TABLE OF CONTENTTABLE OF CONTENT

NNoo SSuubbjjeecctt PPaaggee

11..11 IInnttrroodduuccttiioonn 22

11..22 OObbjjeeccttiivvee 33

11..33 LLeeaarrnniinng g OOuuttccoommee 33

1.1.44 ThTheoeoreretiticacal l BaBackckgrgrouound nd 3-3-88

22..11 rroobblleem m !!ttaatteemmeenntt 88

33..11 ""####aarraattuu$$ %%

33..22 rroocceedduurree %%--11&&

44..11 ''ee$$uulltt 11&&--1122

44..22 ""nnaall(($$ii$ 1$ 133--1144

44..33 ))ii$$ccuu$$$$iioonn 11**

4.44.4 ConclusionConclusion 1+1+

44..** ,,##eerriimmeennttaal l ##rreeccaauuttiioonn 11++

44..++ ''eeeerreennccee$$ 11++

44..// ""####eennddii 11//

EXPERIMENT TITLEEXPERIMENT TITLE

)etermination o 0etacentric height.)etermination o 0etacentric height.

1.1 INTRODUCTION1.1 INTRODUCTION

11

Level 1 laborator( activit( reer$ to condition here the #roblem and a($ mean$ are

guided and given to the $tudent$. oever the an$er$ to the a$$ignment are let to the

$tudent$ to $olve u$ing the grou# creativit( and innovativene$$. The activit( i$ ho#e to

$lol( introduced and inculcate inde#endent learning among$t $tudent$ and #re#are them

or a much harder ta$k o o#en-ended laborator( activitie$.

In thi$ laborator( activit( $tudent$ ill be e#o$ed to the eui#ment that u$ed to mea$ure

the metacentric height o #ontoon. 5or $tatic euilibrium o the #ontoon6 the total eight6

W6 7 hich act$ through the centre o the gravit(6 G mu$t be eual to the buo(anc( orce

hich act$ through the centre o buo(anc(6B6 hich i$ located at the centroid o the

immer$ed cro$$-$ection. 9hen the #ontoon heel$ through a $mall angle6 the metacentreM

i$ identiied a$ the #oint o inter$ection beteen the line$ o action o the buo(anc( orce

7 ala($ vertical and BG etend. 5or $table euilibrium6 M mu$t be above G.

Fgu!e ".#: " loating bod( i$ $table i the bod( i$ 7a. The centre o gravit( G i$ belo

the centroid B o the bod(; 7b The metacentre M i$ above G; 7c <n$table i M belo G.

1.$ OB%ECTI&E

To identi( the #o$ition o the metacentre 70 o a loating bod(6 b( reerring the di$tance

rom the centre o gravit( 7=.

2

1.' LEARNING OUTCOMES

 "t the end o the laborator( activitie$6 $tudent$ ould be able to: i. )etermine the

$uitable laborator( te$t$ to be conducted to addre$$ the given #roblem. ii. "nal($e te$t

data and #re$ent the $olution to the o#en-ended #roblem. iii. 9ork in a grou# to #roduce

the relevant technical re#ort

1.( T)EORETICAL BAC*GROUND

ontoon i$ a term u$ed to denote a lat bottomed ve$$el hich i$ rectangular in cro$$

$ection and in #lan. >on$idering 5igure 26 e have the eight orce6 9 acting verticall(

don through the ce+t!e o, g!a-t6 G6 o the #ontoon. !ince the #ontoon i$ loating in

ater ith a con$tant de#th immer$ion6 it ollo$ that there mu$t be an eual orce acting

the o##o$ing direction o the eight orce6 knon a$ buoa+c ,o!ce6 F 6 hich act$

verticall( u# through the centre o gravit( o the di$#laced ater.

Fgu!e ".1: " #ontoon loating on even keel ith 9 and 5 collinear.

!ince the #ontoon i$ a $im#le rectangle6 the $ha#e o the di$#laced liuid i$ al$o a

rectangle ith it centre at the geometrical centre namel( the ce+t!e o, buoa+c/ B. The

 buo(anc( orce6 5 act$ u#ard$ through B. ?ote that 9 and 5 act collinearl( ith =

$ituated $ome di$tance above B.

3

Fgu!e ".$: " #ontoon loating ith an im#o$ed angle o tilt6 $hoing the righting cou#le

9hen a #ontoon i$ tilted a$ $hon in 5igure 36 9 act$ verticall( don through =

hich maintained at the $ame #o$ition but 5 no act$ through #oint B@ in$tead o B. Thi$

i$ becau$e act$ through the centre o gravit( o the di$#laced liuid hich i$ no

tra#eAoidal in $ha#e ith it$ centre o gravit( at B@. "$ a re$ult 5 and 9 are no longer

collinear6 but a cou#le o orce$ that return the #ontoon to an even keel are ormed. Thi$

i$ knon a$ righting cou#le. In thi$ ca$e the #ontoon i$ ca#able o righting it$el hen

tilted6 hence it i$ $table.

Fgu!e ".'0 " #ontoon ith a rai$ed = and an im#o$ed angle o tilt6 $hoing the

overturning cou#le cau$ed b( 9 acting out$ide

It a relativel( tall #iece o eight i$ #laced on the #ontoon a$ $hon in 5igure 46 the

combined eight6 9 o the #ontoon and it$ load act$ through the centre o gravit(6 =

hich i$ relativel( high. 9hen = become$ higher and the angle o tilt increa$e$6 9 act$

urther and turn urther to the let. Thi$ mean$ that at the $ome #oint the movement o

 buo(anc( orce6 5 rom B to B@ i$ unlikel( to be large enough to #roduce a righting

cou#le. 9hat e no have i$ the $ituation de#icted in 5igure *6 here the line o action

4

o 9 i$ out$ide 7nearer the edge o the #ontoon than the line along hich 5 act$. Thu$ 9

i$ tr(ing to overturn the #ontoon. The to orce$ 5 and 9 orm an overturning cou#le.

Thu$ it i$ un$table.

Fgu!e ".( : The #o$ition o metacentre

" #ontoon loating on an even keel ha$ it$ center o buo(anc( at B and it$ centre o

gravit( at =. " line joining B to = ould be a$ $hon in 5igure 46 that i$ vertical and at

%& to the deck o #ontoon. Imagine line B= etend$ u#ard$ and ho con$ider the

 #ontoon in tilted #o$ition a$ in 5igure +6 the centre o buo(anc( moved rom B to B@. "

line dran verticall( u#ard$ through B@ ill inter$ect the line B= at the #oint labelled

0 in the diagram. Thi$ called the metacentre. rovided the = doe$ not move6 then or all

relativel( $mall angle o tilt;

i. The vertical line through u# B@ through 0. >on$euentl( i the location o B@ can be

calculated6 the #o$ition o 0 can be ound gra#hicall(.

ii. The di$tance o 0 above 0 con$tant.

iii. The di$tance =0 i$ called metacentric height o #ontoon.

9hen con$idering the $tabilit( o loating bod(6 it i$ u$ual to a$$ume that the angle o

tilt θ $mall. Thi$ i$ nece$$ar( to $im#li( the theor( b( making the a$$um#tion that θ

radian$ C $in θ C tan θ C θ radian$.

>on$idering the re$toring moment that right$ a rectangular #ontoon to an even keel

hen it i$ tilted6 the euation:

B0 C I ws / V 

5

9here:

D C the volume o ater di$#laced b( the bod(

 I ws = the $econd moment o the area

Fgu!e ".": lan o the #ontoon here the tilt take$ #lace about the longitudinal ai$ E-E

 I ws C12

3 LB

It $hould be a##arent that B0 de#end$ onl( u#on:

a. I and b6 the dimen$ion$ o the #ontoon hich govern the value o I ws .

b. D6 the volume o di$#laced ater hich de#end$ onl( u#on the eight o the

 #ontoon.

'eerring to 5igure /6 (ou $hould be able to $ee that B0 C B= F =0 or6 =0 C B0-

B=. I e can calculate B=6 then e can obtain =0 and hence determine i the bod( i$

$table or un$table. ?o6 B i$ the center o buo(anc(6 and ith the #ontoon loating on an

even keel B i$ located at a height eual to hal the de#th o immer$ion 7hG2 above the

 #oint O on the bottom o the #ontoon.

6

Fgu!e ". : " #ontoon $hoing the ke( #oint$ and dimen$ion$

It i$ common #ractice to carr( out an e#eriment on ve$$el to a$$e$$ it$ $tabilit( b(

calculating =0. Thi$ i$ a $im#le #rocedure utiliAing moveable eight #o$itioned on the

deck at a##roimatel( the middle o the longitudinal centreline and a #endulum hanging

in$ide the ve$$el. The eight namel( jocke( eight 7j i$ moved rom the centreline

knon di$tance 7 θ ∂ toard$ the $ide a$ $hon in 5igure 8. Thi$ move$ the centre o

gravit( o the #ontoon rom = on the centreline to a ne #o$ition =@ and cau$e$ the

ve$$el to tilt at the angle o θ ∂ .

The magnitude o ==@ de#end$ u#on ho ar the jocke( eight i$ moved and it$ $iAe

relativel( to the total eight o the #ontoon. <$ing the ratio o eight and  x∂ 6

 xW 

wjGG ∂ 

  

  =@

9here 9 i$ the total eight o the #ontoon including the #ontoon

 xGM GG θ tan@ =

>ombining both euation$6

θ d 

dx

W 

wjGM   

  

  =

It i$ im#ortant to remember that θ   i$ in radian.

7

Fgu!e ".20 0ovement o the jocke( eight rom the centreline

$.1 PROBLEM STATEMENT

"n im#ortant a##lication o the buo(anc( conce#t i$ the a$$e$$ment o the $tabilit( o

immer$ed and loating bodie$ hen being #lace in a luid. Hnoing metacentre6 0

location i$ vital and great im#ortance in the de$ign o $hi#$ and $ubmarine$. The bod( i$

$aid $table i 0 i$ above = and un$table i otheri$e. !tudent$ are reuired to #erorm a

relevant e#eriment to ulil the objective $tated above u$ing both method namel(

adju$table #o$ition traver$ed eight e#eriment and ba$ed u#on geometr( and de#th o

immer$ion. 5or com#utation #ur#o$e6 the $tudent$ are a$ked to ind the euation rom the

literature or eiting manual or luid and h(draulic laborator(.

'.1 APPARATUS

a. 0etacentric eight "##aratu$

8

1. ontoon Bod(2. >ro$$ - bar 

3. "dju$table 0a$$

4. 0a$t*. !liding 0a$$

+. lumb - line

/. Linear !cale

Figure 1.0 - Metacentric Height

apparatus

Figure 1.1 - Lael o! apparatus

'.$ PROCEDURE

1. The tran$ver$e adju$table ma$$ i$ eighed.2. The #ontoon i$ a$$embled and eighed.

3. The $liding ma$$ i$ #o$itioned along the ma$t $uch that the center o gravit( occur$ at

the to# o the #ontoon. Thi$ can be determined b( u$ing either a knie edge or b(

$u$#ending rom a light $tring around the ma$t.

4. The ba$in i$ illed ith ater6 the #ontoon i$ loated en$uring that the adju$table

ma$$ i$ in it$ central #o$ition.

*. The D> #late$ #rovided i$ u$ed to level the loating bod( and Aero datum i$

checked beteen #lumb line and $cale.+. The adju$table ma$$ i$ moved to the right o centre in *mm increment$ to the end o

the $cale6 nothing the angular di$#lacement o the #lumb line or each #o$ition.

"

/. The adju$table ma$$ i$ re#eated or movement to the let centre.

8. 9ith the ece#tion o eighing the adju$table eight and em#t(ing and reilling the

volumetric tank6 all the above i$ re#eated or the $liding ma$$ at dierent height$ u#

the ma$t6 i.e. or dierent centre$ o gravit(.

%. "ll the reading i$ recorded in the re$ult $heet.1&. The gra#h o lateral #o$ition o adju$table ma$$ again$t angle o li$t or each $liding

ma$$ height i$ #re#ared. The value oθ d 

dx or each $liding ma$$ height i$ obtained6

the metacentric height6 =0 and di$tance beteen the centre o buo(anc( and the

metacentre i$ calculated.

(.1 RESULTS

Tota3 4eg5t o, ,3oat+g a66e7b3 8W9 2.343 :g

Weg5t o, a;ju6tab3e 7a66 8<9 &.2&8 :g

Weg5t o, 63;+g 7a66 8<19 &.*11 :g

B!ea;t5 o, =o+too+ 8B9 2&& 77

Le+gt5 o, =o+too+ 8L9 3*& 77

Seco+; 7o7e+t o, a!ea 8I9 2.3331&-4 7(

&o3u7e o, 4ate! ;6=3ace; 8&9 2.3431&-3 7'

)eg5t o, 7etace+t!e abo-e ce+t!e o, buoa+c 8BM9 &.&%%+ 7

De=t5 o, 77e!6o+ o, =o+too+ 8IP9 &.&33* 7

De=t5 o, ce+t!e o, buoa+c 8CB9 &.&1+/ 7

)eg5t o, 

a;ju6tab3e

4eg5t/ 1

8779

Rea;+g o, 36t ,o! a;ju6tab3e 4eg5t 3ate!a3 ;6=3ace7e+t ,!o7 6a3 ce+t!e 3+e/

8779

>"# >(# >'# >$# >1# # 1# $# '# (# "# #

# *4 42 3& 2& 1& & -1& -2& -31 -42 -*+ -++

"# - - - +2 32 & -18 -4+ -+& - - -

1## - - - - - & -*& -+8 - - - -

10