Class 42: Free symmetric top Class 42: Free symmetric top Free - - PowerPoint PPT Presentation

Class 42: Free symmetric top Class 42: Free symmetric top

Free symmetric top in body frame

Assume  = =(symmetric) and no external torque acting on the top: Assume 1=2 =(symmetric) and no external torque acting on the top: In the body frame:

e3 B

 precesses about e3 (stationary) with angular velocity B and sweeps out

 e2 Body cone

y

B

p the body cone

  

3 3)

( 

e1

2

   

3 3 B

) (  

Free symmetric top in space frame

Assume  = =(symmetric) and no external torque acting on the top: Assume 1=2 =(symmetric) and no external torque acting on the top:

L z L z e3 S S  Body cone Body cone Space cone e3  Body cone Body cone Space cone B B Body cone y Body cone y y

  ( ll hi )   ( h f )

x x

For >3(Tall thin top) For <3(Short fat top) In the space frame:

 L

S 



p L, , and e3 are in the same plane. L is stationary (conservation of angular momentum),  is precessing about L and t th ith l sweeps out the space cone with an anglar velocity S. The body cone is rolling on the space cone with  at the line of contact.