Chapter 14 PHYSICS 2048C Fluids

What Is a Fluid?  A fluid, in contrast to a solid, is a substance that can flow.  Fluids conform to the boundaries of any container in which we put them.

Density and Pressure  Density  To find the density ρ of a fluid at any point, we isolate a small volume element ΔV around that point and measure the mass Δm of the fluid contained within that element. The density is then

Some Densities

Pressure  We define the pressure on the piston from the fluid as where F is the magnitude of the normal force on area A. The SI unit of pressure is the Newton per square meter, which is given a special name, the Pascal (Pa).

Some Pressures

Fluids at Rest  Here we want to find an expression for hydrostatic pressure as a function of depth or altitude. m = ρV

Fluids at Rest  The pressure at a point in a fluid in static equilibrium depends on the depth of that point but not on any horizontal dimension of the fluid or its container.  To find the atmospheric pressure at a distance d above level 1

Example  The U-tube, example 3 page 364, contains two liquids in static equilibrium: Water of density ρ w (= 998 kg/m 3 ) is in the right arm, and oil of unknown density ρ x is in the left. Measurement gives l = 135 mm and d = 12.3 mm. What is the density of the oil?

Example

 In the right arm, the interface is a distance l below the free surface of the water  In the left arm, the interface is a distance l + d below the free surface of the oil

Measuring Pressure  The Mercury Barometer a device used to measure the pressure of the atmosphere the long glass tube is filled with mercury and inverted with its open end in a dish of mercury, as the figure shows. The space above the mercury column contains only mercury vapor, whose pressure is so small at ordinary temperatures that it can be neglected.

Measuring Pressure  The Open-Tube Manometer  It is a device used to measure the gauge pressure p g of a gas It consists of a U -tube containing a liquid, with one end of the tube connected to the vessel whose gauge pressure we wish to measure and the other end open to the atmosphere. We can find the gauge pressure in terms of the height h

Pascal's Principle  A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.  Example for that in your daily live is When you squeeze one end of a tube to get toothpaste out the other end

Demonstrating Pascal's Principle If you add a little more lead shot to the container to increase P ext by an amount ∆p ext. The quantities ρ, g, h are unchanged.

Pascal's Principle and the Hydraulic Lever  The Equation shows that the output force F o on the load must be greater than the input force F i if A o > A i, as is the case in the figure If we move the input piston downward a distance d i, the output piston moves upward a distance d o, such that the same volume V of the incompressible liquid is displaced at both pistons

Pascal's Principle and the Hydraulic Lever we can write the output work as With a hydraulic lever, a given force applied over a given distance can be transformed to a greater force applied over a smaller distance.

Archimedes' Principle  This net upward force is a buoyant force. It exists because the pressure in the surrounding water increases with depth below the surface. Plastic sack

Archimedes' Principle  When a body is fully or partially submerged in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight m f g of the fluid that has been displaced by the body.

Apparent Weight in a Fluid  A spherical, helium-filled balloon has a radius R of 12.0 m. The balloon, support cables, and basket have a mass m of 196 kg. What maximum load M can the balloon support while it floats at an altitude at which the helium density ρ He is kg/m 3 and the air density ρ air is 1.25 kg/m 3 ? Assume that only the balloon displaces air.

Solution  F g = m f g ρ = m/V

The Equation of Continuity  the speed v of the water depends on the cross- sectional area A through which the water flows

The Equation of Continuity This relation between speed and cross- sectional area is called the equation of continuity for the flow of an ideal fluid. It tells us that the flow speed increases when we decrease the cross-sectional area through which the fluid flows

The Equation of Continuity  R V is the volume flow rate of the fluid Rm is the mass flow rate

Bernoulli's Equation  If the speed of a fluid element increases as the element travels along a horizontal streamline, the pressure of the fluid must decrease, and conversely.

Proof of Bernoulli's Equation  The work done on the system is then p 1 ΔV, and the work done by the system is –p 2 ΔV.

Proof of Bernoulli's Equation  Bernoulli's Equation W p +W g =ΔK

Problem  block of wood floats in fresh water with two-third of its volume V submerged and in oil with 0.90 V submerged. Find the density of (a) the wood and (b) the oil.

Solution  a- Let V be the volume of the block. Then, the submerged volume is V o = 2/3(V)  According to Archimedes' principle the weight of the displaced water is equal to the weight of the block, so b- If ρ o is the density of the oil, then Archimedes' principle yields