General Physics (PHY 2130)
Lecture VIII
• Solids and fluids
 density and pressure
 buoyant force
 Archimedes’ principle
 Fluids in motion
http://www.physics.wayne.edu/~apetrov/PHY2130/
Lightning Review
Last lecture:
1.
Rotational dynamics
 torque and angular momentum
 two equillibrium conditions
Review Problem: A figure skater stands on one spot on the ice (assumed frictionless) and
spins around with her arms extended. When she pulls in her arms, she reduces her
rotational inertia and her angular speed increases so that her angular momentum is
conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy
after she has pulled in her arms must be
1. the same.
2. larger because she’s rotating faster.
3. smaller because her rotational inertia is smaller.
Example:
Given:
Moments of inertia:
I1 and I2
Rotational kinetic energy is
KErot 
1 2 1
I  L
2
2
We know that (a) angular momentum L is conserved
and (b) angular velocity increases
Find:
K2 =?
Thus, rotational kinetic energy must increase!

Solids and Fluids
Question
What is a fluid?
1.
2.
3.
4.
A liquid
A gas
Anything that flows
Anything that can be made to change shape.
States of matter: Phase Transitions
ICE
WATER
Add
heat
STEAM
Add
heat
These are three states of matter (plasma is another one)
States of Matter
► Solid
► Liquid
► Gas
► Plasma
States of Matter
► Solid

Has definite volume

Has definite shape

Molecules are held in specific
location by electrical forces and
vibrate about equilibrium positions

Can be modeled as springs
connecting molecules
► Liquid
► Gas
► Plasma
States of Matter
► Solid

Crystalline solid
Atoms have an ordered structure
 Example is salt (red spheres are Na+
ions, blue spheres represent Cl- ions)

 Amorphous Solid
Atoms are arranged randomly
 Examples include glass

► Liquid
► Gas
► Plasma
States of Matter
► Solid
► Liquid

Has a definite volume

No definite shape

Exist at a higher temperature than solids

The molecules “wander” through the
liquid in a random fashion

► Gas
The intermolecular forces are not
strong enough to keep the
molecules in a fixed position
► Plasma
States of Matter
► Solid
► Liquid
► Gas

Has no definite volume

Has no definite shape

Molecules are in constant random motion

The molecules exert only weak forces on each other

Average distance between molecules is large
compared to the size of the molecules
► Plasma
States of Matter
► Solid
► Liquid
► Gas
► Plasma

Matter heated to a very high temperature

Many of the electrons are freed from the nucleus

Result is a collection of free, electrically charged ions

Plasmas exist inside stars or experimental reactors or
fluorescent light bulbs!
For more information:
http://fusedweb.pppl.gov/CPEP/Chart_Pages/4.CreatingConditions.html
Is there a concept that helps to distinguish
between those states of matter?
Density
►
The density of a substance of uniform composition is
defined as its mass per unit volume:
m

V
►
►
some examples:
4
Vsphere   R 3
3
Vcylinder   R 2 h
Vcube  a 3
The densities of most liquids and solids vary slightly
with changes in temperature and pressure
Densities of gases vary greatly with changes in
temperature and pressure (and generally 1000 smaller)
Units
SI
kg/m3
CGS
g/cm3 (1 g/cm3=1000 kg/m3 )
Sometimes: Specific Gravity
► The
specific gravity of a substance is
the ratio of its density to the density
of water at 4° C
 The
► Specific
density of water at 4° C is 1000 kg/m3
gravity is a unitless ratio
Pressure
►
Pressure of fluid is the ratio
of the force exerted by a
fluid on a submerged object
to area
F
P
A
Units
SI
Pascal (Pa=N/m2)
Example: 100 N over 1 m2 is P=(100 N)/(1 m2)=100 N/m2=100 Pa.
Pressure and Depth
►
►
►
If a fluid is at rest in a container,
all portions of the fluid must be in
static equilibrium
All points at the same depth must
be at the same pressure
(otherwise, the fluid would not be
in equilibrium)
Three external forces act on the
region of a cross-sectional area A
External forces: atmospheric, weight, normal
F  0
 PA  Mg  P0 A  0,
but : M   V   Ah, so : PA  P0 A   Agh
P  P0   gh
ConcepTest 1
You are measuring the pressure at the depth of 10 cm in three
different containers. Rank the values of pressure from the
greatest to the smallest:
1.
2.
3.
4.
1-2-3
2-1-3
3-2-1
It’s the same in all three
10 cm
1
2
3
Please fill your answer as question 1 of
General Purpose Answer Sheet
Pressure and Depth equation
P  Po  gh
► Po
is normal
atmospheric pressure
 1.013 x 105 Pa = 14.7
lb/in2
► The
pressure does not
depend upon the
shape of the container

Other units of pressure:
76.0 cm of mercury
One atmosphere 1 atm =
1.013 x 105 Pa
14.7 lb/in2
Example:
Find pressure at 100 m below ocean surface.
Given:
masses: h=100 m
P  P0   H 2O gh, so



P  9.8 105 Pa  103 kg m 3 9.8 m s 2 100 m 
 10 6 Pa
Find:
P=?
10  atmospheri c pressure 

Pascal’s Principle
►
►
►
A change in pressure applied to
an enclosed fluid is transmitted
undiminished to every point of
the fluid and to the walls of the
container.
The hydraulic press is an
important application of
Pascal’s Principle
F1 F2
P

A1 A2
Also used in hydraulic brakes,
forklifts, car lifts, etc.
Since A2>A1, then F2>F1 !!!
Measuring Pressure
►
►
The spring is calibrated by a
known force
The force the fluid exerts on
the piston is then measured



One end of the U-shaped tube
is open to the atmosphere
The other end is connected to
the pressure to be measured
Pressure at B is Po+ρgh


A long closed tube is
filled with mercury and
inverted in a dish of
mercury
Measures atmospheric
pressure as ρgh
How would you measure blood pressure?
Has to be:
(a) accurate
(b) non-invasive
(c) simple
sphygmomanometer
Question
Suppose that you placed an extended
object in the water. How does the
pressure at the top of this object
relate to the pressure at the
bottom?
1. It’s the same.
2. The pressure is greater at the top.
3. The pressure is greater at the
bottom.
4. Whatever…
Buoyant Force

This force is called the buoyant force.

What is the magnitude of that force?
F  B  P2  P1 A, but :
P2  P1   gh, so :
P1 A
B  P1   gh  P1 A   fluid ghA   fluid gV !
mg
P2 A
Buoyant Force
►
The magnitude of the buoyant force always equals
the weight of the displaced fluid
B   fluidVg  w fluid
The buoyant force is the same for a totally
submerged object of any size, shape, or density
► The buoyant force is exerted by the fluid
► Whether an object sinks or floats depends on the
relationship between the buoyant force and the
weight
►
Archimedes' Principle
Any object completely or partially submerged
in a fluid is buoyed up by a force whose
magnitude is equal to the weight of the fluid
displaced by the object.
This force is buoyant force.
Physical cause: pressure difference between the top and the bottom of the object
Archimedes’ Principle:
Totally Submerged Object
►
►
►
The upward buoyant force is B=ρfluidgVobj
The downward gravitational force is w=mg=ρobjgVobj
The net force is B-w=(ρfluid-ρobj)gVobj
Depending on the direction
of the net force, the object
will either float up or sink!
The net force is B-w=(ρfluid-ρobj)gVobj
►
►
The object is less dense
than the fluid ρfluid<ρobj
The object experiences
a net upward force

The object is more dense
than the fluid ρfluid>ρobj

The net force is downward,
so the object accelerates
downward
Archimedes’ Principle:
Floating Object
►
►
►
The object is in static equilibrium
The upward buoyant force is
balanced by the downward force of
gravity
Volume of the fluid displaced
corresponds to the volume of the
object beneath the fluid level
If B  mg :  fluid gV fluid   object gVobject , or
 obj V fluid

 fluid Vobj
Question 1
Suppose that you have a steel bar. Will
it float on water? Why?
Question 2
Suppose that you have a steel bar. Will
it float on water? Why?
How come that ships (which are made
of steel) can float?
Question 3
Suppose that your friend gave you a necklace
(crown, piece of yellow metal, …). He claims
that this object is made of pure gold. How can
you check his statement (without going through
his credit history)?
Question 3
Suppose that your friend gave you a necklace
(crown, piece of yellow metal, …). He claims
that this object is made of pure gold. How can
you check his statement (without going through
his credit history)?
Idea: determine density! Let’s weight the
object in and outside the water container:
Out:
mg   gV or

mg
gV
In:
" weight "  mg  B
 mg   fluid gV
V
" weight "mg
 fluid g
If  is not the same as gold, your friend is lying…
ConcepTest 2
Two identical glasses are filled to the same level with water.
One of the two glasses has ice cubes floating in it.Which
weighs more?
1. The glass without ice cubes.
2. The glass with ice cubes.
3. The two weigh the same.
Please fill your answer as question 3 of
General Purpose Answer Sheet
ConcepTest 3
Two identical glasses are filled to the same level with water.
One of the two glasses has ice cubes floating in it.When the
ice cubes melt, in which glass is the level of the water
higher?
1. The glass without ice cubes.
2. The glass with ice cubes.
3. It is the same in both.
Please fill your answer as question 5 of
General Purpose Answer Sheet
Fluids in Motion:
Streamline Flow
► Streamline
flow
 every particle that passes a particular point moves
exactly along the smooth path followed by
particles that passed the point earlier
 also called laminar flow
► Streamline
is the path
 different streamlines cannot cross each other
 the streamline at any point coincides with the
direction of fluid velocity at that point
Fluids in Motion:
Turbulent Flow
► The
flow becomes irregular
 exceeds a certain velocity
 any condition that causes abrupt changes in
velocity
► Eddy
currents are a characteristic of
turbulent flow
Fluid Flow: Viscosity
► Viscosity
is the degree of internal friction in
the fluid
► The internal friction is associated with the
resistance between two adjacent layers of
the fluid moving relative to each other
Characteristics of an Ideal Fluid
► The
fluid is nonviscous
 There is no internal friction between adjacent layers
► The
fluid is incompressible
 Its density is constant
► The
fluid is steady
 Its velocity, density and pressure do not change in time
► The
fluid moves without turbulence
 No eddy currents are present
Equation of Continuity
►
►
A1v1 = A2v2
The product of the crosssectional area of a pipe
and the fluid speed is a
constant
 Speed is high where the
pipe is narrow and speed
is low where the pipe has
a large diameter
►
Av is called the flow rate
Bernoulli’s Equation
►
►
►
►
Relates pressure to fluid speed and elevation
Bernoulli’s equation is a consequence of Conservation
of Energy applied to an ideal fluid
Assumes the fluid is incompressible and nonviscous,
and flows in a nonturbulent, steady-state manner
States that the sum of the pressure, kinetic energy
per unit volume, and the potential energy per unit
volume has the same value at all points along a
streamline
1 2
P  v  gy  constant
2
How to measure the speed of the fluid
flow: Venturi Meter
Shows fluid flowing
through a horizontal
constricted pipe
► Speed changes as
diameter changes
► Swiftly moving fluids exert
less pressure than do
slowly moving fluids
►