Lec 4: Fluid statics, buoyancy and stability, pressure
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1 Lec 4: Fluid statics, buoyancy and stability, pressure
2 For next time: Outline: Important points: Read: § 3-1 to 3-4 HW 2 Zeroth law of thermodynamicsPressure and resulting forcesBuoyancy and stabilityImportant points:How to calculate pressure forceHow to calculate application point of pressure forceHow to analyze stability
3 Fluid staticsFluid statics deals with non-flow situations--fluids at rest.It is particularly applicable with pressure measurements in terms of fluid column heights.
4 TEAMPLAYYou accidentally drive your car into a lake and it submerges but does not admit a significant amount of water into the passenger compartment.A. Can you open a door?B. How will you get out?
5 Fluid staticsThe car door may be regarded as a plane surface of area about 10 square feet.In order to study the force on the submerged car door resisting attempts to open it, we must delve intoForce magnitudeForce application point, known as center of pressure.
6 Fluid staticsConsider the effect of a constant pressure at the top of the liquid. This could be Patm or some other pressure P0.We can neglect P0 as long as it acts on both sides.
7 Fluid statics Consider an arbitrary flat shape and orientation: The pressure at any point on the shape
9 Fluid staticsThe integral is related to the y coordinate of the centroid (center)
10 TEAMPLAYYour pickup, named Bigfoot, has a door which is 4 ft high by 3.5 ft wide and all windows are stuck in the closed position. The bottom of the door is 4 ft off the ground. You accidentally drive into a stock tank where it comes to rest on its wheels in water 10 ft deep. Assume the bottom of the stock tank is flat. What is the force on the door? Can you open it?
11 Fluid staticsNow that we know the resultant force on a submerged plane body iswhere yc is the y-coordinate of the centroid.it is necessary to know where the center of pressure is, that is, the point through which it acts.
12 Fluid staticsIn general the location yP of the center of pressure is below the location of the centroid yC because the pressure increases with depth.
13 Fluid staticsEquate the moment of the resultant force FR to the moment of the distributed pressure force about the x-axis.Where is the second moment of area (area moment of inertia).
14 Fluid staticsMost area moments of inertia are given about the centroid of the shape (IXX,C).They are relate to the moment IXX,0 about the x-axis byArea moments of inertia about the centroid are in Fig for some common shapes. Centroids are also given there.
16 BuoyancyA buoyant force FB is caused by increasing pressure with depth, so
17 BuoyancyThe upward force from the bottom is obviously greater, and so the net buoyancy force iswhere is the density of the fluid, not the body, and V is the volume of the body.
18 TEAMPLAYThe previous equation does not depend on the density of the submerged body.What changes about a submarine as it goes up and down (with zero propulsive thrust)?What is the upward force on a submarine as it holds a constant depth?Does this force change as it changes depth (with zero thrust)?
19 Buoyancy and stability The buoyant force for a constant volume system is equal to the weight W of the displaced fluid.
20 BuoyancyThe gravity force downward on a submerged body acts through the centroid.Similarly, the buoyant force upward must act through the centroid or there would be a rolling moment.Thus, we have Archimedes’ Principle:The buoyant force acting on a body immersed in a fluid is equal to the weight of the fluid displaced by the body, and it acts upward through the centroid of the displaced volume.
21 BuoyancyFor floating bodies, the buoyant force is given by the weight of the displaced fluid, or
22 StabilityImmersed bodies must be bottom-heavy to be stable. Thus the center of gravity G must below the center of buoyancy B so that any disturbance will provide a restoring moment about G.
23 StabilityModel a submarine as a horizontal tube with the top half empty and the bottom half filled with engines, crew quarters, and weaponry. Neglect the mass of the shell (tube). Where is the center of gravity G? Where is the center of buoyancy B? Do the two forces act to restore the sub to an upright condition if it starts to roll, or increase its rolling tendency?
24 StabilityRotational stability criteria are similar for floating bodies.However, if the center of buoyancy shifts during rolling motion, it may be possible to have the center of gravity G above the center of buoyancy and still achieve stability.
25 StabilityThe metacenter M is required to be above G. The metacenter height is the vertical distance between G and M.For many hull shapes the metacenter is almost a fixed point for rolling angles up to about 20°.
26 PressureThe normal force exerted on a (small) area. [Small enough that changes over the area are unimportant, and large enough that molecular effects also are unimportant.]
27 Pressures For pressures above atmospheric For pressures below atmosphericP1P1PgagePatmPatmPvacP2PabsolutePabsoluteP=0P=0
28 In the SI system we use 1 Pa = 1 N/m2 1 kPa = 1,000 N/m2 1 bar = 100,000 N/m21 MPa = 1,000,000 N/m2
29 In the USCS system we use lbf/in2 or psipsi is usually written with an “a”suffix (psia) or a “g” suffix, for absolute or gage (psig)