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Homework Statement
A uniform rod of length l rests on a horizontal floor and leans against a vertical wall, making an angle [tex]\theta[/tex] with the floor. It is initially held at rest. At t = 0, the rod is released and falls, sliding on the floor and the wall with no friction. The only forces acting on the rod are gravity and the forces of constraint.
1. Write the kinetic energy of the rod using the Cartesian coordinates of its center of mass and the angle [tex]\theta[/tex]. Note that this calculation will require an integral.
2. Write the Lagrangian and the equations of constraint, again using [tex]\theta[/tex] and the Cartesian coordinates of the center of mass of the rod as your generalized coordinates.
Homework Equations
[tex]L = T - U[/tex]
[tex]\lambda \frac{dg}{dq_i}=\frac{dL}{dq_i} - \frac{d}{dt} \frac{dL}{dq^{.}_i}[/tex]
Where g is the constraint equation and qi is the generalized coordinate.
The Attempt at a Solution
[tex]L = 1/2 m [x^{.}^2 + y^{.}^2 - 1/12 l^2 \theta ^{.}^2] - mgy[/tex]
Where (x,y) is defined as the center of mass of the rod and m is the mass of the rod.
[tex]g_1 (y, \theta ) = l/2 sin( \theta ),
g_2 (x, \theta ) = l cos( \theta ) - 2x = 0 [/tex]
If I do this, I substituted in g1 for y to have the equation just in terms of x and [tex]\theta[/tex]. This gives me a really ugly differential equation, so I feel like I'm doing something wrong. I eventually have to solve the equations to find when the end of the bar leaves the wall, which I would do by setting [tex] \lambda [/tex] equal to 1, but I haven't gotten that far yet because of the nastiness of the differential equations...
Any help (especially with getting the constraint equations) would be greatly appreciated.