Laboratory Exercise in CENGR 3260 - Hydraulics
LABORATORY EXERCISE NO. 4
DETERMINATION OF THE CENTER OF GRAVITY OF THE
FLOATING BODY
I. INTRODUCTION
The stability of any vessel which is to float on water, such as a pontoon or ship, is of
paramount importance. The theory behind the ability of this vessel to remain upright must be
clearly understood at the design stage. Archimedes’ principle states that the buoyant force has
a magnitude equal to the weight of the fluid displaced by the body and is directed vertically
upward. The buoyant force is a force that results from a floating or submerged body in a fluid
which results from different pressures on the top and bottom of the object and acts through
the centroid of the displaced volume.
Theory
Consider a ship or pontoon floating as shown in Figure 1. The center of gravity of the
body is at G and the center of buoyancy is at B. For equilibrium, the weight of the floating
body is equal to the weight of the liquid it displaces, and the center of gravity of the body and
the centroid of the displaced liquid is in the same vertical line. The centroid of the displaced
liquid is called the "center of buoyancy". Let the body now be heeled through an angle 𝜃 as
shown in a subsequent figure, B1 will be the position of the center of buoyancy after
heeling. A vertical line through B1 will intersect the center line of the body at M and this
point is known as the metacenter of the body when an angle 𝜃 is diminishingly small. The
distance GM is known as the metacentric height. The force due to buoyancy acts vertically up
through B1 and is equal to W The weight of the body acts downwards through G.
Stability of submerged objects:
Stable equilibrium: when displaced, it returns to the equilibrium position.
• If the center of gravity is below the center of buoyancy, a righting moment will be
produced and the body will tend to return to its equilibrium position (Stable).
Unstable equilibrium: if when displaced it returns to a new equilibrium position.
• If the center of gravity is above the center of buoyancy, an overturning moment is
produced and the body is unstable.
Note: As the body is totally submerged, the shape of displaced fluid is not altered when the
body is tilted and so the center of buoyancy unchanged relative to the body.
Laboratory Exercise in CENGR 3260 - Hydraulics
Figure 1. An illustrative figure of a flat-bottomed pontoon
Figure 2. Centers of buoyancy of floating and submerged objects
Figure 3. Stability of submerged
objects
Stability of floating objects:
Metacenter point M: the point about which the body starts oscillating.
Metacentric height GM: is the distance between the center of gravity of the floating body and
the metacenter.
• If M lies above G a righting moment is produced, equilibrium is stable and GM
is regarded as positive.
• If M lies below G an overturning moment is produced, equilibrium is unstable and
GM is regarded as negative.
• If M coincides with G, the body is in neutral equilibrium.
Laboratory Exercise in CENGR 3260 - Hydraulics
Figure 4. Stability of floating objects
II. OBJECTIVE
•
To locate analytically the position of the center of gravity of the boat.
III. EQUIPMENT/APPARATUS
The equipment required to carry out this experiment is the following:
1. FME11 unit (Boat)
2. Ruler
Equipment Description
•
•
•
The unit consists of a floating methacrylate prismatic base, with a vertical mast placed
on it. An adjustable mobile mass has been added to alter the position of the center of
gravity.
A weight that can be horizontally and vertically displaced allows for modification of
the floating base heel.
A plumb bob, attached to the upper part of the mast, is used to measure the angle of
heel of the floating base with the aid of a graduated scale.
Maximum angle: +/- 13°
Corresponding linear dimension: +/- 90 mm
Dimension of the float:
Length: 353 mm
Width: 204 mm
Total height: 475 mm
Laboratory Exercise in CENGR 3260 - Hydraulics
Figure 5. FME 11 Metacentric Height
IV. EXPERIMENTAL PROCEDURE
1. Disassemble the vertical weight of the unit.
2. Disassemble the horizontal weight of the unit.
3. Measure the dimensions of these elements using Figure 5 as your
guide: Base
Front
Rear
Right
side
Left
Horizontal
axis
Vertical
axis Rule
Support of the axis
4. Measure the distance from the geometrical center of each one of the previous
elements to the abscissa axis.
5. Measure the distance from the geometrical center of each one of the previous
elements to the ordinate axis.
6. Calculate the volume of each
element. Volume =
Length*Width*Height
7. Calculate the weight of each element using the given density for each type of
material. Weight = Density * Volume
Laboratory Exercise in CENGR 3260 - Hydraulics
8. Measure the values of Xg and Yg of each element using the given axis in the front
side of the unit (Refer to Figure 6).
9. Calculate the center of gravity of the unit following the principle of moment of
areas. Use the formula given in computing for Xg and Yg.
Figure 6. Schematic diagram of the boat for the determination of its center of gravity.
V. CALCULATIONS AND RESULTS
Element
Base
Front Side
Rear Side
Right
Side
Left Side
Ruler
Support
of the axis
Density
[kg/m3]
1190
1190
1190
1190
1190
1395
1395
Length
[cm]
34.3
19.4
19.4
Width
[cm]
20.4
0.5
0.5
Height
[cm]
0.5
7.3
7.3
34.3
0.5
7.3
34.3
19.3
0.5
2.3
7.3
1.5
3
1.9
2.3
Volume
[m3]
3.49 ∗ 10−4
7.08 ∗ 10−5
7.08 ∗ 10−5
1.25 ∗ 10−4
1.25 ∗ 10−4
7.53 ∗ 10−5
1.43 ∗ 10−5
Weight
[kg]
0.416
0.084
0.084
0.145
Xg
[cm]
17.15
9.7
9.7
Yg
[cm]
10.2
0.25
0.25
17.15
0.25
0.145
0.105
17.15
9.65
0.25
1.3
0.020
1.5
0.95
Laboratory Exercise in CENGR 3260 - Hydraulics
Element
Vertical
Axis
Horizontal
Axis
Density
[kg/m3]
7850
Diameter
[cm]
Length
[cm]
0.8
40.9
0.8
19.8
7850
Volume
[m3]
Weight
[kg]
Xg
[cm]
Yg
[cm]
2.06 ∗ 10−5 0.16138
0.4
20.45
9.95 ∗ 10−6 0.07813
0.4
9.9
COMPUTATION:
Element
Base
Volume [m3]
34.3 ∗ 20.4 ∗ 0.5
Front Side
10003
19.4 ∗ 7.3 ∗ 0.5
Rear Side
10003
19.4 ∗ 7.3 ∗ 0.5
Right
Side
Left Side
Ruler
Support of
the axis
Element
Base
Front Side
Rear Side
Right
Side
Left Side
Ruler
Support of
the axis
Element
Vertical
Axis
Horizontal
Axis
10003
34.3 ∗ 7.3 ∗ 0.5
10003
34.3 ∗ 7.3 ∗ 0.5
10003
34.3 ∗ 2.6 ∗ 1.5
10003
3 ∗ 1.9 ∗ 2.3
3
1000
= 3.49 ∗ 10−4
= 7.08 ∗ 10−5
= 7.08 ∗ 10−5
= 1.25 ∗ 10−4
= 1.25 ∗ 10−4
= 7.53 ∗ 10−5
= 1.43 ∗ 10−5
Weight [kg]
3.49 ∗ 10−4 * 1190 = 0.416
7.08 ∗ 10−5 * 1190 = 0.084
7.08 ∗ 10−5 * 1190 = 0.084
1.25 ∗ 10−4 * 1190 = 0.145
1.25 ∗ 10−4 * 1190 = 0.145
7.53 ∗ 10−5 * 1395 = 0.105
1.43 ∗ 10−5 * 1395 = 0.020
Xg [cm]
34.3 * 0.5 = 17.15
19.4 * 0.5 = 9.7
19.4 * 0.5 = 9.7
34.3 * 0.5 = 17.15
Yg [cm]
20.4 * 0.5 = 10.6
0.5 * 0.5 = 0.25
0.5 * 0.5 = 0.25
0.5 * 0.5 = 0.25
34.3 * 0.5 = 17.15
19.3 * 0.5 = 9.65
3 * 0.5 = 1.5
0.5 * 0.5 = 0.25
2.3 * 0.5 = 1.3
1.9 * 0.5 = 0.95
Volume [m3]
Weight [kg]
0.08
𝜋 ∗ ( 100 )^2 40.9
∗
= 2.06 ∗ 10−5
4
100
0.08
𝜋 ∗ ( 100 )^2 19.8
∗
= 9.95 ∗ 10−6
4
100
2.06 ∗ 10−5 * 7850 = 0.16138
9.95 ∗ 10−6 ∗ 7850 = 0.07813
Laboratory Exercise in CENGR 3260 - Hydraulics
Element
Vertical
Axis
Horizontal
Axis
Xg [cm]
Yg [cm]
0.8 * 0.5 = 0.4
40.9 * 0.5 = 20.45
0.8 * 0.5 = 0.4
19.80 * 0.5 = 9.9
Using the formula above the X is 0.12045856m and the Y is 0.0688925
VI. DISCUSSION OF RESULTS
In this activity we must determine the metacentric height is important while
investigating the stability of the Floating bodies such as ship, during the design phase by
theoretical computations and after the ship have been built by inclining experiments. When a
body is partially or fully submerged in fluid, it is forced upward by a vertical force known as
buoyant force. Archimedes principle states that the buoyant force has a magnitude equal to
the weight of the fluid displaced by the body and is directed vertically upward. Buoyant
force is a force that results from a floating or submerged body in a fluid which results from
different pressures on the top and bottom of the object and acts through the centroid of the
displaced volume. The buoyant thrust on a body of weight W and centroid G acts through
the centroid of the displaced fluid volume and this point of application of the buoyant force
is known as the center of buoyancy (B) of the body. For the body in equilibrium, the weight
W must equal the buoyant thrust FB, both acting along the same vertical line. For small
angle of heel, the intersection point of the vertical through the new center of buoyancy B,
and the Line BG produced is known as the metacenter, M, and the body thus disturbed tend
to oscillate about M. The distance between point G and M is the metacentric height (GM). If
the floating body is bottom-heavy and thus the center of gravity G is directly below the
center of buoyancy B, the body is always stable. But unlike immersed bodies, a floating
body may still be stable when G is directly above B. This is because the centroid of the
displaced volume shifts to the side to a point B+ during a rotational disturbance while the
center of gravity G of the body remains unchanged. If point B+ is sufficiently far, these two
forces create a restoring moment and return the body to the original position. A measure of
stability for floating bodies is the metacentric height GM, which is the distance between the
center of gravity G and the metacenter M the intersection point of the lines of action of the
buoyant force through the body before and after rotation. The metacenter may be considered
to be a fixed point for most hull shapes for small rolling angles up to about 20°. A floating
Laboratory Exercise in CENGR 3260 - Hydraulics
body is stable if point M is above point G and thus GM is positive, and unstable if point M is
below point G, and thus GM is negative. In the latter case, the weight and the buoyant force
acting on the tilted body generate an overturning moment instead of a restoring moment,
causing the body to capsize. The length of the metacentric height GM above G is a measure
of the stability: the larger it is, the more stable is the floating body.
VII. CONCLUSION
We corroborated the stability of the floating body theory through the experimental
determination of the metacentric height of a floating body in water. Considering the stability
of the pontoon, buoyancy works according to Archimedes principle, the upward force on an
object that’s fully or partially immersed in a fluid is equal to the weight of the fluid that it
displaces. If the fluid changes in the experiment and has a constant density, the object is
submerged by a fixed amount, the buoyancy force will be proportional to the density of the
fluid been tested. One of the reasons that can explain why the relative error exceeds 5% can
be the fact that we showed difficulty measuring the center of gravity with the jockey
attached, as the pontoon was hanging, and it was tedious to measure the height of center of
gravity in that position. Also, in some situations during the experiment the fluting body
oscillated from stable to unstable once the jockey was moved from positions. Another reason
that could be is the fact that the movement of the plumb – bob dose in fact affect the angular
scale read and can directly affect to the results if not read properly. There is a lot of room to
generate personal errors when reading a measurement, as the pontoon was floating in water
and the plumb that gave the angle of tilt oscillated a considerable amount. We conclude that
a floating body is stable to small angular displacements around a horizontal axis that is in the
plane whenever its metacentric height is positive, that is, GM> 0 is provided that the
metacenter is above the center of gravity. When the metacenter is below GM < 0, it is
unstable. In this situation we have already shown that a floating body is unconditionally
stable for small vertical displacements and it is also quite obvious that this body is neutrally
stable both for horizontal displacements and for angular displacements around a vertical axis
that passes through its center of gravity. It follows that a necessary and sufficient condition
for the stability of a floating body to a small general perturbation composed of arbitrary
linear and angular components is that its metacentric height is positive for angular
displacements around any horizontal axis.
Laboratory Exercise in CENGR 3260 - Hydraulics
VIII. DOCUMENTATION