PHY 711 Fall 2013 -- Lecture 4 19/4/2013

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 4:

Chapter 2 – Physics described in an accelerated coordinate frame

1. Linear acceleration

2. Angular acceleration

3. Foucault pendulum

PHY 711 Fall 2013 -- Lecture 4 29/4/2013

PHY 711 Fall 2013 -- Lecture 49/4/20133

PHY 711 Fall 2013 -- Lecture 4 49/4/2013

PHY 711 Fall 2013 -- Lecture 4 59/4/2013

PHY 711 Fall 2013 -- Lecture 4 69/4/2013

Physical laws as described in non-inertial coordinate systems

Newton’s laws are formulated in an inertial frame of reference 0ˆ ie

ˆ tie

For some problems, it is convenient to transform the the equations into a non-inertial coordinate system

01e

02e

03e

1e

2e

3e

PHY 711 Fall 2013 -- Lecture 4 79/4/2013

Comparison of analysis in “inertial frame” versus “non-inertial frame”

dt

edV

dt

d

dt

d

edt

dV

dt

d

dt

edVe

dt

dVe

dt

dV

dt

d

eVeV

e

e

i

ii

bodyinertial

ii

i

body

i

iii

i

ii

i

i

inertial

ii

iii

i

i

i

ˆ

ˆ :Define

ˆˆˆ

ˆˆ

system coordinate moving a ˆby Denote

system coordinate fixed a ˆby Denote

3

1

3

1

3

1

3

1

03

1

0

3

1

03

1

0

0

VV

V

V

V

PHY 711 Fall 2013 -- Lecture 4 89/4/2013

Properties of the frame motion (rotation only):

dW

dWye

ze

yed ˆ

zed ˆ

eωe

ee

ee

ˆˆ

ˆˆ

ˆˆ

ˆˆ

ˆˆ

dt

ddt

d

dt

d

dd

eded

eded

yz

zy

PHY 711 Fall 2013 -- Lecture 4 99/4/2013

VωVV

VV

bodyinertial

i

ii

bodyinertial

dt

d

dt

d

dt

edV

dt

d

dt

d ˆ3

1

Effects on acceleration (rotation only):

VωωVωV

ωVV

VωV

ωV

dt

d

dt

d

dt

d

dt

d

dt

d

dt

d

dt

d

dt

d

bodybodyinertial

bodybodyinertial

22

2

2

2

Properties of the frame motion (rotation only) -- continued

PHY 711 Fall 2013 -- Lecture 4 109/4/2013

2

2

inertialdt

d

aApplication of Newton’s laws in a coordinate system which has an angular velocity and linear acceleration

Newton’s laws; Let r denote the position of particle of mass m:

rωωrωr

ωa

Fr

Fr

mdt

dm

dt

dm

dt

dm

dt

dm

dt

dm

bodyinertial

ext

body

ext

inertial

2 2

2

2

2

2

2

Coriolis force

Centrifugal force

ω

PHY 711 Fall 2013 -- Lecture 4 119/4/2013

Motion on the surface of the Earth:

'ˆ

rad/s103.72

2

5

FrF

r

mGM eext

rωωrωr

ωFrr

m

dt

dm

dt

dm

r

mGM

dt

dm

earth

e

earth

2 'ˆ

:onscontributiMain

22

2

PHY 711 Fall 2013 -- Lecture 4 129/4/2013

Non-inertial effects on effective gravitational “constant”

θr

rωωrg

gF

rωωFr

rr

rωωrωr

ωFrr

ˆcossinˆsin

ˆ

'

'ˆ0

,0 and 0For

2 'ˆ

2222

2

2

2

2

22

2

ee

e

e

Rr

e

e

earthearth

earth

e

earth

RRR

GM

r

GM

m

mr

mGM

dt

d

dt

d

mdt

dm

dt

dm

r

mGM

dt

dm

e

9.80 m/s2

0.03 m/s2

PHY 711 Fall 2013 -- Lecture 4 139/4/2013

Foucault pendulum http://www.si.edu/Encyclopedia_SI/nmah/pendulum.htm

The Foucault pendulum was displayed for many years in the Smithsonian's National Museum of American History. It is named for the French physicist Jean Foucault who first used it in 1851 to demonstrate the rotation of the earth.

PHY 711 Fall 2013 -- Lecture 4 149/4/2013

rωωrωr

ωFrr

m

dt

dm

dt

dm

r

mGM

dt

dm

earth

e

earth

2 'ˆ22

2

Equation of motion on Earth’s surface

w

z (up)

x (south)

y (east)

q

zxω ˆcosˆsin

PHY 711 Fall 2013 -- Lecture 4 159/4/2013

zyxr

ω

zxω

zyxF

zr

ˆsinˆsincosˆcos

ˆcosˆsin

ˆcosˆsinsinˆcossin'

ˆˆ2

yzxydt

d

TTT

mgr

mGM

earth

e

Foucault pendulum continued – keeping leading terms:

earthe

e

earthdt

dm

R

mGM

dt

dm

rωFr

r2 'ˆ

22

2

PHY 711 Fall 2013 -- Lecture 4 169/4/2013

Foucault pendulum continued – keeping leading terms:

earthe

e

earthdt

dm

R

mGM

dt

dm

rωFr

r2 'ˆ

22

2

symmgTzm

zxmTym

ymTxm

sin2cos

sincos2sinsin

cos2cossin

sinsin

cossin

: thatnote Also

cos2sinsin

cos2cossin

;0 ;1

:ionapproximatFurther

y

x

xmmgym

ymmgxm

mgTz

PHY 711 Fall 2013 -- Lecture 4 179/4/2013

Foucault pendulum continued – coupled equations:

xyg

y

yxg

x

cos2

cos2

g

g

g

iqtiqt

q

Y

X

qqi

qiq

YetyXetx

2

2

2

:solutions trivial-Non

02

2

cos Denote

)( )(

:form theofsolution a find Try to

PHY 711 Fall 2013 -- Lecture 4 189/4/2013

Foucault pendulum continued – coupled equations:

iYX

q

Y

X

qqi

qiq

YetyXetx

g

g

g

iqtiqt

:iprelationsh Amplitude

:solutions trivial-Non

02

2

)( )(

:continuedSolution

2

2

2

Re)(

Re)(

: and amplitudescomplex ith solution w General

titi

titi

DeCety

iDeiCetx

DC

PHY 711 Fall 2013 -- Lecture 4 199/4/2013

g

gg

srad

q

/ cos107 since 5

2

Re)(

Re)(

: and amplitudescomplex ith solution w General

titi

titi

DeCety

iDeiCetx

DC

sincos )(

coscos)(

0 )0( )0( :Suppose

0

0

0

ttXty

ttXtx

yXx

g

g