Chapter 15 Fluids

Pressure The same force applied over a smaller area results in greater pressure – think of poking a balloon with your finger and then with a needle. Pressure is not the same as force! Pressure is force per unit area Pressure is a useful concept for discussing fluids, because fluids distribute their force over an area

Pressure and Depth Pressure increases with depth in a fluid due to the increasing mass of the fluid above it.

Pressure and depth Pressure in a fluid includes pressure on the fluid surface (usually atmospheric pressure)

Pressure depends only on depth and external pressure (and not on shape of fluid column)

Equilibrium only when pressure is the same Unequal pressure will cause liquid flow: must have same pressure at A and B Oil is less dense, so a taller column of oil is needed to counter a shorter column of water

Pascal’s principle An external pressure applied to an enclosed fluid is transmitted to every point within the fluid. Hydraulic lift Assume fluid is “incompressible” F 1 / A 1 = P = F 2 / A 2

Pascal’s principle Hydraulic lift F 1 / A 1 = P = F 2 / A 2 Are we getting “something for nothing”? Assume fluid is “incompressible” so Work in = Work out!

Buoyancy A fluid exerts a net upward force on any object it surrounds, called the buoyant force. This force is due to the increased pressure at the bottom of the object compared to the top. Consider a cube with sides = L

Buoyant Force When a Volume V is Submerged in a Fluid of Density ρ fluid F b = ρ fluid gV Archimedes’ Principle Archimedes’ Principle: An object completely immersed in a fluid experiences an upward buoyant force equal in magnitude to the weight of fluid displaced by the object. Q: Does buoyant force depend on depth? a) yes b) no

Measuring the Density Get the volume from ( T 1 - T 2 ) = V( ρ water g) Get the mass from W = T 1 = mg The King must know: is his crown true gold?

The crown-maker makes a crown for the king. Archimedes weighs the crown and determines that its weight in air is 5.54 N and that its weight in water is 5.05 N. Should the crown-maker maker be paid or ???

Off with his head!!

Applications of Archimedes’ Principle An object floats when it displaces an amount of fluid equal to its weight. equivalent mass of water wood block equivalent mass of water brass block

Can Brass Float? An object made of material that is denser than water can float only if it has indentations or pockets of air that make its average density less than that of water. An object floats when it displaces an amount of fluid equal to its weight. equivalent mass of water brass block

Applications of Archimedes’ Principle The fraction of an object that is submerged when it is floating depends on the densities of the object and of the fluid.

Cartesian Diver Think of a weighted balloon submerged in water How will the balloon change when pressure goes up? Did its weight change when pressure went up? So when pressure goes up: - will it float higher? - or will it sink?

Wood in Water Two beakers are filled to the brim with water. A wooden block is placed in the beaker 2 so it floats. (Some of the water will overflow the beaker and run off). Both beakers are then weighed. Which scale reads a larger weight? a b c same for both

displaces an amount of water equal to its weight weight of the overflowed water is equal to the weight of the blockbeaker in 2 has the same weight as that in 1 The block in 2 displaces an amount of water equal to its weight, because it is floating. That means that the weight of the overflowed water is equal to the weight of the block, and so the beaker in 2 has the same weight as that in 1. Wood in Water a b c same for both Two beakers are filled to the brim with water. A wooden block is placed in the beaker 2 so it floats. (Some of the water will overflow the beaker and run off). Both beakers are then weighed. Which scale reads a larger weight?

Wood in Water II A block of wood floats in a container of water as shown on the right. On the Moon, how would the same block of wood float in the container of water? Earth Moon abc

weight of water equal to the object’s weight less weightalso has less weight A floating object displaces a weight of water equal to the object’s weight. On the Moon, the wooden block has less weight, but the water itself also has less weight. Wood in Water II A block of wood floats in a container of water as shown on the right. On the Moon, how would the same block of wood float in the container of water? Moon abc Earth

A wooden block is held at the bottom of a bucket filled with water. The system is then dropped into free fall, at the same time the force pushing the block down is also removed. What will happen to the block? a) the block will float to the surface. b) the block will stay where it is. c) the block will oscillate between the surface and the bottom of the bucket

A wooden block is held at the bottom of a bucket filled with water. The system is then dropped into free fall, at the same time the force pushing the block down is also removed. What will happen to the block? a) the block will float to the surface. b) the block will stay where it is. c) the block will oscillate between the surface and the bottom of the bucket Bouyant force is created by a change of pressure with depth. Pressure is created by the weight of water being held up. In free-fall, nothing is being held up! No apparent weight!

A wooden block of cross-sectional area A, height H, and density ρ 1 floats in a fluid of density ρ f. If the block is displaced downward and then released, it will oscillate with simple harmonic motion. Find the period of its motion. h

A wooden block of cross-sectional area A, height H, and density ρ 1 floats in a fluid of density ρ f. If the block is displaced downward and then released, it will oscillate with simple harmonic motion. Find the period of its motion. Vertical force: F y = (hA)g ρ f - (HA)g ρ 1 h at equilibrium: h 0 = H ρ 1 / ρ f Total restoring force: F y = -(Ag ρ f )y h = h 0 - y Analogous to mass on a spring, with κ = Agρ f

Fluid Flow and Continuity Continuity tells us that whatever the mass of fluid in a pipe passing a particular point per second, the same mass must pass every other point in a second. The fluid is not accumulating or vanishing along the way. This means that where the pipe is narrower, the fluid is flowing faster Volume per unit time

Continuity and Compressibility Most gases are easily compressible; most liquids are not. Therefore, the density of a liquid may be treated as constant (not true for a gas). mass flow is conserved volume flow is conserved

Bernoulli’s Equation When a fluid moves from a wider area of a pipe to a narrower one, its speed increases; therefore, work has been done on it. The kinetic energy of a fluid element is: Equating the work done to the increase in kinetic energy gives:

Bernoulli’s Equation Where fluid moves faster, pressure is lower

Bernoulli’s Equation If a fluid flows in a pipe of constant diameter, but changes its height, there is also work done on it against the force of gravity. Equating the work done with the change in potential energy gives:

Bernoulli’s Equation The general case, where both height and speed may change, is described by Bernoulli’s equation: This equation is essentially a statement of conservation of energy in a fluid.

Dynamic lift v low P high v high P low Aircraft wing

Applications of Bernoulli’s Equation If a hole is punched in the side of an open container, the outside of the hole and the top of the fluid are both at atmospheric pressure. Since the fluid inside the container at the level of the hole is at higher pressure, the fluid has a horizontal velocity as it exits.