Intermediate MechanicsPhysics 321

Richard Sonnenfeld

Text: “Classical Mechanics” – John R. Taylor

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Lecture #1 of 25

Course goals Physics Concepts / Mathematical Methods

Class background / interests / class photoCourse Motivation “Why you will learn it”

Course outline (hand-out)Course “mechanics” (hand-outs)Basic Vector RelationshipsNewton’s Laws Worked problemsInertia of brick and ketchup III-3,4

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Physics Concepts

Classical Mechanics Study of how things move Newton’s laws

Classical “hard” problems 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

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Mathematical Methods

Vector Calculus 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 Coordinate change / rotations Diagonalization / eigenvalues / principal axes

Lagrangian formulation Calculus of variations “Functionals” and operators Lagrange multipliers for constraints

General Mathematical competence

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Class Background and Interests

Majors Physics-21 ME-1 CS-2 Math-1

Preparation Assume Math 231 (Vector Calc) Assume Phys 242 (Waves) Assume Math 335 (Diff. Eq) concurrent Assume Phys 333 (E&M) concurrent

Year at tech Soph – 2 Junior-17 Senior-6

Greatest area of interest in mechanics?

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Physics Motivation

Physics component Classical mechanics is incredibly useful

Applies to everything bigger than an atom and slower than about 100,000 miles/sec

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

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Mathematics Motivation

Mathematics component 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.

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About instructorSecond year at NM Tech Atmospheric physics / lightning studies Embedded systems for airborne E-field

measurements

15 years post-doctoral industry experience 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

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Course “Mechanics”

WebCT / Syllabus and HomeworkOffice hours, Testing and Grading

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Vectors and Central forces

Vectors Many forces are of

form Remove

dependence of result on choice of origin

1 2r r

1r

2r

Origin 1Origin 2

1 2( )F r r

2r

1r

1 2r r

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Vector relationships

Vectors Allow ready

representation of 3 (or more!) components at once.Equations written

in vector notation are more compact

zdt

dzy

dt

dyx

dt

dx

dt

rdˆˆˆ

x

xx

ˆ rrrr

3

1

)cos(

iiisr

srsr

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Dot product is a “projection” operator

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O

m

h

ˆ ' cos

ˆ ' cos sin

y

x

W W y W

W W x W W

y

x

ˆ 'yˆ 'x

Block on ramp with gravity

Choose coordinates consistent with “constraints”

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Vector Relationships -- Problem #1-1“The dot-product trick”

Given vectors A and B which correspond to symmetry axes of a crystal:

Calculate:

Where theta is angle between A and B

xA ˆ2

zyxB ˆ3ˆ3ˆ3

,, BA

A

B

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Vector relationships II – Cross product

Determinant Is a convenient

formalism to remember the signs in the cross-product

Levi-Civita Density (epsilon) Is a fancy notation

worth noting for future reference (and means the same thing)

1

1

0

ˆˆˆ

det

)sin(

3

1,

ijk

ijk

ijk

kjijkkji

zyx

zyx

srq

sss

rrr

zyx

sr

srsrq

For any two indices equal

I,j,k even permutation of 1,2,3

I,j,k odd permutation of 1,2,3

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1. A Body at rest remains at rest, while a body in motion at constant velocity remains in motionUnless acted on by an external forceDEFINITION 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 framesNewton’s Laws imply momentum conservation

Newton’s Laws

dt

PdF

2112 FF

dt

Pd

dt

Pd 21

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dt

Pd

dt

Pd 21

021 PPdt

d

CPP 21

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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

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Two types of mass?

Gravitational mass mG

W= mGg

Inertial mass mI

F=mIa

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g

mG

mI

a=0a>0

“Gravitational forces and acceleration are fundamentally indistinguishable” – A.Einstein

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Momentum Conservation -- Problem #1-2“A car crash”

Jack and Jill were drinking “Everclear” punch while driving two cars of mass 1000 kg and 2000 kg with velocity vectors and

Their vehicles collide “perfectly inelastically” (i.e. they stick together)

Assume that the resultant wreck slides with velocity vector

Friction has not had time to work yet. Calculate

finalv

final finalv and v

smx /30

smyx /6010

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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.

Neglect air-resistanceHow long does each ball take to hit the

ground if:

a) mI=mG

b) mI =mG * mG

2/8.9 smg

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Lecture #1 Wind-up

.

Buy the book!!First homework due in class Thursday 8/29Office hours today 4-5:30Get on WebCT

dt

PdF