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Home , Centrifugal force, Frame of reference, Reaction (physics)

STiCM

Select / Special Topics in Classical

P. C. Deshmukh

Department of Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 15:

Unit 5 : Non-Inertial Real effects of pseudo-!

PCD_STiCM 1 Unit 5: Inertial and non-inertial reference frames.

Moving coordinate systems. Pseudo forces. Inertial and non-inertial reference frames.

Deterministic cause-effect relations in inertial frame, and their modifications in a non-inertial frame.

Real Effects of Pseudo Forces!

Gaspard Gustave de Coriolis PCD_STiCM Six Flags over Georgia 1792 - 1843 2 Learning goals:

Understand “Newton’s laws hold only in an inertial frame”.

Distinction between an inertial and a non-inertial frame is linked to what we consider as a fundamental physical /interaction.

Electromagnetic/electroweak, nuclear, ) / pseudo-force (centrifugal, Coriolis etc.) /

We shall learn to interpret the ‘real effects of pseudo-forces’ in terms of Newtonian method. Re-activate ‘causality’ in non- inertial frame of reference!

PCD_STiCM 3 Deterministic cause-effect relation in inertial frame,

and its adaptation in a non-inertial frame!

PCD_STiCM 4 The law of enables us recognize an inertial frame of reference as one in which is self-sustaining, determined entirely by initial conditions alone.

PCD_STiCM 5 Just where is the inertial frame? Newton envisaged the inertial frame to be located in deep , amidst distant stars. PCD_STiCM 6 Kabhi kabhi mere dil mein, Khayaal aata hein….. Tu ab se pahile, sitaron mein bas rahi thi kahin,

Tujhe jamin pe, bulaya gaya hei mere liye….

PCD_STiCM 7 Galilean relativity t is the same in the red frame and in the blue frame. rt() Y I rt'( ) r(t) rt'( ) utc F constant OI X I dr dr' OIc O' u t Y’ uc ZI F’ dt dt X’ O’ First derivative 2 2 d r d r ' Z’ with respect to time is different! 2  2 dt dt So what? Remember Galileo? derivative with respect to time

is, however, very muchPCD_STiCM the same! 8 Galilean relativity d 2 r d 2 r '  rt() dt 2 dt 2

YI rt'( ) F constant ‘’, which O I XI measures departure OIc O' u t Y’ from ‘equilibrium’ is ZI F’ X’ essentially the same a  a ' Z’ O’ in the two frames of F ma ma'' F reference. Essentially the same ‘cause’ explains the ‘effect’ (acceleration) according to the same ‘principle of causality’.

Laws of Mechanics : same inPCD_STiCM all INERTIAL FRAMES. 9 Galilean relativity F ma  ma''  F • A frame of reference moving with respect to an inertial frame of reference at constant is also an inertial frame.

• The same force F  ma explains the linear response (effect/acceleration is linearly proportional to the cause/interaction) relationship in all inertial frames.

PCD_STiCM 10 Galileo’s experiments that led him to the law of inertia.

Galileo Galilee 1564 - 1642

I Law

F  ma Linear Response. Effect is proportional to the Cause Principle of causality. II Law PCD_STiCM 11 F ma Linear Response

W mg

Weight =  acceleration due to gravity

PCD_STiCM 12 Lunatic exercise! Is lifting a cow easier on the ?

PCD_STiCM 13 Another lunatic exercise! Is it easier to stop a charging bull on the Moon ?

Ian Usmovets Stopping an Angry Bull (1849)

Evgraf Semenovich Sorokin (born as Kostroma Province) 1821-1892. PCD_STiCM Russian artist and teacher, a master of historical, religious and genre paintings. 14 What do we mean by a CAUSE ?

CAUSE is that physical agency which generates an EFFECT !

EFFECT: departure from equilibrium

CAUSE: ‘forCE’ ‘intErACtion’

fundamental interaction

Electromagnetic, gravitational nuclear weak/strong PCD_STiCM 15 Galilean relativity F ma  ma''  F

• A frame of reference moving with respect to an inertial frame of reference at constant velocity is also an inertial frame. • The same force F  ma explains the linear response (effect/acceleration is linearly proportional to the cause/interaction) relationship in all inertial frames. • “An inertial frame is one in which Newton’s laws hold” PCD_STiCM 16 Galilean relativity Time t is the same in the red frame and in the green, double-primed frame. Physics in an accelerated frame of reference.

What happens to the (Cause, Effect) relationship?

What happens to the Principle of Causality / Determinism? PCD_STiCM 17 constant acceleration f . rt() rt"( ) r( t ) OO " r ''( t ) Y

F 1 2 O X r( t ) ui t  f t  r ''( t ) 1 OO'' u t f t 2 2 Z i 2 Y’’ f :constant F” X’’ O’’ Z’’ Galilean relativity dd 1    r( t ) ui  f 2 t    r ''( t ) dt 2  dt  d  d   d  d    r( t ) f +    r ''( t ) dt  dt   dt  dt  aa " PCD_STiCM 18 Galilean relativity d  d   d  d    r( t )    r ''( t ) f dt  dt   dt  dt  rt() Y rt''( ) F aa " O X 1 OO'' v t f t 2 Z 2 Y’’ F” a a" f X’’ a"  a f O’’ Z’’ ma"  ma mf

F"  ma m f FFF  pseudo

‘Real’/’Physical’ force/interaction PCD_STiCM 19 Galilean relativity F"" ma  ma  mf rt() rt''( ) Y FFF  F pseudo O X 1 2 OO'' ui t f t 2 Y’’ Z F” X’’ Z’’ O’’

Same cause-effect relationship

does not explain the .

PCD_STiCM 20 Galilean relativity FFF  pseudo rt() rt''( ) Y The force/interaction

F which explained the O X acceleration in the Y’’ inertial frame does Z F” 1 OO'' u t f t 2 not account for the i 2 X’’ Z’’ O’’ acceleration in the accelerated frame of not Laws of Mechanics are the same reference. in a FRAME OF REFERENCE that is accelerated with respect to an inertial frame.

PCD_STiCM 21 Galilean relativity  Time t is the same in the FFF pseudo red frame and in the green, double-primed frame. ma" 

FFF" real/ physical pseudo

Real effects of pseudo-forces!

P. Chaitanya Das, G. Srinivasa Murthy, Gopal Pandurangan and P.C. Deshmukh Resonance, Vol. 9, Number 6, 74-85 (2004) http://www.ias.ac.in/resonance/June2004/pdf/June2004Classroom1.pdf

PCD_STiCM 22 Deterministic / cause-effect / relation holds only in an inertial frame of reference.

This relation inspires in our minds an intuitive perception of a fundamental interaction / force (EM, Gravity, Nuclear strong/weak).

Adaptation of causality in a non-inertial frame requires ‘inventing’ interactions that do not exist. – for these ‘fictitious forces’, Newton’s laws are of course not designed to !PCD_STiCM 23 Galilean relativity: Time t is the same in all frames of references.

RELATED ISSUES:

Weightlessness

 What is Einstein’s in an elevator accelerated upward/downward?

Sergei Bubka (Ukrainian: Сергій Бубка) (born December 4, 1963) is a retired Ukrainian pole vaulter. He represented the Soviet Union before its dissolution in 1991. He is widely regarded as the best pole vaulter ever. Reference: http://www.bookrags.com/Sergei_Bubka PCD_STiCM 24 http://www.stabhoch.com/bildserien/20030825_Isinbajeva_465/bildreihe.html What enables the pole vaulting champion, Yelena Isinbayeva, twist her body in flight and clear great heights? PCD_STiCM 2008 Olympics champion: 5.05meters25 We will take a Break… …… Any questions ? [email protected]

Non-Inertial Frame of Reference Next L16 : ‘cause’, where there isn’t one!

Real effectsPCD_STiCM of pseudo -forces! 26 STiCM

Select / Special Topics in

P. C. Deshmukh

Department of Physics Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 16: Unit 5 Non-Inertial Frame of Reference ‘cause’, where there isn’t one! Real effects of pseudo-forces! PCD_STiCM 27 PCD_STiCM 28 PCD_STiCM 29 http://www.lerc.nasa.gov/WWW/K-12/airplane/state.html

 What will be the shape of a tiny little amount of a liquid in a closed (sealed) beaker when this ‘liquid-in-a-beaker’ system is (a) on

(b) orbiting in a satellitePCD_STiCM around the earth. 30 In a liquid the molecular forces are weaker than in a solid.

A liquid takes the shape of its container with a free surface in a gravitational field.

Regardless of gravity, a liquid has a fixed volume.

Other than GRAVITY, are there other interactions that could influence the shape of the liquid’s free surface?

Intra/Inter molecular forces provide the concave or the convex meniscus to the liquid. These interactions become dramatically consequential in microgravity. PCD_STiCM 31 The actual shape an amount of liquid will take in a closed container in microgravity depends on whether the adhesive, or the cohesive, forces are strong.

Accordingly, the liquid may form a floating ball inside, or stick to the inner walls of the container and leave a cavity inside!

PCD_STiCM 32 You can have zero- gravity experience by booking your zero-gravity flight!

http://www.gozerog.com/

Flight ticket (one adult) : little over $5k

PCD_STiCM 33 http://www.grc.nasa.gov/WWW/RT/RT1996/6000/ 6726f.htm http://science.nasa.gov/headlines/y2000/ast12ma y_1.htm 29/09/09

On Earth, gravity-driven buoyant convection causes a candle flame to In microgravity, where be teardrop-shaped and convective flows are carries soot to the flame's absent, the flame is spherical, soot-free, and tip, making it yellow. PCD_STiCM blue. 34 “Microgravity” ? mm G 12 r 2

There is no ‘insulation’ from gravity!

PCD_STiCM 35 Galilean relativity

Observations in a rotating frame of reference.

PCD_STiCM 36 ZI Galilean relativity C Time t is the same in both the frames

ZR

P The green axis OC

YR is some arbitrary FI O axis about which YI OI FR is rotating. OR

FR XI XR Clearly, the coordinates/components of the point P in the inertial frame of reference are different from those in the rotating frame of reference. PCD_STiCM 37 We shall ignore translational motion of the new frame relative to the inertial frame FI.

We have already considered that.

We can always superpose the two (translational and rotating) .

PCD_STiCM 38 Z Galilean relativity I C ZR

P

YR F Y O I I O R First, we consider the position FR XI vector of a particle at rest in the X R rotating frame.

Clearly, its time-derivative in the rotating frame is zero, but not so in the inertial frame. dd        dt IR  dt  PCD_STiCM 39 Circle of radius bsin nˆ

d

b

 dd     bb   dt RI  dt 

The particle seen at bt() ‘rest’ in the rotating frame would appear to have moved to a new location as seen by an b() t dt observer in FI. PCD_STiCM 40 Galilean relativity Time t is the same in the red frame and in the purple, rotating, frame.

NOTE! The time-derivative is different in the rotating frame.

dd        dt dt   IR  

PCD_STiCM 41 Often, one uses the term

‘SPACE-FIXED FRAME OF REFERENCE’ for FI, and ‘BODY-FIXED FRAME OF REFERENCE’ for FR.

We shall develop our analysis for an arbitrary vector b , the only condition being that it is itself not a time- derivative in the rotating frame of some another vector.

d No vector q exists such that  q b dt R d b  0 in the rotating frame FR . dt R d Question: What is  b ? PCD_STiCM dt I 42 db bt(  dt )  bt ( )  | dbu | ˆ nˆ nˆ bˆ bsin where uˆ  . nbˆ  ˆ  (,)nbˆ ˆ d b db db ( b sin )( d ) 

nbˆ  ˆ db  (bd sin )( )  nbˆ  ˆ db These two terms are equal and hence cancel. b() t dt db d nˆ b bt() db dt b d d since   nˆ bb   dt PCD_STiCMdt I 43 Remember! The vector b itself did d bb not have any time-dependence in dt I the rotating frame.

If b has a time dependence in the rotating frame, the following operator equivalence would follow:

dd     b  b    b dt IR  dt 

Operator dd    Equivalence:        dt IR  dt  PCD_STiCM 44  d  d dd           r   r    r dt I  dt R dt IR  dt  Operating twice: dd    d d r rr   dt I  dt I dt R dt R d    rr    dt R d22   d  d  d 22 r   r   r    r dt IR  dt  dt R  dt R     r Multiplying by mass ‘m’, we shall get quantities that have dimensionsPCD_STiCM of ‘force’. 45 d22   d   d  d m22  r m   r  m  r  2 m   r  dt dt dt dt  RI   R R  m    r

d FRI F  F 2 m   r  m      r  dt R

‘Leap second’ term ‘’ ‘’ Gaspard Gustave de Coriolis term 1792 - 1843 PCD_STiCM 46 We will take a Break… …… Any questions ? [email protected]

Next L17 : Coriolis Deflection Foucault Cyclonic storm’s direction Real Effects of Pseudo-forces! d FRI F  F 2 m   r  m      r  dt R

PCD_STiCM 47 STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 17: Unit 5 Non-Inertial Frame of Reference ‘cause’, when there really isn’t one! Real Effects of Pseudo-forces!

PCD_STiCM 48 ‘force’ / ‘fundamental interaction’ in the inertial frame d22   d   d  d m22  r m   r  m  r  2 m   r dt dt dt dt  RI   R R  m    r Pseudoforce in the rotating frame Real Effects of Pseudoforces !

d FRI F  F 2 m   r  m      r  dt R

‘Coriolis force’ ‘Centrifugal force’ ‘Leap second’ term PCD_STiCM 49 Why are Leap Used?

The time taken by the earth to do one differs from day to day and from year to year. The Earth was slower than atomic clocks by 0.16 seconds in 2005; by 0.30 seconds in 2006; by 0.31 seconds in 2007; and by 0.32 seconds in 2008.

It was only 0.02 seconds slower in 2001.

PCD_STiCM http://www.timeanddate.com/time/leapseconds.html ; 14th October, 2009 50 The atomic clocks can be reset to add an extra second, known as the leap second, to synchronize the atomic clocks with the Earth's observed rotation.

The most accurate and stable time comes from atomic clocks but for navigation and astronomy purposes, but it is the atomic time that is synchronized with the Earth’s rotation.

PCD_STiCM 51 ‘Leap second’ term

The International Earth Rotation and Reference System Service (IERS) decides when to introduce a leap second in UTC (Coordinated Universal Time).

On one average day, the difference between atomic clocks and Earth's rotation is around 0.002 seconds, or around 1 second every 1.5 years. IERS announced on July 4, 2008, that a leap second would be added at 23:59:60 (or near midnight) UTC on December 31, 2008. This was the 24th leap second to be added since the first leap second was added in 1972.

PCD_STiCM http://www.timeanddate.com/time/leapseconds.html ; 14th October, 2009 52 Let us consider 3 ‘definitions’ of the ‘vertical’

[1] ‘vertical’ is defined by the radial from the center of the earth to a point on the earth’s surface

[2] ‘vertical’ is defined by a ‘plumb line’ suspended at the point under consideration.

[3] ‘vertical’ is defined by the space curve along which a chalk PCD_STiCM falls, if you let it go! 53 3 ‘definitions’ of the ‘vertical’

d22   d   d  d m22  r m   r  m  r  2 m   r dt dt dt dt  RI   R R  m    r PCD_STiCM 54 7.2921159 × 10−5 radians per second

d mr dt R

d22   d   d  d m22  r m   r  m  r  2 m   r dt dt dt dt  RI   R R  m    r PCD_STiCM http://commons.wikimedia.org/wiki/File:Globespin.gif 55 Earth and moon seen from

First image of Earth & Moon, ever taken from another , from Mars, by MGS, on 8 May 2003 at 13:00 GMT.

MARS GLOBAL SURVEYOR http://mars8.jpl.nasa.gov/mgs/ http://space.about.com/od/pictures/ig/Earth-Pictures-Gallery/Earth-and-Moon-as- PCD_STiCM viewed-from-.htm 56 The moon is attracted toward the earth due to their mutual gravitational attraction. attraction Will there be an eventual collision?

If so, when? If not, why not? PCD_STiCM 57 [1] When travelling in a car/bus, you experience what you call as a ‘centrifugal force’. Which physical agency exerts it ? Is there a corresponding force of ?

[2] How does a work ?

[3] Do the centripetal and the centrifugal forces constitute an ‘action - reaction’ pair of Newton’s 3rd law?

PCD_STiCM 58 The plane of oscillation of the Foucault pendulum is seen to rotate due to the Coriolis effect. The plane rotates through one full rotation in 24 hours at poles, and in ~33.94 hours at a “ Foo-Koh” of 450 (Latitude of Paris is ~490).

d FRI F  F 2 m  r  m     r dt R ‘Leap second’ term ‘Coriolis force’ ‘Centrifugal force’

PCD_STiCM 59 Gaspard Gustave de Coriolis 1792 - 1843 Jean Bernard Léon Foucault PCD_STiCM 1819 - 1868 60 dd22    m22  r m   r dt RI  dt 

d We often ignore mr dt the ‘leap R second’ and the d  2mr  centrifugal term. dt R  m    r

Pseudo Forces PCD_STiCM 61 2 Fcentrifugal  m     r   m  reˆ  e ˆ  e ˆ r  22  m reˆ sin  e ˆ   m  r sin  e ˆ  eeˆˆ  z v eˆ  r

   0.100 at latitude of 45

eeˆˆz  v (along 'local vertical')

PCD_STiCM 62    at North Pole 2   0 at   goes to  at South Pole 2

Terrestrial equatorial radius is 6378 km.

Polar radius is 6357PCD_STiCM km. 63 d FCoriolis  2 m   r   eˆy e ˆ y    e ˆ z e ˆ z dt R

eeˆˆz  v  eˆ  r

eˆy  eˆx e ˆ East  e ˆ y  e ˆ z     at North Pole 2   0 at Equator

eeˆˆz  v ('local vertical')

eeˆˆy  North

 ,,eeˆˆNorth   PCD_STiCM y  e ˆx e ˆ East  e ˆ y  e ˆ z 64 d FCoriolis  2 m   r   eˆy e ˆ y    e ˆ z e ˆ z dt R

eeˆˆz  v ('local vertical')  ,,eeˆˆNorth   y  eeˆˆy  North  ˆ ˆ ˆ ˆ  ex e East  e y  e z   at North Pole 2   0 at Equator  eˆy   goes to  at South Pole 2

eeˆˆz  v

cos is + in both N- and S- hemispheres PCD_STiCM 65 d F 2 m  r Coriolis    eˆy e ˆ y    e ˆ z e ˆ z dt R

Coriolis deflection of an d revR  v R ( ˆ z ) object in ‘free’ ‘fall’ at a dt R point on earth’s surface:

F 2 m  eˆ e ˆ   e ˆ e ˆ  v (  e ˆ ) Coriolis y y z z R z

2m  eˆy e ˆ x  2 m  cos  e ˆ East  0 (N hemisphere)Coriolis deflection: toward 2 East in both the Northern &     0 (S hemisphere ) the Southern Hemispheres 2 cos PCD_STiCMis + in both N- and S-hemispheres 66  Western North Pacific

Coriolis 1792 - 1843

d FCoriolis  2 m   r   eˆy e ˆ y    e ˆ z e ˆ z dt R

Western South Pacific PCD_STiCM http://agora.ex.nii.ac.jp/digital-typhoon/start_som_wnp.html.en 67 http://www.youtube.com/watch?v=ZvLYrZ3vgio&NR=1

This Nanyuki tourist attraction is not due to Coriolis effect

PCD_STiCM 68 PCD_STiCM 69 Rotating 2 Astronauts in a Space Ship R = 10m B Y  = 0.15 rad/sec v = (0,5) m/sec

X Real Effects of Pseudo A Forces

Please read the paperPCD_STiCM at the following internet weblink: http://www.physics.iitm.ac.in/~labs/amp/homepage/dasandmurthy.htm 70 Force = Rate of change of What interaction is making the instantaneous momentum change? R = 10m Very fundamental question!!  = 0.95 rad/sec v = (-10.0325,2.755) m/s The answer

P. Chaitanya Das, G. determines Srinivasa Murthy, our notion of Gopal Pandurangan and P.C. Deshmukh a 'The real effects of ‘fundamental pseudo-forces', Resonance, Vol. interaction’ 9, Number 6, 74- 85(2004)

(http://www.ias.ac.in/resonance/June2004/pdf/June2004Classroom1.pdf)PCD_STiCM 71 What will be the effect on ?

‘Missile Woman of India’ Dr. Tessy Thomas

th On ICBMs? PCD_STiCM17 May, 2010 AGNI II 72 http://daphne.palomar.edu/pdeen/Animations/34_Coriolis.swf

(eˆz )  (  e ˆ y )   e ˆ x (West)

d FCoriolis  2 m   r dt R

  eˆy e ˆ y    e ˆ z e ˆ z

PCD_STiCM Distance at equator: ~111 Kms. per longitude degree 73 We will take a Break… …… Any questions ? [email protected]

Next L18 : Coriolis Deflection Foucault Pendulum Real Effects of Pseudo-forces!

d FRI F  F 2 m   r  m      r  dt R

PCD_STiCM 74 STiCM

Select / Special Topics in Classical Mechanics

P. C. Deshmukh

Department of Physics Indian Institute of Technology Madras Chennai 600036

[email protected]

STiCM Lecture 18: Unit 5 ‘EFFECT’, when there isn’t a cause! ‘CAUSE’, when there really isn’t one! Real Effects of Pseudo-forces! Foucault Pendulum PCD_STiCM 75 d FRI F  F 2 m  r  m     r dt R Physical Finland Oceanography Sweden of the Baltic Sea Matti Leppäranta, Kai Myrberg Springer-Praxis (2009)

Denmark Circulating current in the Baltic Germany sea (1969?) Poland PCD_STiCM 76   eˆy e ˆ y    e ˆ z e ˆ z

d 2mr  dt R

Calculate Coriolis deflection of a plane flying from Srinagar to Kolkata to Mumbai ? Thiruvananthapuram ? Plane : 3550 kms 850 kms/hr Plane speed:

PCD_STiCM 850 kms/hr 77 Coriolis effect Electronics engineers? – only for Physicists? Computer science engineers? Aerospace engineers? Ocean Engineering?

PCD_STiCM Any navigation system on earth….. GPS in cellphones! 78 Use a Cartesian with reference to a point on the earth’s surface.

ˆ ˆ ˆ Choose  e x ,, e y e z  such that e ˆ z is along the local

‘up/vertical’ direction (which is not along eˆ r due to the centrifugal term).

However, eˆˆzr is very nearly the same as e

Choose such that it is orthogonal to , and points eˆy toward the North-pole seen from the point on the earth’s surface under consideration.

Finally, choose e ˆ x  e ˆ y e ˆ z , which will give us the direction of the local ‘East’ at that point. PCD_STiCM 79 d FCoriolis  2 m   r   eˆy e ˆ y    e ˆ z e ˆ z dt R eeˆˆ  ,,eeˆˆNorth   y  z v   eˆ   at N Pole  r 2 eˆy   0 at Equator  eˆx e ˆ East  e ˆ y  e ˆ z    at S Pole 2 

eeˆˆz  v ('local vertical')

eeˆˆy  North

PCD_STiCMe ˆx e ˆ East  e ˆ y  e ˆ z 80 Caution! We use a mix of three coordinate systems!

1. A cartesian coordinate system as defined in the previous slide.

2. A spherical whose ‘polar’ angle is defined with respect to the axis of the earth’s rotation rather than with respect to the cartesian z-axis which is oriented along the local ‘vertical’.

3. Cylindrical polar coordinate system whose radial unit vector is along the radial outward direction with reference to the earth’s axis of rotation.

PCD_STiCM 81 Just follow the 6 steps indicated in the next slide, exactly in the order given!

PCD_STiCM 82 Earth’s eˆv  Choose eˆˆz  ev rotation axis eˆr   eˆx e ˆ y e ˆ z

eeˆˆvv ertical eeˆˆy North   eˆEast eeˆˆ 2 h horizontal Fcf F cf eˆ  ˆ  e Fg 

eˆh The1. Note‘vertical’ earth’s is axis not of rotation.along the radial line, nor 2. Recognize that Note that: ˆ along3. Recognize the axis that of earth’s rotation!Fg F cf  e h  0 eeˆˆvr . (,)eeˆˆ   4. Choose eeˆˆz  v.  h 5. Choose eeˆˆy  North , pointing toward North. PCD_STiCM ˆ ˆ ˆ ˆ 6. Now, eˆx e ˆ y  e ˆ z  e ˆ East   ey e y    e z e z 83 d FRI F  F 2 m  r - m     r dt R

  eˆy e ˆ y    e ˆ z e ˆ z

d Velocity of the object in ˆ ˆ ˆ r  vx ex +v y e y v z e z ROTATING FRAME dt R

 eˆy eˆz  :latitude 

FCoriolis  2 m  eˆ e ˆ   e ˆ e ˆ  v e ˆ +v e ˆ  v e ˆ  y y z zx x y y z z PCD_STiCM 84 Coriolis ˆ ˆ   ˆ ˆ ˆ  F 2 m cos  ey  sin  e z    vx e x + v y e y  v z e z 

ma 2 m cos  v  sin  v eˆ  sin  v e ˆ  (  cos  v ) e ˆ Coriolis z y x x y x z

 eˆy eˆz  :latitude 

PCD_STiCM 85 Pendulum at the N pole

How will the plane of oscillation of the HowThe will observerthe plane is of at the N pendulum oscillationpole. of the pendulum look to an look to an observer at the observer at N pole, but standing the N pole? disconnected with the earth?

PCD_STiCM 86

How will the plane of oscillation of the HowThe will observerthe plane is of at the S pendulum oscillationpole. of the pendulum look to an look to an observer at the observer at N pole, but standing the S pole? disconnected with the earth?

PCD_STiCM Pendulum at the S pole 87 What if the pendulum is at an intermediate latitude?

How will the plane of oscillation of the How will the plane of pendulum oscillation of the pendulum look to an look to an observer at the observer at N pole, but standing that latitude? disconnected with the earth?

PCD_STiCM 88 Northern edge of Northern the floor of the edge of room is closer to the floor the earth’s axis than moves the Southern edge. eastward slower than the Southern edge.

W.B.Somerville Q.Jl. Royal Astronomical Soc.

PCD_STiCM (1972) 13 40-62 89 http://www.youtube.com/watch?v=nB2SXLYwKkM

video compilation of a Foucault pendulum in action at the Houston Museum of Natural Science.

PCD_STiCM 90 Foucault Pendulum   eˆy e ˆ y    e ˆ z e ˆ z

d FRI F  F 2 m  r - m     r dt R d mrR  mg  S  F 2 m  r - m     r dt R S Suˆ

S eˆx()()() e ˆ x  S  e ˆ y e ˆ y  S  e ˆ z e ˆ z  S S ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ SSeeux()()() x   eeu y y   eeu z z  mg ˆ ˆ ˆ S S excos  e y cos   e z cos  PCD_STiCM 91 d Foucault Pendulum mrR  mg  S 2 m  r dt R ma 2 m cos  v  sin  v eˆ  sin  v e ˆ  (  cos  v ) e ˆ Coriolis z y x x y x z ˆ ˆ ˆ S S excos  e y cos   e z cos  neglect z xy, motion:

mx Scos  2 m  cos  z  sin  y S my Scos 2 m  sin  x

mg S mg mx mgcos 2 m  sin  y

myPCD_STiCM mgcos  2 m  sin  x 92 d Foucault Pendulum mrR  mg  S 2 m  r dt R

S Suˆ neglect z S mg mx mgcos 2 m  sin  y my mgcos 2 m  sin  x xgcos 2sin y S ygcos  2sin x sin mg xgcos 2  y ygcos 2  x PCD_STiCM 93 xg cos  2 y Coupled differential ygcos 2 x equations

Solve by transforming to new coordinates x’,y’ such that x x'cos  t  y 'sin   t y  x'sin  t  y 'cos  t

x  x'sin  t  y 'cos  t y  x'cos  t  y 'sin   t 2 x gcos  2  x 'cos  t  y 'sin   t y gcos  2 2  x 'sin  t  y 'cos  t PCD_STiCM 94 2 x gcos  2  x 'cos  t  y 'sin   t 2 y gcos  2  x 'sin  t  y 'cos  t

22 xcos t  g cos cos  t  2  x 'cos  t  y 'sin  t cos  t 22 ysin t  g cos sin  t  2  x 'sin  t  y 'cos  t sin   t

22 xtytgcos sin  cos cos  t 2 x 'cos  tytt 'sin  cos   22 gcos sin  t  2  x 'sin  t  y 'cos  t sin   t

PCD_STiCM 95 22 xtytgcos sin  cos cos  t 2 x 'cos  tytt 'sin  cos   22 gcos sin  t  2  x 'sin  t  y 'cos  t sin   t

Dropping terms in 2 x gcoscos t  y  g cos sin  t  0

 xgcos  0  ygcos  0

PCD_STiCM 96 d mrR  mg  S 2 m  r sin dt R

Direction cosines of S are: x  xgcos  0 cos  l  ygcos  0 y cos   l zl cos  x l xg0 l y yg0 Two-dimensional linear l PCD_STiCM 97 d Foucault Pendulum mrR  mg  S 2 m  r dt R Dropping terms in 2 x gcos cos  t  y  g cos sin  t  0

xy    x g  0;  y  g   0 ll   

S In the earth’s rotating frame, the path is that of an ellipse. mg

PCD_STiCM 98 d Foucault Pendulum mrR  mg  S 2 m  r dt R Dropping terms in 2 x gcos cos  t  y  g cos sin  t  0 xy    x g  0;  y  g   0 ll    The ellipse would precess at an angular S speed sin A number of approximations made!

mg Detailed analysis is rather involved! The “plane” would in fact be a “curved surface”. PCD_STiCM 99 d Foucault Pendulum mrR  mg  S 2 m  r dt R

The ellipse would precess at an angular speed sin  :latitude

Time Period for the rotation of the “plane” of oscillation of the Foucault S Pendulum 1 2 2 T    mg f  sin 2 24 hours  PCD_STiCM2 sin  sin  100 Famous Foucault Pendulum: - at the Pantheon in Paris, France.

http://www.animations.physics.unsw.edu.a u/jw/foucault_pendulum.html

http://www.youtube.com/watch?v=vVg5P6f rHzY&feature=related

PCD_STiCM 101 sin

   0     2 4 4 2

1 2 2 T    f  sin 2 24 hours  2 sin  sin  PCD_STiCM 102 Philosophical questions: What is ‘force’ ? Mass/ Inertial frame? Gravity?

Sir (1838–1916) 1643 - 1727 From a portrait by 1879 – 1955 Enoch Seeman in 1726 Newton: Gravity is the result of an attractive interaction between objects having mass. Curvature of Space-Time, / Dynamics of MatterPCD_STiCM / General 103 We will take a Break… …… Any questions ? [email protected]

• In the next Unit, we shall consider Lorentz Transformations

and Einstein’s

Special Theory of Relativity.

• c: finite!

Next, Unit 6: Special Theory of Relativity

PCD_STiCM 104