Experiment: Stability of Floating Bodies
1.0 Objectives : 1. Determ Determinat ination ion of cente centerr of buoyanc buoyancy y 2. Determ Determinat ination ion of metace metacentr ntric ic height height 3. Investiga Investigation tion of of stability stability of oating oating objects objects
2.0 !eory: Floa loating ting bodi bodies es are are a specia eciall cas case; only only a port portio ion n of the the body body is submer submerged ged !ith !ith the remainde remainderr po"ing po"ing of the free surfac surface. e. #he buoyan buoyancy cy !hich is the !eight of the displaced !ater i.e. submerged body portion is e$ual to its dead !eigh !eight t . #he center of gravity of the displaced !ater mass is referred to as the center of buoyancy and the center of gravity of the body is "no!n as the center of mass . In %$uilibrium position buoyancy force and dead !eight have the same line of action and are e$ual and opposite &see Fig. 2'. ( submerged body is stable if its center of mass locates belo! the center of buoyancy. )o!ever this is not the essential condition for stability in oating objects.
Figure 1* +uoyancy force and center of buoyancy.
( oating object is stable as far as a resetting moment e,ists in the event of deection or tilting from the e$uilibrium position. (s sho!n in Fig. 3 dead !eight !eight and buoyancy form a force couple !ith the lever arm of !hich !hich provi provides des a righti righting ng moment moment.. #he #he dis distan tance ce bet!ee bet!een n the centr centre e of grav gravit ity y and and the the poin pointt of inte inters rsec ectio tion n of line line of acti action on of buoy buoyan ancy cy and and symmetry a,is is a measure of stability. #he point of intersection is referred to as the metacentre and the distance bet!een the centre of gravity and the metacentre is called the metacentric height .
Figure 2* -etacentre and metacentric height. The floating object is stable when the metacentric height is positive, i.e., the metacentre is located above the centre of gravity; else it is unstable. The position of the metacentre is not governed by the position of the centre of gravity. It merely depends on the shape of the portion of the body under water. There are two methods of determining the metacenter position. In the first method, the centre of gravity is laterally shifted by a certain constant distance, , using an additional weight, causing the body to tilt. Further vertical shifting of the centre of gravity alters the heel angle . A stability gradient formed from the derivation / is then defined which decreases as the vertical centre of gravity position approaches the metacentre. If centre of gravity position and metacentre coincide, the stability gradient is equal to ero and the system is stable. This problem is easily solved graphically !see Fig. "#. The vertical centre of gravity position is plotted versus the stability gradient. A curve is drawn through the measured points and e$trapolated as far as it contacts the vertical a$is. The point of intersection with the vertical a$is locates the position of the metacentre.
#he metacentric height can also be evaluated theoretically using the follo!ing relationship
&1' !here and are metacenter center of gravity and center of buoyancy respectively. is the area moment of inertia of of the !aterline footprint of the body about its tilt a,is and is the submerged volume of the body.
Figure 3* /raphical determination of metacenter For the e,perimental setup of Fig 0 the rst step is to determine the position of the overall centre of gravity from the setting positions of the sliding !eights. #he horiontal position is referenced to the centre line
& 2'
#he vertical position is
&3'
Figure * 4osition and sie of sliding !eights (nd the stability gradient is
&'
3.0 Apparatus: The unit shown in Fig. % consists of a pontoon !%# and a water tan& as float vessel. The rectangular pontoon is fitted with a vertical sliding weight !'# which permits adjustment of the height of the centre of gravity and a horiontal sliding weight !(# that generates a defined tilting moment. The sliding weights can be fi$ed in any positions using &nurled screws. The positions !", )# of the sliding weights and the draught !*# of the pontoon can be measured using the scales. A heel indicator !+# is also available for measuring the heel angle.
Figure ) -ifferent parts of e$perimental apparatus.
Figure * Apparatus for stability of floating body of e$periment.
4.0 Procedure
%. oriontal sliding weight is set to position $ / 0cm. '. 1ertical sliding weight is moved to bottom position. (. The tan& is filled with water and floating body is inserted into it. ". 1ertical sliding weight is gradually raised and the tilting angle is noted down.
5.0 Result: Position of horizontal sliding weight x=8c !eight of "ertical sliding 3c %c weight z #c$ '0) '') Angle 2
&c
'(c
'5c
'4)
'8)
(3)
!orizontal Position od center of gra"it* +s=0.4% c !eight of "ertical 3c %c &c sliding weight z #c$ ,enter of gra"it* %.0c %.(c %.3c position zs #c$ Angle 0) ') ') 0 0.4% 0.4% d+sd2
'(c
'5c
%.4c
%.&c
() 0.(3
5) 0.0&(
8tability /radient vs 9ertical :enter of /ravity 4osition 7 5.6 5.5 5. 5.2 5 0.6 0.5 0.
9ertical :enter of gravity position &cm'
stability gradient
".0 #$S%&SS$O'
%
*
3hat will happen if the center of gravity and the center of buoyancy of a floating object are the same4 The object will stay at rest if no force act on the object as the centre of gravity and the center of buoyancy of a floating object are same. 5ut in moving air, there were a trouble that cause a little bit of thrust that will push the object down.
'
3hen and why will the floating object become unstable4 6ompare theoretical e$pectations to your lab observations and discuss any differences.
*
3hen the centre of the gravity of the object is vertically to the centre of buoyancy of floating object, the object will become unstable. The theoretical result was difference with e$periment result because it is due to common error such paralla$ error, human error and surrounding environment.
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(.0 %O'%)&S$O' In a conclusion this e,periment !as successfully carried out !hereas the data have been collected have a large dierence bet!een the value of metacentric height obtained by e,trapolation on a graph and the calculated metacentric height. In this e,periment !e had some common error !hich is paralla, error. #his error due to the e,periment !hen observer ta"e the reading of the depth of pontoon submerged in !ater and the angle tilt. )o!ever !e overcome this error !ith ma"e sure the observer eyes must be perpendicular to measurement to get the accurate result.
8.0 References
7%8 site.iugaa.edu.ps9t++('09files9/xperient'.pdf , :oogle earch , '<%* 7'8 https=99prei.com9s+cwch<$eyo>9stabilityoffloatingbody9 , :oogle earch , '<%* 7(8 http=99www.codecogs.com9library9engineering9fluid>mechanics9floating>bodies9stability andmetacentricheight.php , :oogle earch , '<%*
