- #1
fluidistic
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Homework Statement
A particle of mass m moves under a uniform gravitational field along a rod which moves in a vertical plane with a constant angular velocity [itex]\vec \Omega[/itex]. Write down the motion equations of the particle and calculate the constraint force. Is the energy conserved? Discuss.
Homework Equations
Lagrangian and... not really sure, maybe some modified Lagrange equation?
The Attempt at a Solution
I have a very strong feeling the energy isn't conserved. I understand how the particle will move and without someone to move the rod the motion will eventually end so there's some dissipation of energy I believe. I'm not sure the Lagrangian is still T-V.
Anyway, assuming it's still T-V, I chose polar coordinates r, theta to describe the position of the particle. My reference frame is in the middle of the rod (I assume it's "fixed").
[itex]T=\frac{m}{2}r^2 \dot \theta ^2[/itex] and [itex]V=mgr \sin \theta[/itex].
Now is my bigger problem. I don't know AT ALL how to modify the Lagrange equation!
I mean, I know that [itex]\frac{\partial L}{\partial q_i}- \frac{d}{dt} \left ( \frac{\partial L}{\partial \dot q_i} \right ) \neq 0[/itex], unlike in conservative systems.
I guess I have to use some Lagrange multipliers but I'm not even sure. I'm stuck here. How to calculate the constraint force? I know it's the force the rod exert on the particle so it would be the normal force, namely [itex]mg \cos \theta[/itex] but I'd have to solve for [itex]\theta (t)[/itex]. Hmm... wait it seems rather easy. Since [itex]\dot \theta = \Omega[/itex], I have that [itex]\theta ( t)= \Omega t + \theta _0[/itex].
I guess r(t) would be much harder to find, but do I need it?
Also, how do I prove that the energy is not conserved? Maybe deriving the Lagrangian with respect to time and if it's not worth 0 it means the Lagrangian is not constant and thus (I believe the implication is true), E isn't constant. Is this a good approach? Am I missing stuff?