Transcript Lecture 8: Forces & The Laws of Motion

Lecture 19:
Introduction to
Solids & Fluids
Questions of Yesterday
1) A solid sphere and a hoop of equal radius and mass are both rolled
up an incline with the same initial velocity. Which object will travel
farthest up the inclined plane?
a) the sphere
b) the hoop
c) they’ll both travel the same distance up the plane
d) it depends on the angle of the incline
2) If an acrobat rotates once each second while sailing through the air,
and then contracts to reduce her moment of inertia to 1/3 of what is
was, how many rotations per second will result?
a) once each second
b) 3 times each second
c) 1/3 times each second
d) 9 times each second
Practice Problem
A 10.00-kg cylindrical reel with the radius of 0.500 m and a frictionless
axle starts from rest and speeds up uniformly as a 5.00 kg bucket falls
into a well, making a light rope unwind from the reel. The bucket starts
from rest and falls for 5.00 s.
10.0 kg
0.500 m
What is the linear acceleration
of the falling bucket?
How far does it drop?
What is the angular acceleration of the reel?
5.00 kg
Use energy conservation principles to determine
the speed of the spool
after the bucket has fallen 5.00 m
4 States of Matter
Solid
Definite
Shape &
Volume
Molecules
close together
& slow
Fluid
Gas
Definite
Volume
Takes shape of
container
Molecules
farther apart
& faster
Expands to fill
any volume
Takes shape of
Container
Molecules even
farther apart
& even faster
Plasma
Expands to fill
any volume
Takes shape of
Container
Made up of ions
Fastest of all
matter states
Density
Distance between molecules
Density
M
r=
V
Density
The amount of matter (mass)
in a given volume
Solids
Applying force can change shape & size (deform)
When force is removed -> original shape & size
Definite
Shape &
Volume
Remind you of anything?
x=0
SOLIDS are ELASTIC
FS = -kx
STRESS = ELASTIC MODULUS x STRAIN
Force per
Unit Area
Elasticity of
material
Measure of
Deformation
Length Elasticity
Elastic Modulus and Induced strain depends on type of stress
F
A
F
L0
L0
DL
F
L0
DL
F
L0
STRESS = ELASTIC MODULUS x STRAIN
SI Units =
N/m2 =
Pascal (Pa)
F = Y DL
L0
A
LENGTH
Relative
Length Change
Young’s Modulus
Volume Elasticity
F
DV
F
V0
V0
STRESS = ELASTIC MODULUS x STRAIN
SI Units =
N/m2 =
Pascal (Pa)
F = -B DV
V0
A
VOLUME
Relative
Volume Change
Bulk Modulus
Pressure
Uniform force F is acting over entire
surface area A in a direction
perpendicular to surface area
F
PRESSURE (P)
Perpendicular Force per unit area
P= F
A
A
When would this
situation occur?
Pressure
Uniform force F is acting over entire
surface area A in a direction
perpendicular to surface area
F
PRESSURE (P)
Perpendicular Force per unit area
P= F
A
Fluids are NOT elastic ->
do not return to initial state after being deformed
But…
Fluids do exerted force
A
Pressure
Force exerted by fluid on a
submerged object is always
PERPENDICULAR
to surface of object
F
Fluids exert PRESSURE on
submerged objects and the
walls of their container
Fluids are NOT elastic ->
do not return to initial state after being deformed
But…
Fluids do exerted force
A
Pressure
If a fluid is at rest in a container what
do we know about it?
y
0
It is in EQUILIBRIUM, so…
The net FORCE acting on any
portion of fluid is ZERO
F1(y1) = -F2(y1)
all points at the same DEPTH
must be at the same PRESSURE
P1(y1) = P2(y1)
F1
F2
y1
y2
Pressure
The net FORCE acting on any
portion of fluid is ZERO
y
0
all points at the same DEPTH
must be at the same PRESSURE
P1A
y1
∑Fy = P2A - P1A - Mg = 0
y2
M
r=
V
P2 = P1 + rg(y1 - y2)
Mg
P2A
Pressure
What is the pressure at the surface
of the fluid
(open to the air)?
P0A
y
0
Gas making up atmosphere
exerts pressure on fluid
h
P2 = P1 + rg(y1 - y2)
P = P0 + rgh
y1
PA
Pressure P at depth h below the surface of a liquid
open to the atmosphere is greater than atmosphere pressure
(P0 = 1.013*105 Pa) by the amount rgh
Pressure
What if you change the pressure
exerted at the surface?
F
P = P0 + rgh
PASCAL’S PRINCIPLE
A change in pressure applied to an
enclosed fluid is transmitted
undiminished to every point of the
fluid and the walls of the container
Pressure
PASCAL’S PRINCIPLE
A change in pressure applied to an enclosed fluid is transmitted
undiminished to every point of the fluid and the walls of the
container
What happens if you apply a force F1 to one side of this
apparatus?
F1
A1
A2
F2
Pressure
PASCAL’S PRINCIPLE
A change in pressure applied to an enclosed fluid is transmitted
undiminished to every point of the fluid and the walls of the
container
F1/A1 = F2/A2
F1
Dx1
F2
Dx2
Pressure
F1/A1 = F2/A2
How does Dx1 compare to Dx2?
Fluids have a definite volume (incompressible) -> DV = 0
F1Dx1 = F2Dx2
What does
this tell you
about the
work done
on the fluid?
F1
Dx1
F2
Dx2
Buoyancy
What allows an object to float
in a fluid?
Is the object in equilibrium?
M
V
Buoyancy
What allows an object to float
in a fluid?
Is the object in equilibrium?
∑Fy = B - mobjg = 0
B
m
mg
V
Buoyancy
What allows an object to float
in a fluid?
P1A
Is the object in equilibrium?
∑Fy = B - mobjg = 0
∑Fy = P2A - P1A - Mfluidg = 0
B = P2A - P1A = rfluidVfluidg
BUOYANT
FORCE
M
Mg
P2A
V
Buoyancy
What allows an object to float
in a fluid?
B
Is the object in equilibrium?
m
∑Fy = B - mobjg = 0
∑Fy = P2A - P1A - Mfluidg = 0
B = P2A - P1A = rfluidVfluidg
BUOYANT
FORCE
mg
V
If the object is in
equilibrium with the fluid…
Vfluid
robj
=
Vobj
rfluid
Buoyancy
B = rfluidVfluidg
What if the object is rising or sinking?
B
m
V
mg a = 0
B = mobjg
Vfluid
robj
=
Vobj
rfluid
B
B
a
m
mg
V
B - mobjg > 0
(rfluid-robj)Vobjg > 0
m
mg
a
V
B - mobjg < 0
(rfluid-robj)Vobjg < 0
Properties of an Ideal Fluid
An Ideal Fluid is…
NONVISCOUS
no internal friction between adjacent layers
INCOMPRESSIBLE
constant density
Ideal Fluid Motion is…
STEADY
velocity, density, pressure at each point is constant in time
WITHOUT TURBULENCE
Angular velocity about center of each element is zero
All points can translate but not rotate
Ideal Fluid Motion
How does v1 compare to v2?
v2
A2
Dx2
Mass is conserved
A1
v1
Dx1 = v1Dt
DM1 = DM2
r1A1v1 = r2A2v2
Fluid is incompressible
Volume of fluid leaving 1 =
Volume of fluid entering 2
in the same time interval
A1v1 = A2v2
Equation of Continuity
Ideal Fluid Motion
P2A2
v2
Is energy conserved in an ideal
fluid?
Dx2
P1A1
DM1 = DM2
Dy2
v1
Dx1
Dy1
What is the work done on the fluid?
W1 = P1A1Dx1
W = P1V - P2V
W2 = -P2A2Dx2
Ideal Fluid Motion
W = P1V - P2V
P2A2
v2
A2
Dx2
P1A1
v1
Dx1
Dy2
Dy1
Is the energy of the fluid changing?
What types of energy are present?
Wfluid = DKE + DPE
P1 + (1/2)rv12 + rgy1 = P2 + (1/2)rv22 + rgy2
Ideal Fluid Motion
W = P1V - P2V
P2A2
v2
A2
Dx2
P1A1
v1
Dx1
Dy2
Dy1
BERNOULLI’S EQUATION
The sum of pressure, kinetic energy per unit volume,
and potential energy per unit volume
is equal at all points along the streamline
P1 + (1/2)rv12 + rgy1 = P2 + (1/2)rv22 + rgy2
Questions of the Day
1) Two women of equal mass are standing on the same hard wood
floor. One is wearing high heels and the other is wearing tennis
shoes. Which statement is NOT true?
a) both women exert the same force on the floor
b) both women exert the same pressure on the floor
c) the normal force that the floor exerts is the same for both women
2) A boulder is thrown into a deep lake. As the rock sinks deeper and
deeper into the water what happens to the buoyant force?
a) it increases
b) it decreases
c) it stays the same