Chapter10

# Chapter10 - Chapter 10 Fluids 10.1 10.2 10.2 10.3 10.4 10.5...

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Chapter 10 Fluids 10.1 Phases of Matter 10.2 Static Equilibrium in Fluids: Pressure and Depth 10.2 Pascal’s Principle 10.3 Archimedes’ Principle and Buoyancy 10.4 Continuity of Fluid Flow 10.5 Bernoulli’s Equation 10.6 Surface Tension 10.7 Pressure Difference across a Curved Surface 10.8 Capillary Rise Formula 10.1 Phases of Matter The three common phases or states of matter are solid, liquid, and gas. A solid maintains a fixed shape and a fixed size; and it is relatively incompressible. Like a solid, a liquid is not readily compressible but it does not maintain a fixed shape. And a gas has neither a fixed shape nor a fixed volume. Since liquids and gases do not maintain a fixed shape, they both have the ability to flow; they are thus often referred to collectively as fluids. 10.2 Static Equilibrium in Fluids: Pressure and Depth Pressure is a useful concept for dealing with fluids. The pressure exerted by a fluid is defined as the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts: A F P = SI unit: Pascal (Pa) where 1 Pa = 1 N/m 2 Experimental observation shows that a fluid can exert pressure in any direction. At any point of a given depth in a fluid at rest, the pressure is the same in all directions. If this were not true, the fluid would flow. Moreover, a fluid at rest cannot produce a force parallel to a surface. Otherwise the surface would apply a reaction force to the fluid resulting in flow of the fluid. The force generated by the pressure of a static fluid is always perpendicular to the surface that the fluid contacts. 1

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Let's study how the pressure varies with the depth in a static fluid of uniform density. Consider a column of fluid with cross-sectional area A and height h in a container filled with a fluid of density ρ (see the figure). On the top face, the fluid pressure P 1 generates a downward force of magnitude P 1 A . Similarly, the fluid pressure P 2 generates an upward force of magnitude P 2 A on the bottom face. Because the fluid is at rest, the column must be in equilibrium. Therefore, mg A P A P mg A P A P + = = - - 1 2 1 2 0 where mg is the weight of the fluid within the column. Substituting Ah m ρ = in the above formula, we obtain. gh P P ρ + = 1 2 Note that we have assumed the density ρ to be the same at any vertical distance h (i.e. the fluid is incompressible) in determining the pressure increment ρ gh . Remarks: When you dive in the water, some small air bubbles are probably let out from your mouth. You may note that diameter of the bubble increases as it rises toward the water surface. The reason is simple. It is because the pressure in the surrounding water decreases as the bubble rises. Thus the volume of the air bubble expands. Example 10-1 A U-shaped tube is filled with water and then a small amount of vegetable oil has been added to one side. If the depth of the oil is 5.00 cm, what is the difference in the liquid levels h between the two sides of the U-tube? Given that the density of water is 1000 kg/m 3 , and the density of the vegetable oil is 920 kg/m 3 .
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