- #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?