Transcript Slide 1

Lecture #4
Cassandra Paul
Physics 7A
Summer Session II 2008
Graphing
On Monday we didn’t get to finish talking about
graphing, let’s do that now.
Diving: Potential Energy
From board From floor
2m or
0m or
5m
3m
-2m
or
1m
-3m
or
0m
At highest point, Tricia Woo is 2 meters above the board and 5 meters above the water, how
should we calculate her PE? Where should we measure the height from?
How can it not Matter!?
System: Diver
Initial: Highest point
Final: Just before
hitting water
KEtrans
Speed
+
From board From floor
2m or
5m
0m or
PEgravity
3m
Height
-2m
-
ΔKE +ΔPE= 0
-3m
½ m(vf2-vi2) + mg(hf-hi)= 0
(0 - 5)
(0.5)(50kg)(vf2-0) + (50kg)(10m/s2)(hf-hi)= 0
Δh is the same! Δh=-5 so vf = 10m/s (-3 - 2)
or
or
1m
0m
Instantaneous PE and KE
ΔKE +ΔPE= 0
(KEf – KEi) + (PEf- PEi) = 0
KEf + PEf - KEi - PEi = 0
KEf + PEf = KEi + PEi
= Etot
The sum total of all of the energies at one
point in time is equal to the total energy of
the system. In a closed system that value
is constant throughout the process.
KEanytime + PEanytime = Etot
Let’s graph our instantaneous
energy….
h
height
PE(J)
KE(J)
Etot(J)
(mgh) Etot-PE
sum
2
1000
0
1000
0
0
1000
1000
-2
-1000
2000
1000
-3
-1500
2500
1000
Mass=50kg Say g = 10m/s2
2m
0m
-2m
-3m
PE(J)
KE(J)
Etot(J)
height
(mgh)
Etot-PE
sum
2
1000
0
1000
0
0
1000
1000
-2
-1000
2000
1000
-3
-1500
2500
1000
Practice this!!!!
Energy in Joules
h
X axis is height in meters
Where have I set h=0?
3000
2500
A
2000
Energy in Joules
1500
pe
1000
ke
etot
500
B
0
E
D
C
B
A
-500
5m
-1000
X axis is height in meters
The answer is D!
Potential Energy is 1 meter above the water.
Why? Look at the Graph we can tell that PE is zero
At letter D, and KE is zero at A. From there we should
Be able to work backwards to interpret the graph.
C
D
E
h=0 at board (3 meters above water)
Energy in Joules
Same and different?
X axis is height in meters
h=0 (2 meters above water)
3000
2500
Energy in Joules
2000
1500
pe
1000
ke
etot
500
0
E
D
C
B
-500
-1000
X axis is height in meters
A
•PE can be
negative
•KE can’t be
negative. (½ mv2)
•Etot is NOT the
same for both
cases
•KE line remains
the same! (2500J just
before hitting the ground)
Springs!
Intro to Spring-Mass Oscillator Model
(another model in blue pages in
course notes)
Potential Energy: Springs
• Springs contain energy when you stretch or compress them.
We will use them a lot in Physics 7.
• The indicator is how much the spring is stretched or
compressed, x, from its equilibrium position.
ΔPEspring = (1/2) kΔx2
•k
is a measure of the “stiffness”
of the spring, with units [k] = kg/s2.
x
• x: Much easier to stretch a spring a little bit than a lot!
Mass-Spring Systems
ΔPEmass- spring = (1/2) kΔx2
• k is a property of the spring only
• PEmass-spring does not depend on mass
• PE = 0 arbitrary
x
x
x
Mass-Spring Systems
KE
Speed
initial
PEmassspring
∆y
final
Δx = -2cm
Mass-Spring Systems
System: mass-spring
Initial: mass at rest at 2cm
Final: mass at x=0
initial
(x
KE
Speed
X = 2cm
PEmassspring
∆x
final
Potential Energy and Forces:
Springs, Gravitational
The indicator is how much the spring is stretched or compressed, x, from its equilibrium
position.
x
ΔPEspring = (1/2) kΔx2
The indicator is the change in vertical distance that
the object moved (I.e. change in the distance between the
center of the Earth and the object)
∆PEgrav = mg h
h
PE vs displacement: Force
[-]
Displacement from equilibrium y [+]
PE vs displacement: Force
direction of force
[-]
Displacement from equilibrium y [+]
PE vs displacement: Force
direction of force
[-]
Displacement from equilibrium y [+]
PE vs displacement: Force
On this side
force
pushes
down
Equilibrium
On this side
force
pushes up
[-]
Displacement from equilibrium y [+]
Forces from potentials
point in direction
that (locally) lowers PE
Graphing Energies
What are the x-axis, y axis? Units?
x axis (independent variable: height)
y axis (dependent variable: PEgrav)
Which quantity (energy) is the easiest to graph?
Etot ? PEgrav? What about KE?
Where should the origin (0) be placed? Where
does it most make sense?
Should the floor be 0m?
Practice: Pendulum
Cassandra, how can we do
a practice problem with a
pendulum? We’ve never
learned anything about
pendulums!!!!!!!!
Yes you have! Just
use the Model,
You’ll be surprised
How much it will
tell you.
Practice: Pendulum
A 2kg pendulum swings to a maximum height of
3 meters. At it’s lowest point it is one meter
above the floor. Find the maximum speed of
the pendulum, and then graph Etot, PE and KE
as a function of distance.
A)6.32 m/s
B)7.74 m/s
C)60m/s
Initial
D)None of these
2m
3m
Final (Still in motion)
Let’s use the instantaneous method instead of the Energy Diagram
SUM OF ALL ENERGIES PRESENT = Etot
PEanytime + KEanytime = Etot
2m
PEtop + KEtop = Etot
mgh + 0 = Etot
3m
Set h=0 at floor
(2)(10)(3)J + 0 = Etot =60J
PEbottom + KEbottom = Etot
mgh + ½ mv2 = 60J
Initial
Final (Still in motion)
h (m) PE (J) KE (J) Etot
(J)
3
60
0
60
1
20
40
60
(2)(10)(1) + ½ 2v2 = 60J  v=6.32m/s
Energy 
h (m) PE (J) KE (J) Etot (J)
h (m) PE (J) KE (J) Etot
(J)
3
60
0
60
3
60
0
60
2
40
20
60
1
20
40
60
1
20
40
60
Height 
Initial
2m
3m
Back to Springs…
OK so springs are cool for physicists, but does
understanding the spring help us understand
anything else??
• Three-phase model of matter
What is Ebond anyway??
• Energy-interaction model
r
• Mass-spring oscillator
• Particle model of matter
 Particle model of bond energy
 Particle model of thermal energy
We will model real atoms of liquids
and solids as oscillating masses and springs
Particle Model of Matter
•Thermodynamics
• Ideal gas model
• Statistical model of thermodynamics
Intro to Particle Model of Matter
This model helps us understand how particles
interact with each other at the molecular level.
(another blue page in your course notes)
Particle Model of Matter
Goal : To understand macroscopic phenomena
(e.g. melting, vaporizing) and macrocopic properties of
matter such as phases, temperature, heat capacities,
in terms of microscopic constituents and its behavior.
We will model real atoms of liquids
and solids as oscillating masses
r
and springs
Model Bonded Atoms
as Masses on Spring
~ two atomic size particles interacting
via“pair-wise potential” a.k.a. Lennard-Jones Potential
Atom 1
(anchored)
Atom 2
(bonded or un-bonded)
r0
Ro the distance that the particles are at equilibrium
We observe the system oscillating. At one instance
we take a snapshot of the oscillation and see this:
r > r0
1
2
Which way is the force on particle #2?
A)
B)
C)
D)
E)
To the right because the particle is traveling to the right. 
We can’t tell because we don’t know which way the particle is traveling
To the left because the spring is pulling from the left. 
We can’t tell which way it’s traveling but we know the force is to the right. 
There is no force acting particle 2.
Energy 
Lennard-Jones
Potential
(pair-wise potential)
IT’S JUST A WEIRD SPRING!
Equilibrium
separation
ro
Distance 
Not to scale
(particles are just about touching at equilibrium)
Link to Applet:
• http://polymer.bu.edu/java/java/intermol/ind
ex.html
Don’t worry
about this graph
on the right.
Introduction to the Particle Model
Potential Energy between two atoms
PE
Repulsive:
Atoms push apart as they
get too close
Flattening:
atoms have negligible forces
at large separation.
separation
r
Distance between the atoms
Equations to memorize, and more
importantly know how to use for
Monday’s Quiz
mcΔT = ΔEth
±lΔmΔHl = ΔEb
½ mΔ(v2)= ½ m (vf2-vi2) = ΔKEtrans
½ k (Δxf2-Δxi2) = ΔPEspring
mgΔh = ΔPEgrav
Also, we don’t use an equation for rotational energy, but know how to tell if it’s there.
DL sections
•
•
•
•
Swapno:
11:00AM Everson Section 1
Amandeep: 11:00AM Roesller Section 2
Yi:
1:40PM Everson Section 3
Chun-Yen: 1:40PM Roesller Section 4