067 - Pr 18 - Bead on Rotating Hoop

 A point mass is constrained constrained to move on a massless massless hoop of radius a fixed fixed in a vertical plane that that rotates about its vertical vertical  symmetry axis with constant angular angular speed   .

[I.1] Obtain the Lagrange equations of motion, assuming the only external forces arise from gravity. Kinetic energy is exactly the same as with the spherical pendulum of problem 19,

T

1 2





ma 2  2  sin2    2 

1 2





ma 2  2  sin2    2  

[I.2]

Potential energy is due to gravity,

V  mgz  mga cos   

[I.3]

Then, the lagrangian is,

 L  T  V 

1 2





ma 2  2  sin2    2  mga cos    

[I.4]

Then, the Lagrange equations of motion are,

 L d  L      dt    



1 2

2 2 ma  2 sin  co cos    mga sin  

d 1  2  ma 2  dt  2 

  2 sin  co c os 

g  a

sin 

   [I.5]  

In the angular direction, we have the equation of motion of a physical pendulum. What are the constants of motion? Don’t know how to compute c ompute constants of motion! Hint: look at Hint: is

 L q

, the generalized force, in any given coordinate’s direction.

the Lagrangian explicitly time dependent? If not, then you have an energy- like thing conserved…

 H  T  V 

1





ma 2  2  sin2    2  mga cos    

[I.6]

 ma 2  2  2 sin cos     2   mga sin    

[I.7]

2

See if Hamiltonian is a constant of the motion,

dH  dt 

1

2

Hmm…weird. No vanishing. The Hamiltonian is time dependent?

Show that if

 

 is greater than some critical frequency  0 , there can be a solution in which the particle remains

 stationary on the hoop at a point other than at the bottom, but that if    0 , the only stationary point for the particle is at the bottom of the hoop. What is this critical frequency  0 ? Hint: get

the effective potential.