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Density, Pressure And Buoyancy Iceberg off Newfoundland Density The density of a substance of uniform composition is defined as its mass per unit volume: m V Units are kg/m3 (SI) or g/cm3 (cgs) 1 g/cm3 = 1000 kg/m3 PRESSURE …is force per unit area. F P A N Unit: 1 2 = 1 pascal (Pa) m Look at the force needed to burst a balloon. FLUID PRESSURE The force exerted by a fluid on a submerged object at any point is perpendicular to the surface of the object F P A N Unit: 1 2 = 1 pascal (Pa) m Variation of Pressure with Depth If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium. All points at the same depth must be at the same pressure. Otherwise, the fluid would not be in equilibrium. The fluid would flow from the higher pressure region to the lower pressure region. Examine the darker region, assumed to be a fluid. It has a crosssectional area A and extends to a depth h below the surface. Three external forces act on the region. Look at Pressure (P) vs. Depth (h) data. How does P depend on h? Pressure vs. Depth equation from theory and experiment: P Po gh Po is normal atmospheric pressure. Po 1.013 105 Pa 14.7 lb in 2 The pressure does not depend upon the shape of the container. Absolute vs. Gauge Pressure The pressure P is called the absolute pressure. Remember, P = Po + gh P – Po = gh is the gauge pressure. Pressure Values in Various Units One atmosphere of pressure is defined as the pressure equivalent to a column of mercury exactly 0.76 m tall at 0o C where g = 9.806 65 m/s2 One atmosphere (1 atm) = 76.0 cm of mercury 1.013 x 105 Pa 14.7 lb/in2 Buoyancy Three Observations: 1) Solid object on a spring balance, submerged in a fluid. 2) Balancing water bottles. 3) Sink or float movie clip. More on Buoyancy Iceberg off Newfoundland Archimedes' Principle Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object. Buoyant Force The upward force is called the buoyant force. The physical cause of the buoyant force is the pressure difference between the top and the bottom of the object. The magnitude of the buoyant force (B) always equals the weight of the displaced fluid: B fluidVfluid g Wfluid The buoyant force is the same for a totally submerged object of any size, shape, or density. Buoyancy – Three Possibilities 1. The object is less dense than the fluid. The upward buoyant force is B = ρfluidVobj g The downward gravitational force is w = mg = ρobjVobj g The net force is B-w = (ρfluid-ρobj)Vobj g Buoyancy – Three Possibilities 1. The object is less dense than the fluid. The net force B-w = (ρfluid-ρobj)Vobj g is upward. Now the volume of displaced fluid is less than the volume of the object. The object accelerates upward and floats when the weight of the displaced fluid equals the weight of the object. Buoyancy – Three Possibilities 2. The object is more dense than the fluid. The net force B-w = (ρfluid-ρobj)Vobj g is downward and the object accelerates to the bottom. Buoyancy – Three Possibilitiess 3. Object and fluid densities are equal. The net force is B-w = (ρfluid-ρobj)Vobj g = 0 Now the weight of displaced fluid equals the weight of the object. The submerged object remains stationary wherever it is placed. How much of an Iceberg is Below the Surface? Ask the Captain of the Titanic! B fluidVfluid g Vfluid is the volume of the displaced fluid. B fluidVfluid g Vfluid is the volume of the displaced fluid. For an iceberg, Vfluid .9Viceberg because ice .9 water B fluidVfluid g Vfluid is the volume of the displaced fluid. For an iceberg, Vfluid .9Viceberg So about 9/10 of the volume of the iceberg is below the surface. A static fluid in a container is subject to both atmospheric pressure at its surface and Earth's gravitation. The pressure at the bottom of the container A. is equal to the atmospheric pressure. B. depends on the height of the fluid column. C. depends on the shape of the container. D. depends on the height of the fluid column and the shape of the container. A static fluid in a container is subject to both atmospheric pressure at its surface and Earth's gravitation. The pressure at the bottom of the container A. is equal to the atmospheric pressure. B. depends on the height of the fluid column. C. depends on the shape of the container. D. depends on the height of the fluid column and the shape of the container. The buoyant force on an immersed body has the same magnitude as A. the difference between the weights of the body and the displaced fluid. B. the average pressure of the fluid times the surface area of the body. C. the weight of the fluid displaced by the body. D. the weight of the body. The buoyant force on an immersed body has the same magnitude as A. the difference between the weights of the body and the displaced fluid. B. the average pressure of the fluid times the surface area of the body. C. the weight of the fluid displaced by the body. D. the weight of the body. A block of aluminum and a block of iron have equal volumes. If each block is completely submerged in water, A. the buoyant force on the aluminum will be greater. B. both blocks will experience the same buoyant force. C. the buoyant force on the iron will be greater. A block of aluminum and a block of iron have equal volumes. If each block is completely submerged in water, A. the buoyant force on the aluminum will be greater. B. both blocks will experience the same buoyant force. C. the buoyant force on the iron will be greater. Imagine holding two bricks under water. Brick A is just beneath the surface of the water, while brick B is at a greater depth. The force needed to hold brick B in place is A. smaller than B. larger than C. the same as than the force required to hold brick A in place. Imagine holding two bricks under water. Brick A is just beneath the surface of the water, while brick B is at a greater depth. The force needed to hold brick B in place is A. smaller than B. larger than C. the same as than the force required to hold brick A in place. An ice cube is placed in a glass of water. As the ice melts, the water level on the side of the glass A. stays the same. B. goes up. C. goes down. An ice cube is placed in a glass of water. As the ice melts, the water level on the side of the glass A. stays the same. B. goes up. C. goes down. Buoyancy Two more observations: 4) Barges on the river. 5) Recovering a damaged vessel.