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Hydrostatic forces on curved surfaces (a) Liquid above the surface Rule #1: The vertical component of pressure force on a curved surface is Magnitude: We are interested in computing the hydrostatic force equal to the weight of liquid vertically above the curved surface and acting on the curved surface shown in the above figure. Consider extending up to the free surface or its extension. the differential element in the above figure. Let us analyze the force acting on it. The line of action of this force is a long the center of mass of the fluid volume and act downward. Vertical Component: D C Horizontal Component: dFy dA dFx h dAy dAx dF Fy dAy h Fx h dAy dFx dF , P h Ay yAy A where C Ay is the horizontal projection of the curved surface onto a vertical CP Fx plane. B h y is the distance from the free surface to the centroid of Consider then the forces acting on the differential the projected area (Ay). element shown in the figure. Rule #2: dF PdA The horizontal component of pressure force on a curved surface is equal to the pressure force exerted on the horizontal projection of hdA the curved surface onto a vertical plane normal to the component. dFy hdAx The line of action is determined from the plane surface formula but hdAx d I Fy d ycp y yA Fy W where (acting downward ) I is the second moment of the projected area along the Centroidal axis. A is the projected area, (Ay). 1 above the curved surface and extending up to the free surface or its (b) Liquid below the surface extension. The line of action of this force is a long the center of mass of the To analyze this problem, we vertically project the curved surface imaginary fluid volume and acting upward. onto the free surface extension. Then we imagine that the volume ABCD is filled with an imaginary fluid similar to the real fluid. Horizontal Component: Consider then the differential element shown in the figure. The D C The magnitude and the line of action of the horizontal component is computed exactly as we did in Part A Imaginary fluid P, dF Final Note on the horizontal component dA h P, dF A Closed Surface B Curved surface Curved surface with equal elevation with different elevation pressure at this element point is the same in all direction as was ends ends proved earlier. Hence, The horizontal component of pressure force on a closed surface or P P a curved surface with equal elevation for both ends is always zero, since on opposite sides of the body the area-element projections multiply both sides by dA have opposite signs as shown on in the figures. However, if the ends of the curved surface do not have same dF P dA elevations, then there will be a net horizontal force. h dA where h is the distance from the free surface (or its extension) to the differential element. Now we have a problem similar to part a. Hence, i.e. Fy V W (acting upward ) Rule #3: The vertical component of pressure force due to liquid below a curved surface is equal to the weight of imaginary liquid vertically 2 Buoyancy forces Concept of Hydrometry Having obtained the necessary tools to analyze curved A hydrometer uses the principle of buoyant force to surfaces, we are now able to compute the buoyancy determine the specific gravities of liquids. force acting on a totally (or partially) submerged body. Q M B A C stem h D 0 We split the body into two surfaces, namely, ABC and ADC (due to liquid is above the former and below the FB latter) (Fy)ABC = fluid VABCMQ = VABCMQ Whydrometr (Fy)ADC = fluid VADCMQ = VADCMQ (Fy)net = fluid VABCD = VABCD If we immerse a hydrometer in a fluid then it gives us = weight of the displaced fluid. the reading S (specific gravity, = /water) stem h Astem Q M Q M WABCMQ WADCMQ B A force balance in the y direction gives: FB Whydrometer A C A C D B (0 stem ) Whydrometer A WABCD C (0 stem ) S water Whydrometer D Whydrometer The line of action is along the centroid of the displaced S volume of fluid. (0 stem ) water 3 Stability of Immersed Bodies A body is said to be in a stable equilibrium position if, when displaced, it returns to its Unstable Equilibrium Position equilibrium position. Conversely, it is in an unstable equilibrium position if, when displaced (even slightly), it moves to a new equilibrium If Center of gravity is above the centroid, the body position. is unstable. An overturning restoring moment will bring the body to a new equilibrium position. Stable Equilibrium Position If Center of gravity is below the centroid, the body is stable. A restoring moment will bring the body back to its original position. 4 Stability of Floating Bodies M G G C Righting moment Old position of C Fb the centroid New position of Fb the centroid New position of Center of grav ity M e t a c e n t e r ( M ) is d e fi ne d a s t he intersection point of the line of action of the G buoyant force before and after the tilt. Ov er-turning Moment M M etacenteric Hight (GM ) is defined as the distance from G to M. If M is above G then the body is stable otherwise it is not. Old position of C the centroid Fb 5