(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant 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.

3. The attempt at a solution

[itex]L=T-mgl_{cm}cosθ_{1}[/itex]

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Lagrangian equation from this free body diagram

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