Physics
Buoyancy
Buoyancy is the upward force exerted by a fluid on an object immersed in it. It is a result of the pressure difference between the top and bottom of the object, causing it to float or rise. This force is equal to the weight of the fluid displaced by the object and is a fundamental concept in understanding the behavior of objects in fluids.
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11 Key excerpts on "Buoyancy"
eBook - PDFPhysics of Continuous Matter
Exotic and Everyday Phenomena in the Macroscopic World
- B. Lautrup(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
3 Buoyancy and stability “Buoy”, pronounced “booe”, probably of Germanic origin. A tethered floating object used to mark a location in the sea. Fishes, whales, submarines, balloons, and airships all owe their ability to float to Buoyancy , the lifting power of water and air. The understanding of the physics of Buoyancy goes back as far as antiquity and probably sprung from the interest in ships and shipbuilding in classic Greece. The basic principle is due to Archimedes. His famous law states that the Buoyancy force on a body is equal and oppositely directed to the weight of the fluid that the body displaces. Before his time it was thought that the shape of a body determined whether it would sink or float. Archimedes of Syracuse (287– 212 BC). Greek mathematician, physicist, and engineer. Dis-covered the formulae for area and volume of cylinders and spheres, and invented rudimen-tary infinitesimal calculus. For-mulated the Law of the Lever, and wrote two volumes on hy-drostatics titled On Floating Bod-ies , containing his Law of Buoy-ancy. Killed by a Roman soldier. (Source: Photograph of Fields Medal courtesy Stefan Zachow. Wikimedia Commons.) The shape of a floating body and its mass distribution do, however, determine whether it will float stably or capsize. Stability of floating bodies is of vital importance to shipbuilding, and to anyone who has ever tried to stand up in a small rowboat. Newtonian mechanics not only allows us to derive Archimedes’ principle for the equilibrium of floating bodies, but also to characterize the deviations from equilibrium and calculate the restoring forces. Even if a body floating in or on water is in hydrostatic equilibrium, it will not be in complete mechanical balance in every orientation, because the center of mass of the body and the center of mass of the displaced water, also called the center of Buoyancy, do not in general coincide.
eBook - PDF- Joseph Katz(Author)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
3.7 Buoyancy Calculation of the hydrostatic forces acting on bodies immersed or floating in a liq-uid is important to numerous disciplines and to the design of ships, balloons, etc. Some of these problems are depicted schematically in Fig. 3.24. A ship, for exam-ple, that is floating on the water surface is lifted by the Buoyancy force that is equal 88 Fluid Statics (a) (b) (c) Figure 3.24. Cases of objects in a liquid. to its weight. A submarine or a fish planning to sink or rise in the water must account for Buoyancy forces. The third case, that of a structure on the bottom floor of the sea, is similar to the cases discussed in the previous section, and the normal force on it is equal to the weight of the fluid column above it. Let us start with a simple analysis of a completely submerged object. Assume that a cube is placed inside the fluid such that its upper and lower surfaces are hori-zontal, as shown in Fig. 3.25. It is clear that the left-to-right and fore-and-aft forces on the cube are the same because of symmetry, whereas the forces on the upper and lower surfaces are not identical. The downward-pointing force on the upper surface is F 1 = ρ gh 1 S and the upward-pointing force on the lower panel is F 2 = ρ gh 2 S . The net force L (lift) is therefore L = F 2 − F 1 = ρ g ( h 2 − h 1 ) S . But the cube volume V = ( h 2 − h 1 ) S , and therefore we can conclude that the buoy-ancy force is L = ρ gV , (3.46) where V is the volume of the displaced fluid. This was discovered many years ago by the Greek physicist Archimedes (287–212 b.c.e. ) who made these statements: 1. A body immersed in a fluid experiences a vertical Buoyancy force equal to the weight of the displaced fluid. 2. A floating body experiences a vertical Buoyancy force equal to its weight (which is also equal to the weight of the displaced fluid). 3. The Buoyancy force acts through the centroid of the displaced fluid volume.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
In so doing, we find that the buoyant force equals ρghA. The quantity hA is the volume of liquid that the cylinder moves aside or displaces in being submerged, and ρ denotes the density of the liquid, not the density of the material from which the cylinder is made. Therefore, ρhA gives the mass m of the displaced fluid, so that the buoyant force equals mg, the weight of the displaced fluid. The phrase “weight of the displaced fluid” refers to the weight of the fluid that would spill out if the container were filled to the brim before the cylinder is inserted into the liquid. The buoyant force is not a new type of force. It is just the name given to the net upward force exerted by the fluid on the object. The shape of the object in Interactive Figure 11.16 is not important. No matter what its shape, the buoyant force pushes it upward in accord with Archimedes’ principle. It was an impressive accomplishment that the Greek scientist Archimedes (ca. 287–212 BC) discovered the essence of this principle so long ago. ARCHIMEDES’ PRINCIPLE Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces: F B = W fluid (11.6) The effect that the buoyant force has depends on its strength compared with the strengths of the other forces that are acting. For example, if the buoyant force is strong enough to balance the force of gravity, an object will float in a fluid. Figure 11.17 explores this possibility. In part a, a block that weighs 100 N displaces some liquid, and the liquid applies a buoyant force F B to the block, according to Archimedes’ principle. Nevertheless, if the block were released, it would fall further into the liquid because the buoyant force is not sufficiently strong to balance the weight of the block.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
FIGURE 11.16 The fluid applies a downward force P 1 A to the top face of the submerged cylinder and an upward force P 2 A to the bottom face. h A P 1 A P 2 A We have substituted P 2 − P 1 = gh from equation 11.4 into this result. In so doing, we find that the buoyant force equals ghA. The quantity hA is the volume of liquid that the cylinder moves aside or displaces in being submerged, and denotes the density of the liquid, not the density of the material from which the cylinder is made. Therefore, hA gives the mass m of the displaced fluid, so that the buoyant force equals mg, the weight of the displaced fluid. The phrase ‘weight of the displaced fluid’ refers to the weight of the fluid that would spill out if the container were filled to the brim before the cylinder is inserted into the liquid. The buoyant force is not a new type of force. It is just the name given to the net upward force exerted by the fluid on the object. The shape of the object in figure 11.16 is not impor- tant. No matter what its shape, the buoyant force pushes it upwards in accord with Archimedes’ principle. It was an impressive accomplishment that the Greek scientist Archimedes (ca. 287–212 BC) discovered the essence of this principle so long ago. Archimedes’ principle Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces: F B ⏟ ⏟ ⏟ Magnitude of buoyant force = W fluid ⏟ ⏟⏟ Weight of displaced fluid (11.6) The effect that the buoyant force has depends on its strength compared with the strengths of the other forces that are acting. For example, if the buoyant force is strong enough to balance the force of gravity, an object will float in a fluid. Figure 11.17 explores this possibility. In part a, a block that weighs 100 N displaces some liquid, and the liquid applies a buoyant force F B to the block, according to Archimedes’ principle.
eBook - PDF- W. Muckle(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
Buoyancy, stability and trim Conditions for equilibrium of body floating in still water When a body is floating in equihbrium in still water there is a force acting downwards which is due to gravity, so that if the body is of mass m, this force called the weight of the body is equal to mg. Since the body is in equihbrium it is correct to conclude that there must be a force of the same magnitude acting upwards. This force is generated by the hydrostatic pressures which act normally to the body as shown in Figure 5.1. The forces normal to the surface have Figure 5.1 vertical and horizontal components. The sum of the vertical com-ponents must add up to give a force equal to the weight mg. This force is called the 'Buoyancy'. The horizontal components of the hydrostatic pressures cancel out, giving a zero horizontal force. If ρ is the normal pressure, and Ph the vertical and horizontal components, and da the element of area on which the pressure acts, 66 5 Buoyancy, stability and trim 67 COSÖ Vertical component of force = Normal force χ cos θ = pgy X cos Ö = pgy dx Thus the vertical force is equal to the density of the fluid multiphed by the immersed area of the element. It follows that Total vertical force = Σ.ρgydx = pgY^ydx = pgA per unit length where A is the immersed cross-sectional area of the section. By integrating this along the length of the body the total buoyant force is obtained and is therefore pgY^A dz, if the longitudinal co-ordinate is z. Hence: Total buoyant force = pgV (5.3) where V is the immersed volume of the body. Since this force is equal to the gravitational force on the mass m of the body, then pgV = mg or m = pV (5.4) In other words the mass of the body is equal to the mass of the fluid displaced by the body. This is an important result as far as ship calculations are concerned.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Archimedes’ Principle When a body is fully or partially sub- merged in a fluid, a buoyant force F → b from the surrounding fluid acts on the body. The force is directed upward and has a magni- tude given by F b = m f g, (14-16) where m f is the mass of the fluid that has been displaced by the body (that is, the fluid that has been pushed out of the way by the body). When a body floats in a fluid, the magnitude F b of the (upward) buoyant force on the body is equal to the magnitude F g of the (down- ward) gravitational force on the body. The apparent weight of a body on which a buoyant force acts is related to its actual weight by weight app = weight − F b . (14-19) Flow of Ideal Fluids An ideal fluid is incompressible and lacks viscosity, and its flow is steady and irrotational. A streamline is the path followed by an individual fluid particle. A tube of flow is a bundle of streamlines. The flow within any tube of flow obeys the equation of continuity: R V = Av = a constant, (14-24) in which R V is the volume flow rate, A is the cross-sectional area of the tube of flow at any point, and v is the speed of the fluid at that point. The mass flow rate R m is R m = R V = Av = a constant. (14-25) Bernoulli’s Equation Applying the principle of conserva- tion of mechanical energy to the flow of an ideal fluid leads to Bernoulli’s equation along any tube of flow: p + 1 2 v 2 + gy = a constant. (14-29) Problems 1 g-LOC in dogfights. When a pilot takes a tight turn at high speed in a modern fighter airplane, the blood pressure at the brain level decreases, blood no longer perfuses the brain, and the blood in the brain drains.
eBook - PDF- W. Muckle, D A Taylor(Authors)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
5 Buoyancy , stabilit y an d tri m CONDITIONS FOR EQUILIBRIUM OF BODY FLOATING IN STILL WATER When a body is floating in still water there is a force acting downŮ wards which is due to gravity, so that if the body is of mass m, this force called the weight of the body is equal to mg. Since the body is in equilibrium it is correct to conclude that there must be a force of the same magnitude acting upwards. This force is generated by the hydrostatic pressures which act normally to the body as shown in Figure 5.1 The forces normal to the surface have vertical and horizonŮ tal components. The sum of the vertical components must add up to give a force equal to the weight mg. This force is called the 'Buoyancy'. The horizontal components of the hydrostatic pressures cancel out, giving a zero horizontal force. If/? is the normal pressure, p v and p h the vertical and horizontal components, and da the element of area on which the pressure acts, it follows that w P L Figure 5.1 Xp y da = mg Xp h da = 0 (5.1) (5.2) 80 Buoyancy , STABILIT Y AND TRI M 81 In other words the mass of the body is equal to the mass of the fluid displaced by the body. This is an important result as far as ship The gravitational force mg can be imagined to be concentrated at a point G which is the centre of mass or is more commonly known as the centre of gravity. Similarly the buoyant force can be imagined to be concentrated at a point  called the centre of Buoyancy, which can be considered to be the centroid of the underwater volume. For equilibŮ rium then G and  must lie in the same vertical line. Consider now the hydrostatic force acting on a small element of the surface length ds, a distance y below the free surface: Pressure = Head X Density X Gravitational acceleration = ggy Normal force per unit length = Qgyds If è is the angle of inclination of the surface to the horizontal, then the projection of d^ on the horizontal plane is dx, and dx/ds = cos È.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
When a body floats in a fluid, the magnitude F b of the (upward) buoyant force on the body is equal to the magnitude F g Questions 403 of the (downward) gravitational force on the body. The apparent weight of a body on which a buoyant force acts is related to its actual weight by weight app = weight − F b . (14.5.4) Flow of Ideal Fluids An ideal fluid is incompressible and lacks viscosity, and its flow is steady and irrotational. A streamline is the path followed by an individual fluid particle. A tube of flow is a bundle of streamlines. The flow within any tube of flow obeys the equation of continuity: R V = Av = a constant, (14.6.3) in which R V is the volume flow rate, A is the cross-sectional area of the tube of flow at any point, and v is the speed of the fluid at that point. The mass flow rate R m is R m = R V = Av = a constant. (14.6.4) Bernoulli’s Equation Applying the principle of conserva- tion of mechanical energy to the flow of an ideal fluid leads to Bernoulli’s equation along any tube of flow: p + 1 _ 2 v 2 + gy = a constant. (14.7.2) 1 We fully submerge an irregular 3 kg lump of material in a certain fluid. The fluid that would have been in the space now occupied by the lump has a mass of 2 kg. (a) When we release the lump, does it move upward, move downward, or remain in place? (b) If we next fully submerge the lump in a less dense fluid and again release it, what does it do? 2 Figure 14.1 shows four situations in which a red liquid and a gray liquid are in a U-tube. In one situation the liquids cannot be in static equilibrium. (a) Which situation is that? (b) For the other three situations, assume static equilibrium. For each of them, is the density of the red liquid greater than, less than, or equal to the density of the gray liquid? (1) (2) (3) (4) FIGURE 14.1 Question 2. 3 A boat with an anchor on board floats in a swimming pool that is somewhat wider than the boat.
eBook - PDF- Adrian Biran(Author)
- 2003(Publication Date)
- Butterworth-Heinemann(Publisher)
We conclude that the Buoyancy force passes through the centre of the submerged volume, B (centre of the displaced volume of liquid). 2.3 The conditions of equilibrium of a floating body A body is said to be in equilibrium if it is not subjected to accelerations. Newton’s second law shows that this happens if the sum of all forces acting on that body is zero and the sum of the moments of those forces is also zero. Two forces always act on a floating body: the weight of that body and the Buoyancy force. In this section we show that the first condition for equilibrium, that is the one regarding the sum of forces, is expressed as Archimedes’ principle. The second condition, regarding the sum of moments, is stated as Stevin’s law . Basic ship hydrostatics 33 Further forces can act on a floating body, for example those produced by wind, by centrifugal acceleration in turning or by towing. The influence of those forces is discussed in Chapter 6. 2.3.1 Forces Let us assume that the bodies appearing in Figures 2.1, 2.3(a) float freely. Then, the weight of each body and the hydrostatic forces acting on it are in equilibrium. Archimedes’ principle can be reformulated as: The weight of the volume of water displaced by a floating body is equal to the weight of that body. The weight of the fluid displaced by a floating body is appropriately called displacement. We denote the displacement by the upper-case Greek letter delta , that is ∆ . If the weight of the floating body is W , then we can express the equilibrium of forces acting on the floating body by ∆ = W (2.16) For the volume of the displaced liquid we use the symbol ∇ defined in Chapter 1. In terms of the above symbols Archimedes’ principle yields the equation γ ∇ = W (2.17) If the floating body is a ship, we rewrite Eq. (2.17) as γC B LBT = Σ n i =1 W i (2.18) where W i is the weight of the i th item of ship weight.
eBook - PDFApplied Mathematics
Made Simple
- Patrick Murphy(Author)
- 2014(Publication Date)
- Butterworth-Heinemann(Publisher)
A density bottle is not essential for finding such a volume since we could fill a measuring jar with a known quantity of water, put the body in the jar and note the rise in the level of the water; this would tell us the volume of the body immersed. This method is acceptable for bodies which sink, but what of those which float ? Using a pin, the body could easily be held under the water surface and the volume noted as before. However, on doing this we see that a force is necessary to push the body beneath the surface, and we should ask ourselves immediately whether we always need the same force to hold any body under. For example, does it depend on the size, shape, and weight of the body itself or even per-haps the liquid in which it floats ? Archimedes' principle which answers this question is probably the best-known principle in the whole of applied mathematics. To obtain some preparation for an understanding of the statement of the principle, consider the following simple experiment. Take a heavy body with a shape which can be held in a cotton loop and suspend it from a spring balance to find its weight. Now raise a bowl of water underneath the body until it is completely immersed as shown in Fig. 224 (a). As the volume immersed in the water is increased so the readings of the spring will decrease and will continue to decrease until the body is com-pletely immersed. 268 Applied Mathematics Made Simple Let us look at a diagram of the forces on the body when it is in equilibrium completely immersed in the liquid as shown in Fig. 224 (b). The forces on the body are the tension Tin the spring as registered by the scale on the spring and the weight mg of the body. Since we know that Τ ψ mg there must be another force on the body due to its presence in the water. Furthermore, since Τ < mg (α) (b) (c) Fig. 224 this other force must be vertically upwards. We call this force the upthrust U of the water on the body.
- Robert Ehrlich(Author)
- 2020(Publication Date)
- Princeton University Press(Publisher)
Based on Archimedes' princi-ple, the buoyant force (the drop in scale reading as the weight is submerged) divided into the original scale reading 105 Fluids: Bernoulli's Principle should equal the specific gravity of the weight. Be sure that the weight is completely submerged, but not resting on the bottom. Bernoulli's Principle H.10. Ping-Pong ball near a water stream Demonstration A Ping-Pong ball glued onto a piece of string is attracted to a stream of water from a faucet, according to Bernoulli's principle. Equipment A Ping-Pong ball glued to the end of a piece of string, and water from a faucet (or water poured out of a bottle). Comment According to Bernoulli's principle, a water stream traveling at a high velocity creates a region of low pressure. The Ping-Pong ball moves toward the water stream when placed next to it, since the other side of the ball is acted on by full atmos-pheric pressure. The ball does not move into the center of the stream because of the force of the descending water. H.11. Ping-Pong ball in an inverted funnel Demonstration Blowing with sufficient speed through an inverted funnel that contains a Ping-Pong ball causes the ball not to fall, according to Bernoulli's principle. Equipment A funnel (preferably transparent) and a Ping-Pong ball. Comment You may first want to demonstrate that the ball can be held in place by sucking on the funnel stem, and then show that blowing also keeps the ball from falling, if the ball has been placed near the top of the funnel. The high-velocity airflow past the top right and top left sides of the ball (see illustra-tion) causes a pressure drop at these points, while the region under the ball, where there is no airflow, is at atmospheric pressure. The resultant of these forces is upward and coun-teracts the weight of the ball; thus, the ball doesn't fall. For the demonstration to work, a high-speed airflow is needed to
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