### Testing

```Intermediate Mechanics
Physics 321
Richard Sonnenfeld
Text: “Classical Mechanics” – John R. Taylor
:00
Lecture #1 of 25
Course goals

Physics Concepts / Mathematical Methods
Class background / interests / class photo
Course Motivation

“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


Study of how things move
Newton’s laws
 Classical “hard” problems

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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 (skipped)
Chaotic driven damped pendulum
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

Physics-21
ME-1
CS-2 Math-1
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

Soph – 2 Junior-17 Senior-6
Greatest area of interest in mechanics?
5 :08
Physics Motivation
Physics component

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
Second year at NM Tech
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Atmospheric physics / lightning studies
Embedded systems for airborne E-field
measurements
15 years post-doctoral industry experience
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Automated mechanical, tribological and magnetic
measurements of hard-drives
Bringing a 20-million unit/year product to market
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

r1
r2
r2

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
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
Dot product is a “projection” operator
Block on ramp with
gravity
h


O


Wy  W yˆ '  W cos 
Wx  W xˆ '  W cos   W sin 

yˆ '
yˆ

xˆ
xˆ '
Choose coordinates
consistent with
“constraints”
12 :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
13 :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
Determinant

Levi-Civita Density
(epsilon)
j k ijk
 ijk  0
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
Is a convenient
formalism to
remember the signs
in the cross-product

Is a fancy notation
worth noting for
future reference
(and means the
same thing)
14
Newton’s Laws


dP
F 
dt


F12  F21


dP1
dP2

dt
dt


dP1
dP
 2
dt
dt
d  
P1  P2  0
dt


 
P1  P2  C
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
DEFINITION of Inertial reference frame
2.
The rate of change of momentum is directly proportional to the applied force.
3.
Two bodies exert equal and opposite forces on each other
<--- Using 2 and 3 Together
 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
Newton’s Laws imply momentum conservation
15 :42
Newton’s Laws imply momentum conservation
 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
16 :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
17 :48
Momentum Conservation -- Problem #1-2
“A car crash”
Jack and Jill were drinking “Everclear” punch
while driving two cars of mass
1000 kg and

30 x m/s
2000 kgwith velocity
vectors

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
18 :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
19 :65
Lecture #1 Wind-up


dP
.F 
dt