Mass confined to a rotating hoop with a moment of inertia I3 about the vertical axis

  • Thread starter tphysicsb
  • Start date
  • #1
4
0
1. Homework Statement

A particle of mass can slide without friction on the inside of a small tube bent in a circle of radius r. The tube can rotate freely about the vertical axis, x3, and has a moment of inertia I3 about this axis.


Homework Equations


a) Derive the Hamiltonian for this system. I was able to determine the Hamiltonian to be.

b) List the cyclic coordinates and determine the conserved qty associated with each?

c) Write down Hamilton's equation of motion

The Attempt at a Solution



a) I was able to determine the Hamiltonian to be.
H= (Pθ)^2/2mr^2 + (P∅)^2/(2(I3+mr^2(sinθ)^2)) +mgrcosθ +c

b)I was able to determine the Lagrangian to be
L= 1/2mr^2 (theta dot)^2 + 1/2 mr^2(phi dot)^2(sinθ)^2 +1/2I3 (phi dot)^2 -mgrcosθ -c


Since the Lagrangian does not explicitly depend on phi(∅) which i believe this just ends up telling me that the conjugate momentum P∅ is constant ????

c) I used (q dot) = ∂H/∂pi and -pi = ∂H/∂qi

and found

(theta dot) = pθ/ mr2 = 1
p dot theta = mgrsinθ - mr^2 (phi dot)^2 sinθcosθ


P∅ = constant

(phi dot) = Pθ/ (I3 + mr^2(sinθ) ^2






I want to determine if I am on the right track with part b and part c. Specifically is their anything else to be be determined from part b other then the conjugate momentum (P∅ = constant) ??


Also Now that i determined the equations of motion in partC. What it is the next step to determine the equations of motion.

Any guidance you may have would be greatly appreciated

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
263
0


Cyclic coordinates just mean a coordinate upon which the lagrangian doesnt depend.

In c.) you determined the Hamilton equations of motion, so you dont need to go further.
 

Related Threads on Mass confined to a rotating hoop with a moment of inertia I3 about the vertical axis

Replies
2
Views
6K
Replies
4
Views
4K
Replies
2
Views
3K
Replies
3
Views
2K
Replies
2
Views
8K
Replies
0
Views
2K
Replies
9
Views
6K
Replies
2
Views
2K
  • Last Post
Replies
1
Views
5K
Replies
2
Views
5K
Top