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Physics 172H
Modern Mechanics
Instructor: Dr. Mark Haugan
Office: PHYS 282
[email protected]
TAs: Alex Kryzwda
John Lorenz
[email protected]
[email protected]
Exam II: Wednesday 8:00-9:30pm in FRNY G140
Lecture 14: Matter & Interactions, Ch. 7.10-7.12 and Ch. 8
Dissipative Interactions:
When interactions like friction and aerodynamic drag between a system like this
block-wall system and its surroundings are negligible the system’s energy is
conserved.
1
1
Esys = Kb + U s ≈ mv 2 + ks x 2 = constant
2
2
When this is the case the amplitude of the block’s oscillation back and forth
through its equilibrium position remains constant.
If we measure carefully enough or observe the system for long enough, however,
that is not the case. The oscillation amplitude gradually decreases which means
that the system energy defined above, the macroscopic kinetic energy of the
block plus the spring potential energy, gradually decreases.
In such cases we say that the system’s initial energy is dissipated into
microscopic kinetic and potential energies of the system and its surroundings,
the block and floor getter warmer, for example.
Air Resistance
A small metal ball from our hand
to the ground when we let go as
if the constant gravitational force
exerted on it by the Earth is the
only force acting on it.
A coffee filter released in the same way falls differently. It reaches a nearly constant
‘terminal’ velocity
Approximate Air-Drag Formula
Observations about air resistance:
1.
|Fair| increases with speed.
2.
Increases with area.
3.
Depends on density of air.
4.
Depends on object’s shape (not mass)
5.
Opposes direction of motion.
The force due to air resistance
can be approximated as:
v1
Fair
Fair
mg
mg
v2
1
Fair ≈ − C ρ Av 2 v̂
2
ρ = density of air
A = area of object
v = speed of object
C = shape-dependent parameter
Application: Fuel Efficiency of a Car
1
Fair ≈ − C ρ Av 2 v̂
2
Daihatsu UFE III: C = 0.16
http://www.atmosphere.mpg.de/enid/Information_ss/Velocity___air_drag_507.html
Toyota Prius: C = 0.26
Toyota Tacoma: C = 0.44
Q1. A single falling coffee filter quickly reaches a constant terminal speed
of 1 m/s. At this speed, what do we know about the air resistance force?
1) Fair > Fgrav
2) Fair = Fgrav
3) Fair < Fgrav
4) not enough information
v filter
Q2. Three nested coffee filters fall, quickly
reaching a terminal speed of ~ 1.7 m/s. If the
Magnitude of the air resistance force on a single falling coffee filter was F1,
how large is the air resistance force on the three falling filters?
1) F1
2) 3F1
3) (1/3)F1
4) not enough information to know
Q3. If the cross-sectional area of a coffee filter were doubled, how would
you expect the air resistance force to change?
1) Its magnitude would be larger
2) Its magnitude would stay the same
3) Its magnitude would be smaller
Where does the energy go?
The hand exerts a constant force
which keeps the stretch of the
spring constant and the block
moving at constant velocity.
Apply the Energy Principle: System = Block
The block’s kinetic energy is constant, but work is done by the force
exerted by the of spring on the block. Where does the energy go?
The block and table-top get warmer. Macroscopically, we say that their thermal
energies increase. Microscopically, we say that the kinetic and potential
energies of the block’s and table-top’s atoms increase.
Driven Harmonic Oscillations
Oscillations damped by some viscous fluid:
“s” = the stretch
guess sinusoidal motion
A=
ωF2
(ω
2
F
c

− ωD2 ) −  ωD 
m

2
2
D
With an interesting amplitude
Resonance
A=
ωF2
(ω
2
F
−ω
) −  mc ω
2 2
D

D 

2
D
D=driving motor amplitude
A=amplitude of object
ωF=free oscillation (natural) frequency
ωD=driving frequency
c=friction constant
Want a big response? Drive it at the Resonant Frequency.
Don’t want a big response? Stay away from the Resonant Frequency!
Energy Quantization
Our study of energy to this point makes it seem that a system can possess any
amount of energy and that its energy will vary continuously as it interacts with
its surroundings.
It turns out that this is not the case and that this is most obvious for atomic
and other microscopic systems. In this class we will not be able to predict the
specific energies that a microscopic system can possess (you will study
quantum mechanics later to learn how we make such predictions), however,
we can use the fact of energy quantization to understand important phenomena
that are easily observed.
In chapter 8 we focus on the implications of energy quantization for the
emission and absorption of light energy by atoms and other microscopic
systems. Later, we will use it to better understand the microscopic physics
underlying thermal phenomena.
Spectroscopy
Spectrum of “white” light
is essentially continuous.
Spectrum of hydrogen
gas is clearly discrete.
What’s going on here?
Light and Energy
“cooler”
“hotter”
Different colors of light = different photon energies
E photon = p photon c =
h
λ photon
c
the wavelength λ determines
color of light
Energy Quantization in Atoms
Consider a hydrogen atom
(1 proton and 1 electron)
It turns out that the
electron seems only to be
able to execute certain
orbits.
Then U + Kelectron takes only
certain values.
N=1
N=2
N=3
NOTE: A satellite seems able to orbit the Earth at any height
and, so, U + Ksatellite can take on any value whatsoever.
Energy Quantization in Atoms
1 eV = 1.6 x 10-19 J
−13.6 eV
EN ≡ K e + U e =
N2
N = 1, 2,3, etc
electronic energy levels of hydrogen atom
(no other atom has these specific levels!)
Emission and Absorption of Photons
emitted
photon
absorbed
photon