to te point of ori!in is zero+ 2

9+ a

2

+ b =c 2

2

cosθ =

11.67

Anser. 66+E m+ 31° SE

,. Re(erences "terneim, '+,  Kane, J+ <66=+ -eneral   physics. e Jersey. >iley  "ons+

2

−1 10.02

+ 6  7+8  66+E m+

θ=31 ° SE



2

c  +6 m

)  6+  +78  6:+:7

10.02

¿ =c 8.25 ¿ +¿ ¿ 4.0

)  7+8 cos @8

θ= cos

(

11.67

)