Tiêu đề 5 Buoyancy and Stability.pdf ✅

HGE 5: Buoyancy and Stability 1. Buoyancy Archimedes’ principle states that when a body is placed in a static fluid, it is buoyed up by a force that is equal to the weight of the fluid that is displaced by the body. BF = F1 − F2 BF = γLV1 −γLV2 BF = γL (V1 − V2 ) BF = γLVs 2. Stability of Floating Bodies The forces acting on a floating body are its own weight and buoyant forces. The point about which the buoyant forces form an overturning or righting moment is called the metacenter. It is found at the point of intersection of the original and final lines of action of the buoyant force. To locate the metacenter, equate the moment of the buoyant force to the moment of the shifted wedge about the center of buoyancy. BF z = Fs γVs (MB0 sin θ) = γvs MB0 = vs Vs sin θ For a rectangular object, v = 1 2 ( B 2 ) ( B 2 tan θ) (L) Also from the figure, s = 2 3 B cos θ
Therefore, MB0 = 1 2 ( B 2 ) ( B 2 tan θ) (L) ( 2 3 B cos θ) Vs sin θ MB0 = 1 12 LB 3 Vs But 1 12 LB 3 is the moment of inertia at the waterline about the axis of rotation. Thus, MB0 = I Vs For a rectangular body, MB0 = 1 12 LB 3 LBD MB0 = B 2 12D If tilted by an angle θ, MB0 = B 2 12D (1+ tan2 θ 2 ) These values of MB0 indicate the distance between the center of buoyancy and the metacenter. To locate the metacenter from the center of gravity, called the metacentric height, MG = MB0 ± GB0 3. Righting and Overturning Moments If the metacenter is above the center of gravity, buoyant forces will cause the body to stabilize. RM = BF(MG sin θ)