Intermediate Mechanics
Physics 321
Richard Sonnenfeld
New Mexico Tech
:00
Lecture #1 of 25
Course goals

Physics Concepts / Mathematical Methods
Class background / interests / class photo
Course Motivation
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“Why you will learn it”
Course outline (hand-out)
Course “mechanics” (hand-outs)
Basic Vector Relationships
Newton’s Laws
Worked problems
Inertia of brick and ketchup III-3,4
2 :02
Physics Concepts
Classical Mechanics
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Study of how things move
Newton’s laws
Conservation laws
Solutions in different reference frames (including
rotating and accelerated reference frames)
Lagrangian formulation (and Hamiltonian form.)
Central force problems – orbital mechanics
Rigid body-motion
Oscillations lightly
Chaos
3 :04
Mathematical Methods
Vector Calculus
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Differential equations of vector quantities
Partial differential equations
More tricks w/ cross product and dot product
Stokes Theorem
“Div, grad, curl and all that”
Matrices
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Coordinate change / rotations
Diagonalization / eigenvalues / principal axes
Lagrangian formulation
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Calculus of variations
“Functionals” and operators
Lagrange multipliers for constraints
General Mathematical competence
4 :06
Class Background and Interests
Majors
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Physics?
EE?
CS? Other?
Preparation
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Assume
Assume
Assume
Assume
Math 231 (Vector Calc)
Phys 242 (Waves)
Math 335 (Diff. Eq) concurrent
Phys 333 (E&M) concurrent
Year at tech
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2nd 3rd 4th 5th
Graduate school?
Greatest area of interest in mechanics
5 :08
Physics Motivation
Physics component
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Classical mechanics is incredibly useful
 Applies to everything bigger than an atom and slower
than about 100,000 miles/sec
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Lagrangian method allows “automatic” generation
of correct differential equations for complex
mechanical systems, in generalized coordinates,
with constraints
Machines and structures / Electron beams /
atmospheric phenomena / stellar-planetary
motions / vehicles / fluids in pipes
6 :10
Mathematics Motivation
Mathematics component
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Hamiltonian formulation transfers DIRECTLY to
quantum mechanics
Matrix approaches also critical for quantum
Differential equations and vector calculus
completely relevant for advanced E&M and wave
propagation classes
Functionals, partial derivatives, vector calculus.
“Real math”. Good grad-school preparation.
7 :12
About instructor
15 years post-doctoral industry experience
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Materials studies (tribology) for hard-drives
Automated mechanical and magnetic
measurements of hard-drives
Bringing a 20-million unit/year product to market
Likes engineering applications of physics
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Will endeavor to provide interesting problems that
correspond to the real world
8 :16
Course “Mechanics”
WebCT / Syllabus and Homework
Office hours, Testing and Grading
9 :26
Vectors and Central forces
r1  r2
r1  r2
Vectors
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r1
r2
r2
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Many forces are of
form F ( r1  r2 )
Remove dependence
of result on choice of
origin
r1
Origin 1
Origin 2
10 :30
Vector relationships

Vectors
dr dx
dy
dz

xˆ 
yˆ  zˆ
 Allow ready
dt dt
dt
dt
representation of 3


 
(or more!)
x
r  r  r r
xˆ 
components at once.
x
 Equations written in
 
r  s  r s cos( )
vector notation are
more compact
3
  ri si
i 1
11 :33
Vector Relationships -- Problem #1-1
“The dot-product trick”
Given vectors A and B which correspond to
symmetry axes of a crystal:
B

A  2 xˆ

B  3xˆ  3 yˆ  3zˆ
Calculate:

A, B , 
A
Where theta is angle between A and B
12 :38
Vector relationships II – Cross product
  
q  r  s  r s sin(  )
 xˆ
 

r  s  det  rx
sx

qi 
yˆ
ry
sy
zˆ 

rz 
s z 
3
r s 
j , k 1
 ijk  0
j k ijk
For any two indices equal
 ijk   1
I,j,k even permutation of 1,2,3
 ijk   1
I,j,k odd permutation of 1,2,3
Determinant
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Is a convenient
formalism to
remember the signs
in the cross-product
Levi-Civita Density
(epsilon)
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Is a fancy notation
worth noting for
future reference
(and means the
same thing)
13
Newton’s Laws
1. A Body at rest remains at rest, while a
body in motion at constant velocity
remains in motion
Unless acted on by an external force


dP
F 
dt
2. The rate of change of momentum is
directly proportional to the applied
force.


F12   F21
3. Two bodies exert equal and opposite


dP1
dP2

dt
dt
<--- Using 2 and 3 Together
forces on each other
14 :42
Newton’s Laws imply momentum conservation


dP1
dP2

dt
dt
d  
P1  P2   0
dt
 
P1  P2  C
 In absence of external force,
momentum change is equal
and opposite in two-body
system.
 Regroup terms
 Integrate.
Q.E.D.
Newton’s laws are valid in all
inertial (i.e. constant velocity)
reference frames
15 :45
Two types of mass?
Gravitational mass mG
mG
W= mGg
Inertial mass mI
g
F=mIa
mI
a=0
a>0
“Gravitational forces and acceleration are
fundamentally indistinguishable” – A.Einstein
16 :48
Momentum Conservation -- Problem #1-2
“A car crash”
James and Joan were drinking straight tequila
while driving two cars of mass 1000
 kg and
2000 kg with velocity
vectors 30 x m/s

and 10 x  60 y m/s
Their vehicles collide “perfectly inelastically” (i.e.
they stick together)
Assume that the resultant
wreck slides with

velocity vector v final
Friction has not had time to work yet. Calculate
v final and v final
17 :55
Two types of mass -- Problem #1-3 a-b
“Galileo in an alternate universe”
A cannonball (mG = 10 kg) and a
golf-ball (mG = 0.1 kg) are
simultaneously dropped from
a 98 m tall leaning tower in
Italy.

g  9.8m / s 2
Neglect air-resistance
How long does each ball take to
hit the ground if:
a) mI=mG
b) mI =mG
*
mG
18 :65
Lecture #1 Wind-up


dP
.F 
dt
Buy the book!!
First homework due in class Thursday
8/29
Office hours today 3-5
Get on WebCT
19