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Homework Statement
I am having trouble understanding how to apply Lagrange's equation. I will present a simplified version of one of my homework problems.
Imagine an inverted pendulum, consisting of a bar attached at a hinge at point A. At point A is a torsional spring with spring constant K. The bar has length L and a moment of inertia of I about point A. The free end of the bar experiences a load P that is always pointing downward. Find the equation of motion for this system.
Homework Equations
L= T-V
The Attempt at a Solution
To have something to check against, I first did the problem using angular momentum balance. My coordinate of choice is the angle the bar forms with respect to vertical, with positive θ being counterclockwise.
I[itex]\ddot{θ}[/itex]-PLsin(θ)-Kθ=0
OK. Now:
T= 1/2 I[itex]\dot{θ}[/itex]2
V= 1/2 K[itex]\dot{θ}[/itex]2
L= T-V
Eventually I get:
I[itex]\ddot{θ}[/itex]-Kθ=0
Clearly something is not right. I need somehow to take into account load P, but I don't know where in Lagrange's equation it would show up. I thought maybe instead of setting Lagrange's equation equal to zero I should set it equal to the forcing term? But then how do I take into account the direction of the force? The other thought was, maybe I somehow need to take the load into account through the potential term, but I really have no clue how.
I am pretty sure this system is conservative, though considering how much I suck at physics I could be completely wrong...
I would appreciate any help... I am so lost.