PHY 711 Fall 2012 -- Lecture 5 19/7/2012

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 5:

1. Chapter 2 – Physics described in an accelerated coordinate frame

PHY 711 Fall 2012 -- Lecture 5 29/7/2012

#5

PHY 711 Fall 2012 -- Lecture 5 39/7/2012

SILICENE

Special seminar today: 11-11:50 AM Olin 103

PHY 711 Fall 2012 -- Lecture 5 49/7/2012

2

2

inertialdt

d

a

Application 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 2012 -- Lecture 5 59/7/2012

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 2012 -- Lecture 5 69/7/2012

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 2012 -- Lecture 5 79/7/2012

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 2012 -- Lecture 5 89/7/2012

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

PHY 711 Fall 2012 -- Lecture 5 99/7/2012

zyxr

ω

zxω

zyxF

zr

ˆcosˆsincosˆcos

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 2012 -- Lecture 5 109/7/2012

Foucault pendulum continued – keeping leading terms:

earthe

e

earthdt

dm

R

mGM

dt

dm

rωFr

r2 'ˆ

22

2

cos2cos

sincos2sinsin

cos2cossin

ymmgTzm

zxmTym

ymTxm

sinsin

cossin

: thatnote Also

cos2sinsin

cos2cossin

;0 ;1

:ionapproximatFurther

y

x

xmmgym

ymmgxm

mgTz

PHY 711 Fall 2012 -- Lecture 5 119/7/2012

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 2012 -- Lecture 5 129/7/2012

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 2012 -- Lecture 5 139/7/2012

g

g

srad

q

/ cos107 5

2

Re)(

Re)(

: and amplitudescomplex ith solution w General

titi

titi

DeCety

iDeiCetx

DC