Experiment 4
Buoyancy
1.0
Objective
i) To i) To obtain the value of metacentric height by the Buoyancy floating study
2.0
Introduction When a body is immersed in a fluid partially or fully, an upward force which tends to
lift or float the body is subjected. This tendency of a body to immerse or float in fluid is called buoyancy. There is a force exerted by the fluid on the floating body against the gravitational force and it is known by ‘buoyant force’. In the third century B.C., Archimedes formulated the well-known principle of floating body which is used to determine the magnitude of the buoyant force. Base on Archimedes principle, it states that buoyant force is equal weight of the liquid displaced by the body and which is an upward force that is opposite direction to the force of gravity. For metacentric height, is defined as the distance between the meta-center of a floating body and the center of gravity of the body. For meta-center, it is defined as a point, which a body is start to oscillating when this body is tilted by an angle. Meta-center can be also defined as the point at which the line of action of the force of buoyancy will intersect the normal axis of the body when the body is tilted with a small angle.
3.0
Methodology 3.1
Apparatus and Materials i)
Pontoon
ii)
Adjustable weight weight
iii)
Plumb weight
iv)
String
v)
Red container with water
3.2
Experimental Procedures
1.
A center line that goes through the center hole of the pontoon to the 0
0
is
drawn with a white board marker pen. 2.
The red container is filled with water. The pontoon is let to be float on water.
3.
The adjustable weight is placed at the 2nd bottom row. (135mm away from the base)
4.
The plumb weight is ensured to point at the zero reading on the angular scale.
5.
The adjustable weight is then moved in a stop across the width of the pontoon, the corresponding angle of tilt is recorded each a step. Tip of the scale for the angle reading and the sign convention for the reading should be observed carefully to obtain accurate reading.
6.
For the other three pontoons rows above, step 5 is repeated. The readings are recorded into the table provided.
4.0
7.
A graph of θ against x 1 is plotted. Gradient of all the cases are computed.
8.
All the table provided are computed.
Result Data analysis Weight of pontoon (excluding adjustable weight) (Wp)
= 1.280 kg
Adjustable weight (ω)
= 0.219 kg
Total weight, W = Wp + ω
= 1.499 kg
Breadth of pontoon (D)
= 0.220m
Length of Pontoon (L)
= 0.305m
Second moment of area, I = LD 3/12
= 2.706x10 -4 m4
Pontoon displacement, V = W/ρ
= 1.499x10 -3 m3
Area of pontoon in water, A = LD
= 0.0671 m 2
Depth of immersion, OC = V/A
= 0.0223m
Height of centre to centre of buoyancy =OB=BC=OC/2 ρ = density of water
``
= 0.0112 m = 1000 kg/m 3
Table 1: y1 and OG of the experiment. Height
of
adjustable
weight from base plate
305
260
200
135
80
99
90
81
64
58
(mm) Center
of
gravity
(OG)
(mm)
Table 2: Result of data obtained. of List (θ°) of Angle Adjustable For adjustable weight lateral displacement from sail center line , X 1 (mm) Height
Weight, y1 (mm)
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
60
70
135
-10
-9
-7
-5
-3
-2
0
2
4
5
7
9
10
11
-10
-8
-6
-4
-3
0
3
4
6
7
265
-7
-5
-3
0
2
3
4
6
315
-8
-5
-3
0
3
200
Table 3: Result from graph.
Height
of
Height
of
G
Adjustable
above
Water
weight, y1 (mm)
Surface,
Meta-centric dx1/dθ (mm/rad)
Height,
GM
BM (mm)
(mm)
OG
(mm)
135
64
9.73
1.42
52.8
80
200
81
51.30
7.49
69.8
265
90
53.02
7.75
78.8
315
99
36.21
5.29
87.8
dx1/dθ (mm/rad) = (Gradient of the graph) -1 x 180o / π
Meta-centric height, GM =
BM = BG + GM = OG - OB + GM BM = BG = OG - OB
Graph 1: θ° vs X 1 θ° against X1 80 y = 5.8894x - 0.0481
60 Series1 40
y = 1.5822x - 0.0685 y = 1.0806x - 1.0806
20
-10
-5
0
y = 1.1197x - 0.2042
5
10
° θ
e l g n 15 -20 A
0
-15
Series2
-40
Series3 Series4 Linear (Series1) Linear (Series2) Linear (Series3)
-60
x1(mm)
-80
Linear (Series4)
Line equation for series 1 : y = 1.1197x – 0.2042 Line equation for series 2 : y = 1.0806x – 1.0806 Line equation for series 3 : y = 1.5822x – 0.0685 Line equation for series 4 : 7 = 5.8894x – 0.0481
Calculation I = LD3/12 = [(0.305m) (0.220m) 3] / 12 = 2.706x10 -4 m4 V = W/ρ = 1.499kg / 1000 kg/m 3 = 1.499x10 -3 m3 A = LD = (0.305m) (0.220m) = 0.0671 m 2 OC = V/A = 1.499x10 -3 m3 / 0.0671 m 2 = 0.0223 m OB =BC = OC/2
``
= 0.0223 m / 2 = 0.0112 m
dx1/dθ = (Gradient of the graph) -1 x 180o / π = (5.8894mm) -1 x 180o / π = 9.73 mm/rad
GM = = (0.219/1.499) kg x 9.73 mm/rad = 1.42 mm BM = BG = OG - OB
= 64 mm – (0.0112x10 3) mm = 52.8 mm
5.0
Discussion Based on the result and graph we get through this experiment, for the 315mm height
of the adjustable weight (y 1), we get 5.29mm of the metacentric height (GM), and 87.8mm of BM. This show that larger the value of height of the adjustable weight (y 1), it yields larger value of metacentric height (GM) and BM. Metacentric height (GM) is the measurement of the initial static stability, and it is calculated by the distance between the centre of gravity of the body and its metacenter. Higher value of metacentric height will implies greater stability against overturning. We get different result when compare to other group’s result. When we are carrying out the experiment, due to our parallax error, when we observed the reading of the plumb weight. Besides, due to the environment which is windy in the laboratory, the plumb weight is keep on shaking and cannot stop so we are not able to get the result accurately. The state of equilibrium of a body that is fully or partially immersed in a liquid is defined as floating. The theory of floating is concerned by determining the equilibrium position of the body immersed in water and the condition of stability of the equilibrium. The basic concept of Archimedes concept about the floating are the displacement of a body is the weight of the liquid that displaced by the body and this weight is equal to the weight of body. The vertically downward force which is the weight is balanced by the vertically upward force which is the buoyant force. (Zhukovskii, 1952) It is important to determine the stability of floatation for the ship building. Builder of ship have to determine how far the ship can bear with the inclined force. It is an obvious requirement that a floating body such as boat that does not topple when slightly disturbed. (Aerospace, Mechanical & Mechatronic Engg. 2005). Ships still can be in equilibrium state during the inclined movement as the volume immersed on the right hand side of the ship increase while that part on left hand side is decrease. Therefore the centre of buoyancy moves towards the right to its new position. This is the importance of determining the stability of the floating body.
6.0
Conclusion
At last of the experiment, we had achieved our objective that the value of metacentric height is obtained. Metacentric height for each height of adjustable weight (y 1) 135mm, 200mm, 265mm, 315mm are 1.42mm, 7.49mm, 7.75mm, 5.29mm.
7.0
References i)
Aerospace,
Mechanical
&
Mechatronic
Engg.
2005
(http://www-
mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/statics/node2 4.html) ii)
Zhukovskii,
N.
E.
1952,
Floating
of
Body
(http://encyclopedia2.thefreedictionary.com/Floating+of+Bodies ) iii)
DAS.
M.
M.,
2008,
FLUID
MECHANIC
AND
TURBO
MACHINE
(http://books.google.com.my/books?id=EY87AER8f4C&pg=PA63&dq=metacentric+buoyancy&hl=en&sa=X&ei=zksIU7HHAoTOiAez3YCAA g&redir_esc=y#v=onepage&q=metacentric%20buoyancy&f=false ) iv) Bansal. R. K., 2005, A text Book of Fluid Mechanic and Hydraulic Machines (http://books.google.com.my/books?id=nCnifcUdNp4C&pg=PA128&dq=metacentric+buo yancy&hl=en&sa=X&ei=JVMIU_joHcubiQf104HoBQ&redir_esc=y#v=onepage&q=metac entric%20buoyancy&f=false )
