CHAPTER 02PRESSURE & Fluid statics 1 PRESSURE 1. Defined as a normal force exerted by a fluid per unit area P = F/A 2. The pressure at any point in a fluid is the same in all direction. 3. The only forces between adjacent surface are pressure forces normal to the surfaces. 4. There are no shearing stresses present in a fluid at rest 2 Cont. PRESSURE A C B Acosθ Asinθ P2 P3 P1 A θ Consider a small triangular element ABC at a point in a fluid, if area AC is A, then area AB=Asinθ and area BC=Acosθ. For horizontal equilibrium, For vertical equilibrium, Thus, the pressure at any point in a fluid is the same in all direction. 2 1 2 1 sin sin p p A p A p = × = × u u 3 1 2 1 cos cos p p A p A p = × = × u u 3 Pressure increases with depth: Relationship Between Pressure and Depth Cont. PRESSURE z g P A = A µ and 0 1 1 P gz P + = µ 0 2 2 P gz P + = µ So, we obtain: 4 Cont. PRESSURE Pressure At a Point 5 Cont. PRESSURE 6 Cont. PRESSURE 7 1. The pressure at a point within a fluid mass will be designated as either an absolute pressure or a gage pressure. 2. Absolute pressure is measured relative to a perfect vacuum (absolute zero pressure), where as gage pressure is measured relative to the local atmospheric pressure. Cont. PRESSURE 8 Cont. PRESSURE 9 A barometer is a scientific instrument used to measure atmospheric pressure. It can measure the pressure exerted by the atmosphere by using mercury. (using water might be difficult) BAROMETER 10 ~ One of the apparatus we use to measure the pressure in a tank. ~ Mainly consist of glass or plastic U-Tube (different fluid : oil, mercury, water). MANOMETER C B P P = ρ fluid ρ 1, fluid 1 A B C h 1 h 2 D ρ 2, fluid 2 Air 1 1 gh P P A B µ + = 2 2 gh P P D C µ + = Air 11 A pressure gage connected to a chamber reads 65kPa at a location where the atmospheric pressure is 1.013 bar. Determine the absolute pressure in the chamber. If the pressure give reading -10 kPa. Also determine the absolute pressure EXAMPLE 01 12 Figure show a manometer with an angle of 20 0 that used to measure air pressure in a closed tank. The specific gravity of manometric fluid that had been used in this type of manometer is 0.75. If atmospheric pressure is 200kPa, find the absolute pressure in closed tank in unit of kPa. [ g = 9.81 m/s 2 ] EXAMPLE 02 A In gas, pressure is equal regardless of elevation except for big difference (hundred meters) of elevation 13 BUOYANCY The difference between two forces, F B (upward) and F T (downward) is a net upward force called Buoyant Force. The buoyant force is caused by the increase of pressure in a fluid with depth. ( ) B Bottom Top f f f f F F F g s h A ghA ghA gV µ µ µ µ = ÷ = + ÷ = = 14 STABILITY A floating body is: • Stable if the center of gravity G is below the centroid B of the body or if the metacenter M is above point G. • Unstable if the point M is below point G. B G 15 A crane is used to lower weights into the sea (density 1025 kg/m 3 ) for an underwater construction project. Determine the tension in the rope of the crane due to a rectangular 0.4m x 0.4m x 3m concrete block (density 2300 kg/m 3 ) when it is (a) suspended in the air and (b) completely immersed in water. EXAMPLE 03 (a) (b) 16 HYDROSTATIC FORCES SUBMERGED PLANE SURFACES SUBMERGED CURVED SURFACES Fluid static (hydrostatic forces) is used to determine the force acting on submerged bodies. It is necessary to calculate the magnitudes and locations of forces that act on plane and curved surfaces. 17 SUBMERGED PLANE SURFACES A gh F c µ = u sin c c y h = c xx c p Ay I y y + = 18 SUBMERGED PLANE SURFACES 19 SUBMERGED PLANE SURFACES Area and Moments of Inertia for Some Common Geometries 20 EXAMPLE 04 1.2m wide and 2m high rectangular gate shown below is laying underwater at angle of 30° with the horizontal water surface. Determine the force F acting on the gate and the vertical center of pressure, h P 21 30º y C y P h C h P 8.6 m Gate Water 2 m EXAMPLE 05 2.0m wide rectangular gate shown in figure is hinged at B and making an angle of 45° with the horizontal. Determine the minimum force F required to open the water gate which act on the gate center. 2m 5m 22 SUBMERGED CURVED SURFACES A h g Fh ÷ = µ gV Fv µ = 2 2 V H R F F F + = | | . | \ | = ÷ H V F F 1 tan u 23 SUBMERGED CURVED SURFACES 24 EXAMPLE 06 Determine the forces acting on a curved surface. i) The horizontal force, F H ii) The vertical force, F V and iii) Resultant force, F R Width=2m 25 EXAMPLE 07 Determine the forces acting on a curved surface. i) The horizontal force, F H ii) The vertical force, F V and iii) Resultant force, F R Width=1m 4 2 x y = 26