Phys 061 Physics 1

Chapter 7 Fluid Mechanics

“Liquids and gases have the ability to flow; hence they are called fluids. Because they are
both fluids, they obey similar mechanical laws.”

Density – is the measure of compactness. It is a measure of how much mass is squeezed into a given
space, the amount of matter per unit volume:
mass 𝑚
Density = or D=
volume 𝑉

Units: kg/m3, kg/L, g/mL, g/cc

A quantity known as weight density, commonly used when discussing liquid pressure, is weight per unit
volume:
weight
Weight density=
volume
Units: N/m3, dyn/cm3

Quick Questions:
1. Which has greater density – 1 kg of water or 10 kg of water?
2. Which has greater density – 5 kg of lead or 10 kg of aluminum?

Pressure is defined as the ratio of the magnitude of a force applied perpendicular to a surface and the area
over which the force is exerted:
force
Pressure=
area
Quick Questions:
3. Two identical books are positioned on the table as shown. Which of these
exert greater pressure against the table?
A. lying down B. standing upright

Figure 5.1 Books on table

Units: Pa (pascal) lb/in2 (or psi, pound per square inch)

atm (atmosphere) mmHg (millimeter mercury)
Pressure is a scalar quantity and has no direction.

Pressure in Liquid
The Pressure exerted by a liquid varies with depth; the deeper an object is in a
fluid, the greater the pressure acting on the object.
The pressure in liquid can be expressed as:

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where p0 is the pressure at the surface of liquid, ρ is the density of liquid, g is acceleration due to gravity
and h is depth. If the surface of liquid is exposed to surrounding air, then p0 is equal to the pressure in the
air. If the liquid is inside a container and there is no gas on top of the liquid, then p0 is zero.
The magnitude of the force exerted by a fluid on as surface area is
Force = Pressure × area or F = pA
The direction of the force caused by the pressure is perpendicular and towards the
surface.
Absolute Pressure and Gauge Pressure
The excess pressure above atmospheric pressure is usually called gauge pressure, and the total pressure
is called absolute pressure.
Ex 7.1 Water stands 12.0 m deep in a storage tank whose top is open to the atmosphere. What are the
absolute and gauge pressures at the bottom of the tank?

Archimedes’ Principle
“An immersed body is buoyed up by a force equal to the weight of the liquid it displaces.”

The net upward force exerted by the liquid is called buoyant force.
A completely submerged object always displaces a volume of liquid equal to its
own volume. (Refer to Figure 5.10)
The buoyant force can be calculated using the formula:

B   f Vsub g or B   f Vdisp g Figure 7.2 The raised level due

to placing a stone in the
container is the same as if a
where ρ f is the density of fluid, g is acceleration due to gravity, Vsub is the volume of water equal to the
volume of the submerged portion of object (for completely submerged object, volume of the stone were
of course, this is just equal to the volume of the whole object), and Vdisp is the poured in.

volume of the displaced fluid.

Note: A little thought tells us that the volume of the stone—that is, the amount of space it occupies or its
number of cubic centimeters—is equal to the volume of water displaced. Place any object in a container
partially filled with water, and the level of the surface rises (Figure 7.2). How high? That would be to
exactly the level that would be reached by pouring in a volume of water equal to the volume of the
submerged object.

Ex 7.2 A 1-L container completely filled with lead has a mass of 11.3 kg and is submerged in water. What
is the buoyant force acting on it? The density of water is 1 kg/L?

Ex 7.3 A 15.0-kg solid gold statue is raised from the sea bottom. What is the tension in the hoisting cable
(assumed massless) when: (a) the statue is at rest and completely underwater and (b) at rest and
completely out of the water. The densities of gold and seawater are 19,300 kg/m 3 and 1,025 kg/m3.

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Comparing densities of object and of fluid

When the density of the object is greater than the density of fluid, the object sinks.
When the density of the object is less than the density of fluid, the object floats.
Flotation
A floating object displaces a weight of fluid equal to its own
weight.

Atmospheric Pressure
We live at the bottom of an ocean of air. The atmosphere,
much like the water in a lake, exerts a pressure. Just as water
pressure is caused by the weight of water, atmospheric pressure is
caused by the weight of air. Whereas water in a lake has the same
density at any level (assuming constant temperature), the density
of air in the atmosphere decreases with altitude. The average
atmospheric pressure at sea level is 101.3 kPa. The pressure of the Figure 7.3 A barometer
atmosphere is not uniform. Besides altitude variations, there are
variations in atmospheric pressure at any locality due to moving fronts and storms.
Instruments used for measuring the pressure of the atmosphere are called barometer.

Pascal’s principle:
“A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all
points in the fluid.”
This principle was discovered in 17th century by theologian and scientist Blaise Pascal.
This principle is especially useful in hydraulic systems.

Ex 7.4 How large a force is needed on a small piston of area 2 cm2 to support a 1000-N weight resting on
a piston of area 20 cm2?

Buoyancy in a Gas
Archimedes’ principle holds for air just as it does for water:
“An object surrounded by air is buoyed up by a force equal to the weight of the air displaced.”
Fluid Flow
An ideal fluid is a fluid that is incompressible (that is, its density cannot change) and has no internal friction
(called viscosity). Liquids are approximately incompressible in most situations, and we may also treat a gas
as incompressible if the pressure differences from one region to another are not too great.
The path of an individual particle in a moving fluid is called a flow line. If the overall flow pattern does not
change with time, the flow is called steady flow.

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1. Laminar flow
adjacent layers of fluid slide smoothly past each other and the flow is steady. (A lamina is a thin sheet

Figure 7.4 A laminar flow

2. Turbulent flow
when boundary surfaces cause abrupt changes in velocity, the flow can
become irregular and chaotic. This is called turbulent flow. This usually
occurs at sufficiently high flow rates.

Continuity Equation
The mass of a moving fluid doesn’t change as it flows. This leads to an important quantitative relationship
called the continuity equation.

where v1 is the speed of fluid when flowing through the cross sectional area A1 and v2 is the speed of same
fluid when flowing through the cross sectional area A2.
The product of cross sectional area A and speed of fluid v is the volume flow rate the rate at which volume
crosses a section of the tube:

Bernoulli’s principle
“When the speed of a fluid increases, pressure in the fluid decreases.”

Review Questions:
1. A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom
of the lake. The sphere has a volume of 1.5 m3 and the tension in the cord is 900 N. (a) Calculate the
buoyant force exerted by the water on the sphere. What is the mass of the sphere? (b) The cord breaks
and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be
submerged?
2. A hot-air balloon has a volume of 2300 m3. The balloon fabric (the envelope) weighs 900 N. The basket
with gear and full propane tanks weighs 1700 N. If the balloon can barely lift an additional 3200 N of
passengers, breakfast, and champagne when the outside air density is 1.23 kg/m 3,

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(a) How much is the buoyant force exerted on this hot-air balloon by the surrounding air? (b) What is
the weight of the heated gas inside the hot-air balloon? (c) What is the average density of the
heated gases in the envelope?
3. (a) If you swim twice as deep in water, how much more water pressure is exerted on your ears? (b) If
you swim in salt water, is the pressure greater than in fresh water at the same depth? Why or why not?
4. Why does buoyant force act upward on an object submerged in water?
5. (a) Does the buoyant force on a fully submerged object depend on the weight of the object or on the
weight of the fluid displaced by the object? (b) Does the force depend on the weight of the object or on
its volume? Defend your answer.
6. Does Bernoulli’s principle refer to internal pressure changes in a fluid, or to pressures that a fluid can
exert on objects in the path of the flowing fluid? or both?