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 A2 0 2 3 dt 2 d 2 0 2C 1cos tC 2 sin t Bound (stable) A dt 2 2 d 2 t t Unbound 2 2Ce 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 3 2 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 F2 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 F2 mv Z coriolis Fcoriolis ( magnitude ) 2 m v v Fcoriolis • Coriolis force points radially outward.