Summary of Euler’s Equation

COMPLETE ANALYSIS FOR STABILITY OF A RIGID BODY IN MOTION

SOLUTION OF EULER’S EQUATIONS GIVES WHETHER MOTION OF RIGID BODY IS BOUND OR UNBOUND

BOUND MOTION = STABLE UNBOUND MOTION = UNSTABLE

STABILITY ANALYSIS FROM EULER’S EQUATIONS ARE VALID FOR SMALL PERTURBATIONS ONLY

d I1  II   0 Small and negligible 1dt 3 2 3 2

2 d  I I I  I 2 2  1 3 1 2  2 d  2 1  2  0 2 dt I I 2 A2  0 2 3 dt

2 d 2   0 2C 1cos  tC  2 sin  t Bound (stable) A  dt 2 2 d 2 t  t Unbound 2 2Ce 2  Ce 3 A  2  0 (unstable) dt 2 Fictitious Forces Plan

Inertial and Non-Inertial frames

Rotating frame of reference

Relating the Frames

Fictitious Forces in rotating frame

Practice Problems Rotating or Accelerating Frame of Reference

Newton’s laws hold only in inertial frames of reference. However, there are many non-inertial (that is, accelerated or rotating) frames of reference that we might reasonably want to study (such as elevators, merry-go-rounds, as so on).

Is there any possible way to modify Newton’s laws so that they hold in non-inertial frames, or do we have to give up entirely on F = ma?

Inertial frame: Non-Rotating frame Non-Inertial frame: Rotating frame

In this chapter we consider effects in rotating frame only DERIVATION OF FICTITIOUS FORCES IN ROTATING FRAME OF REFERENCE

Lab frame Vs Body frame

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1

Lab frame: Non-Rotating frame or Inertial Frame Body frame: Rotating frame or Non-Inertial frame y1 Relating the Co-ordinates

r Angular velocity  

O1 x1

Lab frame: Non-Rotating frame Body frame: Rotating frame

  dr dr    Velocity as observed in Non-Rotating frame/Lab frame    r  dtNR dt  R Relating the Co-ordinates NON-ROTATING TO ROTATING

  dr dr       r  dtNR dt  R

How to get this relation? Relating the Co-ordinates   dr dr       r   dtNR dt  R r dr P2 r NR

 r P1  r dr r r  r   reˆ NR NR R  r  r     reˆ tNR  t R  t  t  0 r  r  reˆ dr dr   R r   reˆ dtNR dt  R   r eˆ dr dr       r  dtNR dt  R Vector nature of angular velocity and angular momentum

 dr d  r sin dt dt

Direction of velocity is tangential to the circle 

dr dt r( t )  r rt( t ) r( t )   dr d dA  nrˆ  r    A dt dt dt Whaty1 will be the acceleration in Non-Rotating frame?   dr dr       r  r dtNR dt  R

 dr   O1    x1 dt NR

2  d r d d    dt2 dt dt NR NR NR

  d d   d d            dt dt   dtNR dt R NR R Acceleration

2  d r d d          dt2 dt dt   NR R R   dr      dt R     r 

2 2     dr dr   dr   d  r     2 2       r   dt dt dt dt NR  R  R  R

2 2    d r   d r   dr d    dr    2   2   r     r   dt NR  dt  R  dtR dt  dt R Acceleration and Force in rotating frame 2 2    d r   d r   dr d      2   2  2  r    r   dt NR  dt  R  dtR dt 

   d      aa  2  v   r  r   NR R R dt 

d   aa    r   2   v   r  RNR       R    dt  

d   F  Fm    rmvm  2     r        R    R NR dt   Inertial frame: Non-Rotating frame Non-Inertial frame: Rotating frame y1 Force d   F  Fm    rmvm  2     r        ni    ni i dt  

O 1 Earth in rotating frame x1 d   F  Fm    rmvm  2     r        ni    ni i dt  

Centrifugal force Coriolis force Azimuthal force Accelerated Frame of Reference

Newton’s laws hold only in inertial frames of reference. However, there are many non-inertial (that is, accelerated) frames of reference that we might reasonably want to study (such as elevators, merry-go-rounds, as so on). Is there any possible way to modify Newton’s laws so that they hold in non-inertial frames, or do we have to give up entirely on F = ma?

F   ma    i F   mani i  ni

d   aa    r   2   v    r  ni  i     ni    dt   Practice Problems 1. Consider a person standing motionless with respect to a carousel, a distance r from the center. Let the carousel rotate in the X-Y plane with angular velocity = ez.

(1)What are the fictitious forces present? (2)What is the direction of centrifugal force? (3)What is the magnitude of the centrifugal force felt by the person? Z

r (1)What are the fictitious forces present?

• On a Carousel 

r Only Centrifugal force will be present If person is not moving with respect to carousel, and if  is constant, then Centrifugal force is the only non-zero fictitious force.

d   F  Fm    rmvm  2     r        ni    ni i dt  

Centrifugal force Coriolis force Azimuthal force =0 =0 (2)What is the direction of centrifugal force? • On a Carousel    r

    r r

F   m     r    centrifugal  

   What is the direction of m   r   ?

 2 2 F   mremre( ˆr )   ˆ r  centrifugal

Points radially outwards 2a. Consider a person moving radially inward on a carousel with a velocity v. Let the carousel rotate in the X-Y plane with angular velocity = ez.

(1)What are the fictitious forces present? (2)What is the magnitude and direction of coriolis force?

Z What are the fictitious forces present?

Z

v

Centrifugal force and Coriolis force What is the magnitude and direction of Coriolis force? • Moving radially on a carousel F2 mv   Z coriolis

Fcoriolis ( magnitude ) 2 m v

v

• Coriolis force points tangentially inward 2b. Consider a person moving tangentially inward on a carousel with a velocity v. Let the carousel rotate in the X-Y plane with angular velocity =

ez.

(1)What are the fictitious forces present? (2)What is the magnitude and direction of coriolis

force? Z What are the fictitious forces present?

Z

v

Centrifugal force and Coriolis force What is the magnitude and direction of Coriolis force? • Moving tangentially inward on a carousel F2 mv   Z coriolis

Fcoriolis ( magnitude ) 2 m v

v

Fcoriolis

• Coriolis force points radially outward