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Lecture 7

Lecture 7 discusses the concepts of Coriolis and centrifugal forces in rotating frames of reference, emphasizing their effects on moving objects due to Earth's rotation. It explains how these forces cause deflections in motion, varying by hemisphere and latitude, and introduces the Foucault pendulum as a demonstration of Earth's rotation. The lecture also includes examples and questions regarding radial and tangential acceleration in inertial frames.

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12 views31 pages

Lecture 7

Lecture 7 discusses the concepts of Coriolis and centrifugal forces in rotating frames of reference, emphasizing their effects on moving objects due to Earth's rotation. It explains how these forces cause deflections in motion, varying by hemisphere and latitude, and introduces the Foucault pendulum as a demonstration of Earth's rotation. The lecture also includes examples and questions regarding radial and tangential acceleration in inertial frames.

Uploaded by

neelaero31
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Lecture 7

Punit Parmananda

PH111

Email : punit@phy.iitb.ac.in
Office: Room 203, 2nd Floor,
Physics Department
RECAP

Note: no subscript in r
   
 dvin   dvrot   d   dr 
       r    RECAP
 dt in  dt  rot  dt dt  rot
   
   (vrot    r )
Let us assume a constant angular velocity. Then the above equation can be
written as
      
ain  arot  2  vrot    (  r )
Having calculated the accelerations, we can calculate the pseudo force as
before. In S frame:
 
F  main

where F is sum of all real forces
In S’, on the other hand, the observed acceleration would be

      
arot  ain  2  vrot    (  r ) RECAP

Thus the force found by an observer in S’ would be

       
Frot  marot  m( ain  2  vrot    (  r ))
  
This force can be written as: Frot  F  F fict
where
RECAP      
F fict  2m  vrot  m  (  r )
The first term in above expression is called Coriolis force. This force is
present only when a particle is observed to move in the rotating frame of
reference.
The second term on the other hand is called Centrifugal force. This force is
present whenever the particle is at a non zero distance from origin.
Note: In order to apply these forces we must know the angular speed of
S’ relative to S like we have to know the acceleration in the case of
uniformly accelerating frames of reference.
Gaspard-Gustave Coriolis

• May 21, 1792- September


19, 1843
• French mathematician,
mechanical engineer, and
scientist
• kinetic energy and work
to rotating systems like
waterwheels
The Coriolis effect
• The Coriolis effect
• Is a result of Earth’s rotation
• Causes moving objects to follow curved paths:
• In Northern Hemisphere, deflection is to right ?
• In Southern Hemisphere, deflection is to left
• Changes with latitude:
• No Coriolis effect at Equator ?
• Maximum Coriolis effect at poles
The Coriolis effect

• As Earth rotates,
counterclockwise, different
latitudes travel at different
speeds
• The change in speed with latitude
causes the Coriolis effect
Coriolis Force

• If the earth is a cylinder shape and


rotating about its axis, will there be
any Coriolis effect ?
OBSERVATIONS(REPEAT)

• The magnitude of the Coriolis force increases from


zero at the Equator to a maximum at the poles.
• The Coriolis force acts at right angles to the direction
of motion, so as to cause deflection to the right in the
Northern Hemisphere and to the left in the Southern
Hemisphere.
Example: A particle of mass m is rotating in S’ with angular velocity1
about the same axis of rotation as S’ and in the same direction.
According to S: The particle rotates with angular velocity (+).
1
Therefore, some real force must provide a centripetal force.

 Freal   m(   1) 2 rr̂ 
 vrot
According to S’: The particle rotates with .
r
 m
angular velocity 1.   
 Freal  F fict   m 12 rr̂
But Please take care where  1is to be
 used and where  is to be used.
2
Fcentrifuga l  m rrˆ

-
Fcoriolis  2m vrot rˆ  2m1rrˆ
According to observer in S 
   
Fnet  Freal  Fcentrifuga l  Fcoriolis
2 2
  m(   ) rrˆ  m rrˆ  2m1 rrˆ
1

2
  m rrˆ
1

as expected
EXAMPLE
Neglect Gravity

v
r
N
Top View F centrifugal

F coriolis
-
0
Tangential acceleration is zero

If
Questions:
1. In inertial frame is radial acceleration zero?
2. In inertial frame is tangential acceleration zero?

Answers:
1.Yes, there is no radial force
2. No, there is a real force normal reaction acting. It is
interesting because  is constant
Motion on the Rotating Earth
Intuitively ?

Kleppner and Kolenkow


Foucault Pendulum
Foucault’s Pendulum r

Demonstrates that the earth is rotating. The plane of oscillation of .


O

a simple pendulum changes at a rate determined by the latitude.

Let a Foucault’s pendulum be set at north pole. Let the position of the bob
of the pendulum be described by r,  co-ordinates in the horizontal plane
v

Fcoriolis
.
Drift direction Drift direction

. Fcoriolis
v Intuitively

Left of Equilibrium Right of Equilibrium


1

Foucault’s pendulum set at north pole


1

1 1

Intuitively
(Angle along the horizontal plane)
As the pendulum oscillates, a tangential Coriolis force is felt by the bob,
being zero at the ends and maximum at equilibrium (WHY ?). This causes the
pendulum to continuously change the plane of oscillation.

Intuitively
pendulum
TIDAL WAVES
(Effective Gravity so
 centrifugal
 ignored:
 N.H)
   
arot  ain  2  vrot    (  r )

Board

DEFLECTION <<< HEIGHT


Motion on rotating earth

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