Intermediate MechanicsPhysics 321

Richard SonnenfeldNew Mexico Tech

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Lecture #1 of 25Course 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 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

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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? EE? CS? Other?

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 2nd 3rd 4th 5th

Graduate school?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 instructor15 years post-doctoral industry experience 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 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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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 motion

Unless acted on by an external force

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

Newton’s Laws

dt

PdF

2112 FF

dt

Pd

dt

Pd 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

dt

Pd

dt

Pd 21

021 PPdt

d

CPP 21

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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”

James and Joan were drinking straight tequila 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 3-5Get on WebCT

dt

PdF