Fluids Fluids are materials that can flow, gases and liquids. Air is the most common gas, and moves from place to place as wind. Water is the most familiar liquid.

New Symbol  “Rho” The mass density  is the mass m of a substance divided by its volume V: SI Unit of Mass Density: kg/m 3

Density Density of water is –1, 000 kg /m 3 –1 g/ml –1 pound/pint

Example 1 Blood as a Fraction of Body Weight The body of a man whose weight is 690 N contains about 5.2x10 -3 m 3 of blood. The density of human blood is g/ml (a) Find the blood’s mass and (b) express it as a percentage of the body weight.

Example 1 Blood as a Fraction of Body Weight The body of a man whose weight is 690 N contains about 5.2x10 -3 m 3 of blood. The density of human blood is g/ml (a) Find the blood’s mass and (b) express it as a percentage of the body weight.

11.1 Mass Density (a) (b)

New Symbol “P” The pressure P exerted by a fluid is defined as the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts: The SI unit for pressure: newton/meter 2 = (N/m 2 ) = pascal (Pa).

11.2 Pressure Example 2 The Force on a Swimmer Suppose the pressure acting on the back of a swimmer’s hand is 1.2x10 5 Pa. The surface area of the back of the hand is 8.4x10 -3 m 2. (a)Determine the magnitude of the force that acts on it. (b) Discuss the direction of the force.

11.2 Pressure Since the water pushes perpendicularly against the back of the hand, the force is directed downward in the drawing.

Other units of pressure 1 atm = the pressure at sea level at 0 c. 1 atm = 1.013*10 5 Pa

Pressure – Not a vector. Can push any direction. Forces result from pressure differentials.

New Terms Absolute Pressure – The pressure at a point relative to the vacuum of space. Gauge Pressure – The pressure at a point relative to Earth’s surface.

New Terms Absolute Pressure – The pressure at a point relative to the vacuum of space. Gauge Pressure – The pressure at a point relative to Earth’s surface. –What Pressure gauges read.

Absolute and gauge pressure P abs = P gauge + 1 atm P abs = P gauge * 10 5 Pa

Pressure in a liquid In a liquid, the force pressing down on a surface is the weight of all the fluid above the surface (fluid column)

Pressure in a liquid Weight = mg = ρVg Volume of the column is the area times height. = ρAhg

Pressure in a liquid Weight = ρAhg P = Weight/Area = ρAhg/A P gauge = ρgh

Pressure Difference The pressure difference between different depths reduces to ΔP = ρgΔh

ΔP ΔP will be same between Gauge pressure and absolute pressure. The two are just offset by a constant, so differences are preserved.

Solar water heater. A solar water heater works by pumping water through panels on the roof where the sun warms it, and then storing and circulating it as needed. The storage tank is in the basement 12 m below. What is the gauge pressure and absolute pressure in the water tank?

Pressure in an irregular container The shape of the container doesn’t matter. Only depth.

Pressure in an irregular container The shape of the container doesn’t matter. Only depth.

11.3 Pressure and Depth in a Static Fluid Example 4 The Swimming Hole Points A and B are located a distance of 5.50 m beneath the surface of the water. Find the pressure at each of these two locations.

Pascal’s Law Pressure applied to an enclosed fluid is transmitted undiminished to every surface of the fluid.

Hydraulics Pressure at points depend only on elevation. So A & B have same pressure.

Hydraulics If you’re dealing with a liquid (incompressable), there is no change in Volume.

Hydraulics The input piston has a radius of m and the output plunger has a radius of m. The combined weight of the car and the plunger is N. Suppose that the input piston has a negligible weight and the bottom surfaces of the piston and plunger are at the same level. What is the required input force?

Hydraulics The input piston has a radius of m and the output plunger has a radius of m. If the input piston is pushed 6 m, how far is the Output plunger lifted?

Pressure Measurements: Manometer One end of the U- shaped tube is open to the atmosphere The other end is connected to the pressure to be measured If P in the system is greater than atmospheric pressure, h is positive –If less, then h is negative

Manometer Problem A sample of gas connected to a water filled manometer, and Δh between the sides is millimeters. What is the absolute pressure of the sample?

Pressure Measurements: Barometer Invented by Torricelli (1608 – 1647) A long closed tube is filled with mercury and inverted in a dish of mercury Measures atmospheric pressure as ρgh

Pressure Measurements: Barometer A mercury barometer is used to measure the atmospheric pressure on an unidentified planet. A Hg (ρ = g·cm −3 ) barometer shows a column height of 92 mm. What is the atmospheric pressure?

Quick Question How deep would a scuba diver need to go for the absolute pressure to be 2 atms?

Question A person is standing in the water breathing through a snorkel. The water level is just over the tip of his head. Estimate the force his diaphragm would need exert in order for him to breath.

Question A wooden cube (ρ =.8) 1 meter to a side is submerged in water. List all the forces acting on it. What is the net force?

Buoyancy 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:

11.6 Archimedes’ Principle

Aluminum! A 1 kg block of aluminum (ρ= 2700 kg /m3) is under water. What will it’s free “fall acceleration” be?

If the object is less dense than the fluid, it will float. It will displace as it’s own weight in fluid, with the rest above the surface. The magnitude of the buoyant force is equal to the magnitude of its weight.

11.6 Archimedes’ Principle Example 9 A Swimming Raft The raft is made of solid square pinewood. Determine whether the raft floats in water and if so, how much of the raft is beneath the surface.

11.6 Archimedes’ Principle

The raft floats!

11.6 Archimedes’ Principle If the raft is floating:

An ideal fluid is incompressible and has zero viscosity. Water is really close to an ideal fluid. We have “steady-state” flow. When the flow is steady, streamlines are often used to represent the trajectories of the fluid particles.

New Term “Mass Flow Rate” How much mass of a fluid passes through a given cross section of area per second. For a well behaved fluid, it is constant.

11.8 The Equation of Continuity

EQUATION OF CONTINUITY The mass flow rate has the same value at every position along a tube that has a single entry and a single exit for fluid flow. SI Unit of Mass Flow Rate: kg/s

11.8 The Equation of Continuity Example 12 A Garden Hose A garden hose has an unobstructed opening with a cross sectional area of 2.85x10 -4 m 2. It fills a bucket with a volume of 8.00x10 -3 m 3 in 30 seconds. Find the speed of the water that leaves the hose through (a) the unobstructed opening and (b) an obstructed opening with half as much area.

11.8 The Equation of Continuity (a) (b)

BERNOULLI’S EQUATION In steady flow of a nonviscous, incompressible fluid, the pressure, the fluid speed, and the elevation at two points are related by:

Blood flow Blood flows out of the heart into the Aorta (r =.009 m) where the speed is.33 m/s. From there it flows into a capillary where the speed is m/s. Find the cross sectional area of the capillary.

11.10 Applications of Bernoulli’s Equation Conceptual Example 14 Tarpaulins and Bernoulli’s Equation When the truck is stationary, the tarpaulin lies flat, but it bulges outward when the truck is speeding down the highway. Account for this behavior.

11.10 Applications of Bernoulli’s Equation

Example 16 Efflux Speed The tank is open to the atmosphere at the top. Find and expression for the speed of the liquid leaving the pipe at the bottom.

Flow of an ideal fluid. Flow of a viscous fluid.

Problem The oil in a lubrication system has a density of 850 kg/m3. It flows through a cylindrical pipe of 8 cm diameter at 9.5 liters/second. Find the velocity of the oil. Find the velocity if the pipe narrows to 4cm diameter.

Question The watermain going into a home is 2cm in diameter, and the influx speed is 1.5 m/s. The pipe going to the shower in on the 2 nd floor (5m up) is 1cm in diameter. Find the pressure on the second floor if it is the only thing using water in the home. Find the volume flow rate of the showerhead.

A water tank has a radius R, is attached to a piston with radius r. How far (∆x) does the piston need to be pushed to raise the water level an amount (∆y)

If the piston is height H below the water level, how much force and how much pressure need to be applied by the piston to move the piston by the same ∆x as before?

A spigot is two meters below water level. IT starts off with a radius of 5 cm, then tapers to 2 cm. How long will it take for 1 liter to drain? (assume waterlevel doesn’t change much)

Find the velocity of water at the efflux.

Question An icecube is floating in a drinking glass of water. As it melts, what happens to the water level? ρ = 917 kg/m3