2.6 Force reacts with planar object in fluid

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1 2.6 Force reacts with planar object in fluid Fluid surface Specific weight (γ) => Object sinks in fluid => C is center of gravity or Centroid => P is center of pressure (always under C) => x axis is cross point of fluid surface and upper planar object => y is sider of planar object => 0 is the origin

2 Fluid surface Specific weight (γ) h = depth of small area d A h C = depth of object centroid h P = depth of center of pressure x = length from small area (d A ) in x axis x C = length from centroid in x axis x P = length from center of pressure in x axis y = length from small area (d A ) in y axis y C = length from centroid in y axis y P = length from center of pressure in y axis d x = width of small area in x axis d y = width of small area in y axis d A = area of small area on planar object surface d F = small force reacts with planar object F = Total force reacts with planar object

3 2.6.1 How to determine force From P = F ; F = P A A γ ysinθ is a constant then F = γsinθʃ y d A = γsinθ y C A And P = γh then But h C = y C sinθ F = γha Consider on small force (d F ) which reacts on small area (d A ) d F = γhd A But h = ysinθ then d F = γ ysinθ d A Integrate d F to determine the force of whole object F =ʃ γ ysinθ d A Finally F = γh C A F = force reacts with planar object (N) γ = specific weight of fluid (N/m 3 ) h C = centroid depth (m) A = cross section area of planar object (m 2 ) *** This force does not depend on slope angle, but it depends on the depth of centroid***

4 2.6.2 How to determine the position of force In air, force will react at center of gravity or centroid, but in fluid, it has pressure reacting to planar object. Therefore, force will react at center of pressure. It can consider from moment around x axis. Moment = Force x Perpendicular length of its force

5 Fluid surface Specific weight (γ)

6 F. y P is equal to summation of small force (d F ). y F. y p = ʃyd F But F = γh C A and d F = γ ysinθ d A then γhca. y p = ʃy γ ysinθ d A = ʃy 2 γ sinθ d A But γ sinθ is a constant then γhca. y p = γ sinθ ʃy 2 d A But h C = y C sinθ then y C sinθ.a. y p = sinθ ʃy 2 d A y C.A. y p = ʃy 2 d A y P = ʃy2 d A y C.A ʃy 2 d A is moment of inertia or called second moment of area around x axis which is Ay 2 C+I C then y P = Ay C 2 +IC y C.A y P = Ay C 2 y C.A + y P = y C + I C y C.A I C y C.A y P = length from center of pressure in y axis or slope (position of action force) (m) y C = length from centroid in y axis (m) I C = moment of inertia of area (m 4 ) A = cross section area of planar object (m 2 ) ***It can use for object which sinks in plumb line.*** h C = y C sin90⁰; h C = y C then h P = y P (sin90⁰=1)

7 Moment of inertia of each plane b A = bh y C yc = h 2 I C h I C = bh2 12 Rectangle A = bh 2 I C h y C = h 3 Triangle b y C I C = bh3 36

8 I C b h y C A = πbh 4 y C = h 2 Ellipse I C = πbh3 64 I C d y C A = πd2 4 y C = d 2 Circle I C = πd4 64

9 2.7 Force reacts with curve object in fluid => This force can be calculated by integration of small area, but it is very complicated. => For convenience, it needs to separate force into x axis (Fx) and y axis (Fy)

10

11 From P = F ; F = P A A Consider on small force (d F ) which reacts on small area (d A ) F x =γh C A X When A x is Projection area d F = P.d A Visualize df to x component. We got d Fx =d F sinθ d Fx =P.d A sinθ Projection area of d A but P = γh then d Fx =γhd A sinθ Integrate to summation forces in x axis F x =γʃhd A sinθ When ʃhd A sinθ is first moment of area of the projection area = h C A X Position of force in x axis y P = y C + I C y C.Ax I C is moment of inertia of projection area

12 Visualize d F to y-component. We got d Fy = d F cosθ d Fy = P.d A cosθ but P = γh Thus d Fy = γh.d A cosθ Integrate to summation all forces in y-component F y = γʃh.d A cosθ but ʃh.d A cosθ = volume of fluid above d A = ʃd V Thus F y = γʃdv = γv When Thus F y = W W = weight of fluid in volume above curve of object to fluid surface F y = down direction when force when fluid above curve object F y = up direction force when fluid under curve object Position of F y always will pass the center of fluid volume above the curve object. Resultant force reacts with curve object F = F x 2 + F y 2 From γ = W V γv = W Position of resultant pressure react with curve object θ = tan 1 F x F y

13 2.8 Buoyant Force Buoyancy Force - Object which sinks or floats in fluid will has the reacted force at the upward direction. - The reacted force will equal to the object weight. - The Archimedes principle => the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces. F y = W = γv Thus F B = W When F B = Buoyant force (N) W = γv = Weight of fluid (N)

14 When F B = W therefore ρ object = ρ fluid W > F B => object will sink, it will sink until W object = W fluid or fluid ground W< F B => object will float, it will float until W object = W fluid or fluid surface Object will compensate the fluid volume that equal to object weight

15 2.8.2 Hydrometer Hydrometer => Use for measure specific gravity (S) => Use buoyancy force principle F B = W = γv When W = hydrometer weight (W) γ = specific weight of fluid (N/m 3 ) V = volume of liquid (m 3 )

16 If hydrometer has specific weight (γ) = 1 (water) Fluid level = 1 Fluid = water Fluid level > 1 Fluid lighter than water Fluid level < 1 Fluid heavier than water

17 Value of h can be found by h = V W(S 1) S. A When h = length from fluid to scale 1 (m) V W = volume of liquid is replaced (m 3 ) S = specific gravity of fluid (no unit) A = cross section area of glass tube (m 2 ) h = 0 ; S = 1 h = - ; S < 1 h = + ; S > 1 => fluid = water => fluid = easy sink fluid => fluid = hard sink fluid

18 2.8.3 The of object in fluid The 3 things that need to concern about the balance of object in fluid 1. Center of gravity; C 2. Center of buoyancy; B 3. Meta center; M

19 The balance of object that sinks in fluid Fluid surface - Center of buoyancy (B) > Center of gravity (C) B C F B W C B W F B - Stable Equilibrium - Hanging bridge, Rocking doll Revert moment

20 Fluid surface - Center of gravity (C) > Center of buoyancy (B) C B W B C W - Unstable Equilibrium F B F B Turnover moment - Wine glass

21 Fluid surface - Center of gravity (C) = Center of buoyancy (B) C W B C W B - Neutral Equilibrium F B F B - Rolling cylinder, Rolling bottle

22 The balance of object that floats in fluid - The object floats in fluid surface F B = W C - F B and W are in the same vertical - C is higher than B - When object rolls or shakes, the volume of part of object that sinks in fluid is still the same.

23 Axis of Symmetry - When the object is rolling, B will change to B. F B and W are not in the same vertical. - F B will cross to the axis of symmetry. The cross point called Meta center (M) (M at B) - If M higher than C => Revert moment - If M lower C => turnover moment - Length between M to C called Metacenter height

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