9/7/2012 PHY 711 Fall 2012 -- Lecture 5 1 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
Dec 28, 2015
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 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