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