- #1
Strukus
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Homework Statement
A hoop of mass m and radius R rolls without slipping down an inclined plane of mass M, which makes an angle [tex]\alpha[/tex] with the horizontal. Find the Lagrange equations and the integrals of the motion if the plane can slide without friction along a horizontal surface.
Homework Equations
L = T - U
T = kinetic energy
U = potential energy
Inertia tensor of hoop = mr[tex]^{2}[/tex]
The Attempt at a Solution
Here's what I believe to be the situation.
I know that I have to break up the kinetic energy into its separate components, but I'm not entirely sure how. I have 2 examples to work with, one being a block sliding down a moving ramp and the other being a hoop rolling down a stationary ramp.
Here's the block.
Here's the hoop.
With the block, we break it down into velocity along the x-axis and the y-axis for both the block and the ramp (with the ramp's y-velocity being 0). But with the hoop, we break up the velocity into an x'-axis (axis at the same angle as the ramp) and a [tex]\theta[/tex]. Both of the examples make sense to me, though combining them in this way leaves me a little confused.
If I were to just solve with what I knew now, I would combine the rotational velocity, the velocity along the x-axis, and the velocity along the y-axis into the kinetic energy.