CIEG-306 Fluid Mechanics Laboratory
2. Stability of a Floating Body
The objectives of this experiment are:
1. to measure the angle of inclination at which an eccentrically loaded body floats, 2. to observe the circumstances under which a floating body is unstable, and 3. to compare the observed results with theoretical predictions. Apparatus
The apparatus consists of an open plastic box (‘barge’) which floats in water and carries a mast (Figure 1). A plumb-bob suspended from the mast provides a means of measuring the angle of inclination of the barge. The vertical position of the center of gravity is controlled by a weight Wv which may be moved to different heights on the mast. The horizontal position of the center of gravity is controlled by a second weight Wh which may be moved to different horizontal positions on the barge. The following information is necessary: length of barge L= 34.9 cm., width of barge b = 20.3 cm., vertically moving weight Wv = 2.79 N, horizontally moving weight Wh = 3.11 N, total weight of assembled apparatus W = 13.21 N. For the barge without the weights, the vertical position of the center of gravity, zb, is 5.2 cm. from the outer bottom of the barge. In the following all z distances are measured from the outer bottom of the barge.
Figure 1: A schematic plot of the barge and experimental apparatus 1
The barge in the inclination test is stable and the purpose is to determine the relationship between load that brings the barge to tilt and the angle of the tilt. The theory behind the inclination test goes as follows. A floating body shall experience net vertical buoyancy force B (upward) that balance with its weight W (downward), i.e., B=W. The weight of the floating body W acts through its center of gravity (xG, zG) while the buoyancy force B acts through the centriod of the displaced volume (called buoyancy center). When the barge is inclined at an angle , the balance of moments about the origin O (see Figure 2) requires (xB cos θ + zB sin θ) – (xG cos θ + zG sin θ) = 0
In the above derivation, we have used the condition that the buoyant force B pushing up on the barge is equal to the total weight W, as is required for a floating body. The position of the center of buoyancy can be derived from the geometry of the submerged part of the barge:
where d is the submerged depth of the barge which can be evaluated as d = W/(ρgLb), with ρ is the water density and g is the gravitational acceleration. For given d and b, xB and zB depend on the angle only.
The gravity center, with weights Wv and Wh are found from the locations of the weights
where Wb = the barge weight given by Wb = (W – Wv – Wh). (xh,zh), (xv,zv) and (xb,zb) are respectively the coordinates for the location of the center of gravity of the horizontal weight, the vertical weight, and the barge. In Equation (3a), we have specified xb = 0 and xv = 0 appropriate for this experiment. The values of zv, xh and zh are determined by the locations of the weights, and zb = 5.2 cm as stated above. Substitution of Equation (2) into Equation (1) yields the following 3rd order algebraic equation for tan:
Figure 2: Definition of the coordinate system, , (xG, zG) and (xB, zB).
This test aims to verify the stability criterion that can be derived from the theory above. The stability criterion for the barge may be described as follows. First, with the barge in horizontal position, a small tilt of is provided. The weight of the floating body provides a rotating moment and tend to destabilize the system. On the other hand, the buoyancy force provides a counter moment that tends to stabilize the system. If Eq. (1) is positive for a small value of , then the buoyancy force acting on the barge due to the tilt can bring the barge back to an upright position, implying the barge is stable. However, if zG becomes too large, the gravity force acting on the barge rotates...
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