pressure, depth, and pascals Law




If the weight of the fluid can be ignored, the pressure in a fluid is the same throughout its volume. We used that approximation in our discussion of bulk stress and strain in Section 11.4. But often the fluid’s weight is not negligible, and pressure variations are important. Atmospheric pressure is less at high altitude than at sea level, which is why airliner cabins have to be pressurized. When you dive into deep water, you can feel the increased pressure on your ears.

We can derive a relationship between the pressure p at any point in a fluid at rest and the elevation y of the point. We’ll assume that the density r has the same value throughout the fluid (that is, the density is uniform), as does the acceleration due to gravity g. If the fluid is in equilibrium, any thin element of the fluid with thickness dy is also in equilibrium (Fig. 12.4a). The bottom and top surfaces each have area A, and they are at elevations y and y+dy above some reference level where y=0. The fluid element has volume dV=A dy, mass dm=r dV=rA dy, and weight dw=dm g=rgA dy.

pressure, depth, and pascals Law

 

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What are the other forces on this fluid element (Fig 12.4b)? Let’s call the pressure at the bottom surface p; then the total y-component of upward force on this surface is pA. The pressure at the top surface is p+dp, and the total y-component of (downward) force on the top surface is -(p+dp2). The fluid element is in equilibrium, so the total y-component of force, including the weight and the forces at the bottom and top surfaces, must be zero:

ΣFy=0  so  pA+(p+dp)A-rgA dy=0

When we divide out the area A and rearrange, we get

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This equation shows that when y increases, p decreases; that is, as we move upward in the fluid, pressure decreases, as we expect. If p1 and p2 are the pressures at elevations y1 and y2 , respectively, and if r and g are constant, then

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It’s often convenient to express Eq. (12.5) in terms of the depth below the surface of a fluid (Fig. 12.5). Take point 1 at any level in the fluid and let p rep-resent the pressure at this point. Take point 2 at the surface of the fluid, where the pressure is p0 (subscript zero for zero depth). The depth of point 1 below the surface is h=y2-y1 , 

  and Eq. (12.5) becomes 

  p0-p=-rg1y2-y12=-rgh  or

 

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The pressure p at a depth h is greater than the pressure p0 at the surface by an amount rgh. Note that the pressure is the same at any two points at the same level in the fluid. The shape of the container does not matter (Fig. 12.6).Equation (12.6) shows that if we increase the pressure p0 at the top surface, possibly by using a piston that fits tightly inside the container to push down on the fluid surface, the pressure p at any depth increases by exactly the same amount. This observation is called Pascal’s law.

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pressure, depth, and pascals Law

 



Frequently Asked Questions

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Ans: A fluid exerts a force perpendicular to any surface in contact with it, such as a container wall or a body immersed in the fluid. This is the force that you feel pressing on your legs when you dangle them in a swimming pool. Even when a fluid as a whole is at rest, the molecules that make up the fluid are in motion; the force exerted by the fluid is due to molecules colliding with their surroundings view more..
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Ans: A fluid is any substance that can flow and change the shape of the volume that it occupies. (By contrast, a solid tends to maintain its shape.) We use the term “fluid” for both gases and liquids. The key difference between them is that a liquid has cohesion, while a gas does not. The molecules in a liquid are close to one another, so they can exert attractive forces on each other and thus tend to stay together (that is, to cohere). That’s why a quantity of liquid maintains the same volume as it flows: If you pour 500 mL of water into a pan, the water will still occupy a volume of 500 mL. The molecules of a gas, by contrast, are separated on average by distances far larger than the size of a molecule. Hence the forces between molecules are weak, there is little or no cohesion, and a gas can easily change in volume. If you open the valve on a tank of compressed oxygen that has a volume of 500 mL, the oxygen will expand to a far greater volume. view more..
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Ans: Fluids play a vital role in many aspects of everyday life. We drink them, breathe them, swim in them. They circulate through our bodies and control our weather. The physics of fluids is therefore crucial to our understanding of both nature and technology view more..
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Ans: If the weight of the fluid can be ignored, the pressure in a fluid is the same throughout its volume. We used that approximation in our discussion of bulk stress and strain in Section 11.4. But often the fluid’s weight is not negligible, and pressure variations are important. Atmospheric pressure is less at high altitude than at sea level, which is why airliner cabins have to be pressurized. When you dive into deep water, you can feel the increased pressure on your ears. view more..
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Ans: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. view more..
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Ans: If the pressure inside a car tire is equal to atmospheric pressure, the tire is flat. The pressure has to be greater than atmospheric to support the car, so the significant quantity is the difference between the inside and outside pressures. When we say that the pressure in a car tire is “32 pounds” (actually 32 lb>in.2 , equal to 220 kPa or 2.2 * 105 Pa), we mean that it is greater than atmospheric pressure (14.7 lb>in.2 or 1.01 * 105 Pa) by this amount. view more..
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Ans: The simplest pressure gauge is the open-tube manometer . The U-shaped tube contains a liquid of density r, often mercury or water. The left end of the tube is connected to the container where the pressure p is to be measured, and the right end is open to the atmosphere view more..
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Ans: A body immersed in water seems to weigh less than when it is in air. When the body is less dense than the fluid, it floats. The human body usually floats in water, and a helium-filled balloon floats in air. These are examples of buoyancy, a phenomenon described by Archimedes’s principle: view more..
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Ans: We’ve seen that if an object is less dense than water, it will float partially submerged. But a paper clip can rest atop a water surface even though its density is several times that of water. This is an example of surface tension: view more..
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Ans: We are now ready to consider motion of a fluid. Fluid flow can be extremely complex, as shown by the currents in river rapids or the swirling flames of a campfire. But we can represent some situations by relatively simple idealized models. An ideal fluid is a fluid that is incompressible (that is, its density cannot change) and has no internal friction (called viscosity). view more..
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Ans: The mass of a moving fluid doesn’t change as it flows. This leads to an important relationship called the continuity equation view more..
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Ans: According to the continuity equation, the speed of fluid flow can vary along the paths of the fluid. The pressure can also vary; it depends on height as in the static situation (see Section 12.2), and it also depends on the speed of flow. We can derive an important relationship called Bernoulli’s equation, view more..
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Ans: To derive Bernoulli’s equation, we apply the work–energy theorem to the fluid in a section of a flow tube. In Fig. 12.23 we consider the element of fluid that at some initial time lies between the two cross sections a and c. The speeds at the lower and upper ends are v1 and v2. In a small time interval dt, the fluid that is initially at a moves to b, a distance ds1 = v1 dt, and the fluid that is initially at c moves to d, a distance ds2 = v2 dt. The cross-sectional areas at the two ends are A1 and A2, as shown. The fluid is incompressible; hence by the continuity equation, Eq. (12.10), the volume of fluid dV passing any cross section during time dt is the same. That is, dV = A1 ds1 = A2 ds2. view more..
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Ans: HERE ARE SOME EXAMPLES TO DEAL WITH view more..
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Ans: Viscosity is internal friction in a fluid. Viscous forces oppose the motion of one portion of a fluid relative to another. Viscosity is the reason it takes effort to paddle a canoe through calm water, but it is also the reason the paddle works. Viscous effects are important in the flow of fluids in pipes, the flow of blood, the lubrication of engine parts, and many other situations view more..
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Ans: When the speed of a flowing fluid exceeds a certain critical value, the flow is no longer laminar. Instead, the flow pattern becomes extremely irregular and complex, and it changes continuously with time; there is no steady-state pattern. This irregular, chaotic flow is called turbulence view more..
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Ans: SUMMARY OF EVERY TOPIC OF FLUID MECHANISM. view more..
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Ans: Some of the earliest investigations in physical science started with questions that people asked about the night sky. Why doesn’t the moon fall to earth? Why do the planets move across the sky? Why doesn’t the earth fly off into space rather than remaining in orbit around the sun? The study of gravitation provides the answers to these and many related questions view more..




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