TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES COLLEGE OF ENGINEERING MECHANICAL ENGINEERING DEPARTMENT DEPARTMENT
EXPERIMENT NO. 3
Measurement of areas of irregularly shaped planar regions
Llorera, Jeshua B ME !L" BSME # $D
July 8, 2016
I. OBJECTIVES:
Fa%&l&ar&'e s(u)e*(s +&(h (he %easure%e*( o areas &* (he la-ora(or. a*) +or/sho0 Measure a*) (a-ula(e (he area o &rre1ular sha0es &* 2ar&ous s&'es us&*1 (he 0la*&%e(er
•
De(er%&*e (he a33ura3. o %easure%e*(s &* (er%s o 0er3e*( error a*) s(a*)ar) )e2&a(&o*
II. THEOETIC!" B!C#$O%&': The (lanimeter:
The 0olar 0la*&%e(er &s a %e3ha*&3al )e2&3e or %easur&*1 areas o re1&o*s &* (he 0la*e +h&3h are -ou*)e) -. s%oo(h -ou*)ar&es The %easure%e*( &s -ase) )&re3(l. o* Gree*4s (heore% &* %ul(&52ar&a-le 3al3ulus6 (he 0la*&%e(er &*(e1ra(es a l&*e &*(e1ral o a 2e3(or &el) +h&3h has 3o*s(a*( 3url Pla*&%e(ers are use) &* %e)&3&*e or e7a%0le (o %easure (he s&'e o (he 3ross5se3(&o*s o (u%ors or or1a*s, &* -&olo1. (o %easure (he area o lea2es or +&*1 s&'es o &*se3(s, &* a1r&3ul(ure (o %easure (he area o ores(s, &* e*1&*eer&*1 &( &s use) (o %easure (he s&'e o 0ro&les Types of (lanimeters: (olar and "inear (lanimeters The %a&* +or/&*1 0ar(s o 0olar a*) l&*ear 0la*&%e(ers are (he sa%e Ea3h has a ro), 3alle) (he (ra3er ar%, o*e e*) o +h&3h &s (he (ra3&*1 0o&*( T, a*) a +heel a((a3he) (o (he ro) +&(h &(s a7&s 0arallel (o (he ro) A s3ale a((a3he) (o (he +heel re3or)s ho+ %u3h (he +heel (ur*s
Polar Pla*&%e(er
L&*ear Pla*&%e(er The (+o 0la*&%e(ers )&er &* (he %o2e%e*( o (he e*) o (he (ra3er ar% o00os&(e (he (ra3er 0o&*( I* (he l&*ear 0la*&%e(er, (ha( e*) &s res(r&3(e) (o %o2e alo*1 a s(ra&1h( l&*e The )ra+&*1 su11es(s (ha( &( ru*s alo*1 a (ra3/, -u( a %ore 3o%%o* +a. (o 3ause (h&s %o(&o* &s (o ha2e (h&s e*) a((a3he) (o a se( o +heels (ha( are &7e) (o a* a7&s so (ha( (he +heels (ur* (o1e(her I* (he 0olar 0la*&%e(er, (he e*) o00os&(e (he (ra3er 0o&*( &s res(r&3(e) (o %o2e alo*1 a 3&r3le Th&s &s )o*e -. %a/&*1 (ha( 0o&*( (he h&*1e -e(+ee* (he (ra3er ar% a*) a se3o*)ar. ar%, o*e e*) o +h&3h 8(he 0ole9 &s &7e) The or&1&*al 0olar 0la*&%e(er +as &*2e*(e) &* :;!$ -. Ja3o- A%sler , a S+&ss %a(he%a(&3&a* a*) &*2e*(or o %a*. %easur&*1 &*s(ru%e*(s I( +as so %u3h s&%0ler, eas&er (o use, a*) %ore a33ura(e (ha* 0re2&ousl. &*2e*(e) 0la*&%e(ers (ha( (he la((er Esser 0olar 0la*&%e(er I ha2e 8%o)el ?@:!9 Th&s +as %a)e -. (he Ger%a* 3o%0a*. Ha &* a-ou( :
The 0&3(ure o* (he le( sho+s (he 0la*&%e(er rea). (o %easure (he area o a re1&o* O* (he r&1h( &s a )&sasse%-le) 2&e+ The (ra3er 0o&*( &s e
The a7&s o (he +heel &s eas&l. see* o* (he u*)ers&)e o (he 3arr&a1e The +heel &(sel 8&*)&3a(e) -. (he 0o&*(er9 &s a s(eel )&s/ a((a3he) )&re3(l. (o (he 3.l&*)er +&(h (he s3ale The 0las(&3 +heel o* (he sa%e a7&s (ha( &s 2&s&-le &* -o(h 0&3(ures &s sl&1h(l. s%aller (ha* (he s(eel o*e a*) )oes *o( 3o*(a3( (he 0a0er I( &s use) (o se( (he s3ale (o 'ero
Methods of use
Se( u0 (he 0la*&%e(er so (ha( (he (ra3&*1 ar% a*) le*s are (o+ar)s .ou A((a3h (he 0ole ar% (o (he roller hous&*1 a*) (o (he 0ole 0la(e The 0ole ar% shoul) -e a00ro7&%a(el. a( r&1h( a*1les (o (he (ra3&*1 ar%, +&(h (he 0ole o* (he r&1h( 8F&1:9
F&rs( %o2e (he 'ero se((&*1 +heel a*) +a(3h (he s3ale a*) (he )&al S(o0 as soo* as (he o* (he )&al &s 3o2ere) -. (he &*)&53a(or a*) (he o* (he s3ale &s o00os&(e (he o* (he 2er*&er The &*s(ru%e*( &s *o+ &* &(s 'ero 0os&(&o* 8F&1 @9 No+ %o2e (he (ra3&*1 le*s 2er. sl&1h(l. (o (he r&1h( a*) s(o0 -eore (he o* (he 2er*&er has rea3he) (he &rs( o (he 3al&-ra5(&o*s o* (he s3ale I *o+, or e7a%0le (he $(h 3al&-ra(&o* o* (he 2er*&er %a(3hes a 3al&-ra(&o* o* (he s3ale, (he* (he s3ale has %o2e) our Ver*&er U*&(s 8VU9 (o+ar)s (he &rs( 3al&-ra(&o* 8F&1 9 Ne7( %o2e (he (ra3&*1 le*s a l&((le ur(her (o (he r&1h( u*(&l (he o* (he 2er*&er &s o00os&(e (he &rs( 3al&-ra(&o* o* (he s3ale The s3ale has *o+ %o2e) : 2er*&er u*&(s 8: VU9 or o*e &1 :, @, 3al&-ra(&o* 8F&1 $9
I (he s3ale ro(a(es u*(&l (he : o* &( &s o00os&(e (he o* (he 2er5*&er, &( has (he* (ur*e) (hrou1h : 2er*&er u*&(s 8: VU9 or (e* 3al&-ra(&o*s 8F&1 !9 he* (he s3ale has %a)e a 3o%0le(e ro(a(&o* 80ass&*1 all (he &1ures ro% : (o 9 a*) has re(ur*e) (o , &( has (ur*e) (hrou1h : 2er*&er u*&(s 8: VU9 or : 3al&-ra(&o*s The )&al *o+ &*)&3a(es : &*s(ea) o 8F&1 ?9 Ea3h o (he (e* &1u5res o* (he )&al 3orres0o*)s (o a 3o%0le(e re2olu(&o* o (he s3ale # : VU To 3ou*( (he (o(al *u%-er o u*&(s, +e rea) (he (housa*)s o* (he )&al, (he hu*)re)s a*) (e*s o* (he s3ale, a*) (he u*&(s o* (he 2er*&er A &*al e7a%0le shoul) %a/e (he %e(ho) o rea)&*1
s<3% : s<3% : s<3%
:
s<3% O*e re2olu(&o* o (he s3ale or o*e 3al&-ra(&o* o* (he : )&al : s<3% O*e re2olu(&o* :, : o* (he )&al s<3% The HAFF Pla*&%e(ers No : a*) :! ha2e a)us(a-le (ra53&*1 ar%s, (he le*1(h o +h&3h 3a* -e 2ar&e) (o sele3( (he %os( useul 2er*&er u*&( 2alue sho+* &* Ta-le III a*) IV 80a1e :!9 Measur&*1 +&(h Pole 0la(e ou(s&)e (he &1ure 8usual9 For (hese +or/&*1 areas6
The 0ole 0la(e %a. -e se( )o+* &* a*. 0os&(&o* ou(s&)e (he &1ure +h&3h allo+s (he (ra3&*1 le*s (o -e 1u&)e) rou*) (he e*(&re ou(l&*e I* (he 3ase o lar1er &1ures 3o*sul( Se3(&o* E@ The le*1(h o (he 0ole ar% has *o ee3( o* (he %easure%e*( +he* (he 0ole 0la(e &s se( )o+* ou(s&)e (he &1ure
&1 ; Tra)ing an area
F&1
Beore (ra3&*1 a* area (he (ra3&*1 le*s %a. -e se( )o+* &* (he %&))le o (he area The 0ole ar% shoul) -e a00ro7&%a5(el. a( r&1h( a*1les (o (he (ra3&*1 ar% Mar/ (he s(ar(&*1 0o&*( 8+h&3h +&ll also -e (he &*&sh&*1 0o&*(9 +&(h a l&*e a( r&1h( a*1les (o (he ou(l&*e 8F&1 ;9 Pos&(&o* (he s%all 3e*(re r&*1 o (he (ra3&*1 le*s 8+h&3h shoul) -e hel) &* (he r&1h( ha*)9 e7a3(l. o2er (he s(ar(&*1 0o&*( 8F&1 9 A( (he sa%e (&%e, +&(h (he le( ha*), (ur* (he 'ero se((&*1 +heel u*(&l (he )&al a*) (he s3ale -o(h re(ur* (o 'ero 8F&1 :9
Hol)&*1 (he (ra3&*1 le*s as sho+* &* F&1 , (ra3e (he ou(l&5*e &* a 3lo3/+&se )&re3(&o* +&(h (he s%all 3e*(re r&*1 =ee0 loo/&*1 &* (he )&re3(&o* &* +h&3h (he le*s &s (o (ra2el a*) (r. (o /ee0 (he l&*e &*s&)e (he r&*1 I( &s &%0oss&-le (o hol) (he s%all r&*1 e7a3(l. o2er (he 3e*5(re o (he l&*e all (he (&%e, so 3o%0e*sa(e or errors 3ause) -. 1o&*1 o (he l&*e (o o*e s&)e -. 1o&*1 o a* e
III. I&ST%ME&TS !&' M!TEI!"S: Irre1ular sha0e 0la*e areas Pla*&%e(er IV. METHO'S !&' (OCE'%ES: : Chose : s(a*)ar)s a*) re1ularl. sha0e) o 0la*er se3(&o* o 2ar&ous s&'es @ Measures ea3h areas ! (&%es us&*1 (he 0la*&%e(er Cal3ula(e (he a2era1e o (he &2e %easure%e*(s $ Cal3ula(e or (he 0er3e*( error &* (he 3ase o ea3h area ! De(er%&*e (he s(a*)ar) )e2&a(&o* o (he %easure%e*(s ? Ta-ula(e (he %easure%e*(s"o-ser2a(&o*s a*) resul(s o 3al3ula(&o* D&s3uss (he o-ser2a(&o*s a*) o(her resul(s
V. OBSEV!TIO&S: h&le )o&*1 (he e70er&%e*( I sa+ %a*. errors a*) 3o*s&)era(&o*s &* 3o*)u3(&*1 our e70er&%e*( e7a%0le o (h&s errors are ho+ (he +a. +e )o (he s(r&00&*1 or (he &rre1ular sha0es (he +&)(h o (he s(r&0s )e(er%&*es (he a33ura3. o (he 3o%0u(e) area Also +e shoul) ha2e a +ell /*o+le)1e &* )&ere*( or%ulas o sha0es
VI. '!T! !&' ES%"TS: U*&(s are &* 3% a*) u*&(s or (r&al are &* 3% @
sha0e Re3(a*1le +&(h se%&3&r3le a3(ual @!:: (r&a*1le +&(h se%&3&r3le Actual= 9.384 Irre1ular sha0e *o : Actual=68. 075 Irre1ular sha0e *o @ Actual=54. 525
Trial *
Trial +
Trial ,
Trial -
Trial
Trial /
Trial 0
Trial 1
!2e
@!$ 3%@
@! 3%@
@!? 3%@
@! 3%@
@!? 3%@
@!? 3%@
@!! 3%@
@! 3%@
@!? 3%@
@3%@ 3%@
@ 3%@
@ 3%@
$ 3%@
3%@
3%@
$ 3%@
@;; 3%@
?;@ 3%@
?;: 3%@
?; 3%@
? 3%@
?; 3%@
?; 3%@
? 3%@
?;: 3%@
?;@ ! 3%@
!$ 3%@
!! 3%@
!!: 3%@
!!? 3%@
!$ 3%@
!$: 3%@
!$! 3%@
!$ 3%@
!$ ; 3%@
Computation of area 3or e)tangle 4ith semi)ir)le:
A(o(al Are3 As3 8$9 88!$9 @K"@9"$9 :;? 3% @ ::$!:3% @ @!:: 3% @ 3or Triangle 4ith semi)ir)le:
A(o(al AT As3 879"@ 889 @K"@9"$9 !;!3% @ !$3%@ ;$ 3%@ 3or Irregular shape no. *:
A +8!.o.:.*5:!.*9 !8!8@!9$!!@!?!;! ;!;!?@;$@ ????@!?$@8!99 !5/1.60 )m +
Irregular shape no. +
A +8!.o.:.*5:!.*9 !88!9;8:?9$:?:!;@; ;?;?;?$!;$? $?:@:@?@;@!8!9 !5-.+ )m + Computation of per)entage error e)tangle 4ith semi)ir)le 25.311
−25.6
25.311
X 100 =1.141
Triangle 4ith semi)ir)le 9.384− 9.288 9.484
X 100 =1.023
Irregular shape no. * 68.075
−68.025
68.075
X 100 =0.073
Tr&al @ 8@!5@!?9 @ : Tr&al 8@!?5@!?9 @ Tr&al $ 8@!5@!?9 @ : Tr&al ! 8@!?5@!?9 @ Tr&al ? 8@!?5@!?9 @ Tr&al 8@!!5@!?9 @ : Tr&al ; 8@!5@!?9 @ : 8
Computation for standard de2iation 3or e)tangle 4ith semi)ir)le : µ= @!?
( Xi −µ ) ∑ = k
@
1
;
:"; 8;9 :
@
Tr&al : 8@!$5@!?9 $
√ 0.01 :a*s+er
3or triangle 4ith semi)ir)le : µ= @;;
3or Irregular shape no. * 6 ?;@!
Tr&al : 8@5@;;9 @ ;$7: 5$
Tr&al : 8?;@5?;@!9 @ :
Tr&al @ 85@;;9 @ !:;$7: 5
Tr&al @ 8?;:5?;@!9 @ !?@!7: 5
Tr&al 8@5@;;9 @ ;$7: 5$
Tr&al 8?;5?;@!9 @ ?@!7: 5$
Tr&al $ 8@5@;;9 @ ;$7: 5$
Tr&al $ 8?5?;@!9 @ :?
Tr&al ! 8$5@;;9 @
Tr&al ! 8?;5?;@!9 @ ?@!7: 5$
Tr&al ? 85@;;9 @ !:;$7: 5
Tr&al ? 8?;$5?;@!9 @ :$:
Tr&al 85@;;9 @ !:;$7: 5
Tr&al 8?:5?;@!9 @ :?
Tr&al ; 8$5@;;9 @
Tr&al ; 8?;:5?;@!9 @ !?@!7: 5 8
8
( Xi −µ ) ∑ = k
1
@
;
:"; 8;9 ;
√ 0.098 : A*s+er
( Xi −µ ) ∑ = k
1
@
@:
:"; 8@:9 @
√ 0.027 :?$ A*s+er 3or Irregular shape no. + 6 !$;
Tr&al : 8!$5!$;9 @ :@
Tr&al @ 8!!5!$;9 @ :!
Tr&al ? 8!$:5!$;9 @ $
Tr&al 8!!:5!$;9 @ ::
Tr&al 8!$!5!$;9 @ !
Tr&al $ 8!!?5!$;9 @ $
Tr&al ; 8!$5!$;9 @ :$$$7: 5
Tr&al ! 8!$5!$;9 @ :@
8
( Xi −µ ) ∑ = k
1
@
@;
:"; 8@;9 @!!
√ 0.255 !! a*s+er
VII.
CO&C"%SIO& !&' ECOME&'!TIO&S:
The a33ura3. o our e70er&%e*(s )e0e*)s o* (+o a3(ors o*e &s (he e*2&ro*%e*(al a3(or a*) (he o(her &s hu%a* error •
En2ironmental fa)tor The +or/&*1 area 3a* ae3( (he a33ura3. o (he e70er&%e*(al 2alue I (he sura3e &s *o( le2ele) (he roller &* (he 0la*&%e(er Also & (he (es(&*1 su-e3( &s *o( &* 1oo) 3o*)&(&o*s (he resul( +&ll -e +ro*1 Human error So%e 0eo0le o-(a&* -e((er resul(s (ha* o(hers A 1oo) e.e, a s(ea). ha*) a*) 0a(&e*3e +he* (ra3&*1 (he ou(l&*e are %os( &%0or(a*( or 1oo) resul(s Goo) l&1h( a*) a 3o%or(a-le +or/&*1 0os&(&o* also 0la. a )e3&s&2e role e)ommendation To ha2e 0re3&se resul(s &* 3o*)u3(&*1 (he e70er&%e*( +e shoul) 3o*s&)er (he e*2&ro*%e*( a*) (he hu%a* error (ha( +&ll o33ur &* (he e70er&%e*( (o lesse* (he )&s3re0a*3. Also +e shoul) 3o*s&)er (he 3o*)&(&o* o (he (es( su-e3( (o ha2e a 0ro0er e2alua(&o* &* 3o*)u3(&*1 (he e70er&%e*(
