Fluids and Density
Mass Density
Example 1
The body of a
man whose weight is 690 N contains about 
(5.5
qt) of blood. (a) Find the blood's weight and (b) express
it as a percentage of the body weight. (Blood density of 
).
Pressure
Pressure in Liquids
Pressure of a liquid with density, at depth.
Hydrostatic Pressure
Example 2
Find the pressure
exerted on the skin of a balloon if you press with a force of 2.1 N using (a) your
finger or (b) a needle. Assume the area of your fingertip
is 
,
and the area of the needle tip is 
. (c) Find
the minimum force necessary to pop the balloon with the needle, given
that the balloon pops with a pressure of 
.
Example 3
The Titanic was found in 1985 lying on the bottom of the North Atlantic at a depth of 2.5 miles. What is the pressure at this depth?
Is this possible?
A connected liquid in hydrostatic equilibrium rises to the same height in all open regions of the container.
The pressure is the same at all points on a horizontal line through a connected liquid in hydrostatic equilibrium.
Pascal's Principle – If the pressure at one point in an incompressible fluid is changed, the pressure at every other point in the fluid changes by the same amount.
Measuring and Using Pressure
Solving Hydrostatic Problems
Example 4
A cubical box 20.00 cm on a side is completely immersed in a fluid. At the top of the box the pressure is 105.0 kPa; at the bottom the pressure is 106.8 kPa. What is the density of the fluid?
Example 5
One day, while swimming below the surface of the ocean, you let out a small bubble of air from your mouth. As the bubble rises toward the surface, does its diameter (a) increase, (b) decrease, or (c) stay the same?
Example 6
A U-shaped tube is filled
mostly with water, but a small amount of vegetable oil has been added to
one side, as shown in sketch. The density of the water is 
,
and the density of the vegetable oil is 
.
If the depth of the oil is 5.00 cm, what is the difference in level h between
the top of the oil on one side of the U and the top of the water on the other
side?
Buoyancy
Archimedes' Principle – A fluid exerts an upward
buoyant force 
on
an object immersed in or floating on the fluid. The magnitude of the buoyant
force equals the weight of the fluid displaced by the object.
Example 7
Archimedes supposedly was asked to determine whether a crown made for the king consisted of pure gold. Legend has it that he solved this problem by weighing the crown first in air and then in water as shown. Suppose the scale read 7.84 N in air and 6.86 N in water. What should Archimedes have told the king?
Example 8
A raft is constructed
of wood having a density of 
.
Its surface area is 
,
and its volume is 
.
When the raft is placed in fresh water of density 
,
to what depth does the raft sink in the water?
Fluids in Motion
Assumptions about ideal fluids
1. The fluid is incompressible.
2. The flow is steady. (Laminar Flow – A smooth, streamline type of viscous flow in which the fluid behaves as a system of orderly layers, with no eddies or irregular fluctuations.
3. The fluid is nonviscous.
Equation on Continuity
The volume of an incompressible fluid entering one part of a tube or pipe must be matched by an equal volume leaving downstream.
Flow is faster in narrower parts of a tube, slower in wider parts
Volume Flow Rate – 
Example 9
Water travels through a 9.6 cm diameter fire hose with a speed of 1.3 m/s. At the end of the hose, the water flows out through a nozzle whose diameter is 2.5 cm. What is the speed of the water coming out of the nozzle?
Fluid Dynamics
Pressure Gradient – In atmospheric sciences (meteorology, climatology and related fields), the pressure gradient (typically of air, more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure changes the most rapidly around a particular location.
An ideal fluid accelerates whenever there is a pressure gradient
The pressure is higher at a point along a streamline where the fluid is moving slower, lower where the fluid is moving faster. (Bernoulli Effect)
Bernoulli's Equation
Example 10
Using the information from the previous example, suppose the pressure in the fire hose is 350 kPa. What is the pressure in the nozzle?
Example 11
Water flows with constant speed through a garden hose that goes up a step 20.0 cm high. If the water pressure is 143 kPa at the bottom of the step, what is its pressure at the top of the step?
Example 12
Repeat the previous example with the following additional information: (a) the cross-sectional area of the hose at the top of the step is half that at the bottom of the step, and (b) the speed of the water at the bottom of the step is 1.20 m/s.
Torricelli's Result
Example 13
In designing a backyard water fountain, a gardener wants a stream of water to exit from the bottom of one can and land in a second one, as shown. The top of the second can is 0.500 m below the hole in the first can, which has water in it to a depth of 0.150 m. How far to the right of the first can must the second one be placed to catch the stream of water?
Example 14
A nearsighted sheriff fires at a cattle rustler with his trusty six-shooter. Fortunately for the cattle rustler, the bullet misses him and penetrates the town water tank and causes a leak. If the top of the tank is open to the atmosphere, determine the speed at which the water leaves the hole when the water level is 0.500 m above the hole.
