Topic 2.2 Extended J – Fluid statics : Pascals Principle FYI: In America we use PSI and PSF. As we learned in the previous lesson, pressure is force per unit area: Definition of Pressure F A p = The SI unit for pressure is the pascal (Pa) in honor of the French scientist and philosopher Blaise Pascal (1623-1662). 1 Pa = 1 n/m2. A nickel (diameter 2.1 cm) has a 16-lb bowling ball placed on it. What is the pressure, in Pa, between the nickel and the floor? © 2006 By Timothy K. Lund d2 4 0.0212 4 F = 16 lb 1 kg 2.2 lb 9.8 n kg A =  =  F = 71 n A = 0.00035 m2 F A 71 0.00035 Question: If we used a dime instead of a nickel would the pressure increase, decrease, or remain the same? p = = p = 2.03105 Pa

Topic 2.2 Extended J – Fluid statics : Pascals Principle FYI: Water has a density of  = 1000 kg/m3. PRESSURE AND DEPTH Consider the tank of water shown here: The tank has a height h and a bottom area A: The area A is supporting the full weight W of the water. Since W = mg, we see that the water pressure on the bottom of the tank is just h F A mg A p = = © 2006 By Timothy K. Lund Since density  = m/V, and since volume V = Ah, we can write m = V = Ah so that mg A Ahg A p = = A Pressure Due to Weight of Fluid p = gh

Topic 2.2 Extended J – Fluid statics : Pascals Principle PRESSURE AND DEPTH What is the pressure due to the weight of water if you are diving to a depth of 30 m? p = gh p = 1000(10)(30) p = 300000 Pa What is the pressure due to the weight of blood in your feet if you are standing up and you are 175 cm tall. Assume the density of blood is that same as that of water. © 2006 By Timothy K. Lund p = gh Question: Is your blood pressure higher in your arm, or in your feet? p = 1000(10)(1.75) p = 17500 Pa

Topic 2.2 Extended J – Fluid statics : Pascals Principle Question: If you are swimming horizontally, and your upper area is about 0.5 m2, the force acting on your back is F = pA = (400000)(0.5) = 200000 n (=50000 lbs!). Why aren't you driven downward by this force? Topic 2.2 Extended J – Fluid statics : Pascals Principle PRESSURE AND DEPTH FYI: We call p the ABSOLUTE PRESSURE and gh the GAUGE PRESSURE. We can write pgauge = p - pa. When you measure the pressure in your tire you are measuring gauge pressure. We have only considered the weight of the water as our source of pressure, but on the surface of that water lies the weight of a column of atmosphere having the same area. FYI: Since we are always at the bottom of the air column, we can just add the pressure of the atmosphere pa to the weight of the water column. © 2006 By Timothy K. Lund Pressure-Depth Relationship p = pa + gh FYI: Atmospheric pressure (at sea level) is about 101.325 kPa (= 1.01325105 n/m2 = 14.7 lb/in2) A What pressure do you feel if you are swimming at a depth of 30 m? p = pa + gh = 1.01105 + 1000(10)(30) p = 4.01105 Pa

Topic 2.2 Extended J – Fluid statics : Pascals Principle FYI: If you are swimming horizontally, and your upper area is about 0.5 m2, the force acting on your back is F = pA = (400000)(0.5) = 200000 n (=50000 lbs!). Why aren't you driven downward by this force? Topic 2.2 Extended J – Fluid statics : Pascals Principle Answer: Because of Pascal's principle, you feel an equal pressure from below, and hence an equal upward force to balance the downward force of 50000 lbs. PASCAL'S PRINCIPLE Why aren't you crushed!? Blaise Pascal studied, among other things, pressure in liquids. He observed the following effect, which we call Pascal's principle: "Pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid and to the walls of the container." Ao Fo Fi Ai Ai Fi = pi © 2006 By Timothy K. Lund Fo Ao = po p pi = po Pascal's principle Fi Ai Fo Ao = p p The Lift Formula Ao Ai Fo = Fi

Topic 2.2 Extended J – Fluid statics : Pascals Principle If the lift below is needed to raise a metric ton with an input force of 200 n, and the diameter of the input piston is 10 cm, what must the diameter of the output piston be? 1000(10) 200 Ao Ai do2 = 102 Fo = Fi do = 70.7 cm Ao Fo Fo Fi Ao = Ai Ai Fi © 2006 By Timothy K. Lund Fo Fi (/4)do2 = (/4)di2 p Fo Fi do2 = di2 p p