- #1
Hyperian
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Homework Statement
Here's the free body diagram with variables.
I am looking for the lagrangian mechanics equation.
[itex]M[/itex] is mass of the bottom wheel.
[itex]m[/itex] is the mass of the top wheel.
[itex]R[/itex] is the radius of the bottom wheel.
[itex]r[/itex] is the radius of the top wheel.
[itex]θ_{1}[/itex] is the angle from vertical of the bottom wheel.
[itex]θ_{2}[/itex] is the angle from vertical of the top wheel.
[itex]\dot{θ}_{1}[/itex] is the angular velocity of the bottom wheel.
[itex]\dot{θ}_{2}[/itex] is the angular velocity of the top wheel.
[itex]x[/itex] is the linear distance.
[itex]\dot{θ}[/itex] is linear velocity of the whole contraption.
Homework Equations
Here are some relationships of these variables according to the free body diagram.
[itex]l_{cm}=\frac{m(R+r)}{M+m}[/itex] is the distance to center of mass from center of the bottom wheel.
[itex]\dot{θ}_{1}R=-\dot{θ}_{2}r[/itex] is just the relationship of the two wheel's angular velocity.
[itex]I=\frac{2}{5}MR^{2}[/itex] is the moment of inertia of the bottom wheel.
[itex]I=\frac{1}{4}MR^{2}[/itex] is the moment of inertia of the top wheel.
[itex]\dot{θ}_{1}R=\dot{x}[/itex] just means that there is no slipping.
I am looking for mechanical Lagrangian equation of [itex]L=T-V[/itex].
While i know [itex]V=mgl_{cm}cosθ_{1}[/itex], I am not sure what T would look like, I know it would have to do with at least 2 terms, transitional kinetic energy and rotational energy terms, but I am not sure how the interaction of the two wheels would play out.
The Attempt at a Solution
[itex]L=T-mgl_{cm}cosθ_{1}[/itex]