Lagrange equations of motion for hoop rolling down moving ramp.

[Strukus]

Homework Statement

A hoop of mass m and radius R rolls without slipping down an inclined plane of mass M, which makes an angle $$\alpha$$ 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$$^{2}$$

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 $$\theta$$. 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.

 

[kuruman]

When you do Lagrangian problems, a good starting point is to decide on your generalized coordinates and then write any constraints relating them. So what are your generalized coordinates and what are your constraints?

 

[Strukus]

I'm going to use the hoop on the non-moving ramp as my reference for this.

It has the equation of constraint being what describes the rolling without slipping, so
f = dx' - r d$$\theta$$ = 0
f = $$\dot{x}$$' - r $$\dot{\theta}$$ = 0

And y-ramp = 0.

And the generalized coordinates are x' and $$\theta$$.

That makes sense to me, but since the ramp is moving I'll have to describe the x-axis movement in terms of the x'-axis movement or describe the x'-axis movement in terms of the x-axis and y-axis, which is where I'm getting stuck.

And I'm not sure why but whenever I preview the post it's not what I typed (in terms of the greek letters).

 

[kuruman]

It is better to use x for the horizontal position of the ramp (more conventional) and s for the coordinate down the plane, relative to the plane. Keep angle θ as you have it. Can you write expressions for the kinetic energy T and the potential energy U in terms of these coordinates?

 

[Strukus]

Wow, that really helps. I forget what example it was when I was taking notes in class but I forgot with this approach you can kind of just use whichever axes you wish and it will still work.

That being said, this is what I have come up with.

T = 1/2 M $$\dot{x}$$$$^{2}$$ + 1/2 m $$\dot{s}$$$$^{2}$$ + 1/2 m r$$^{2}$$ $$\dot{\theta}$$$$^{2}$$
U = m g s sin$$\alpha$$

 

[Strukus]

After much working it through, looking at sample problems, and asking for help, I finished it! Thank you for your help kuruman!

 

[kuruman]

I hope you wrote the translational kinetic energy of the center of mass of the rolling hoop correctly. It is not simply

$$\frac{1}{2}m\dot{s}^2$$

It is

$$\frac{1}{2}m(\dot{s}cos\alpha+\dot{x})^2+\frac{1}{2}m(\dot{s}sin\alpha)^2$$