Lagrangian for Rolling Cylinder on Inclined Plane | Homework Help

[kala]

Homework Statement

Write down the Lagrangian for a cylinder mass m, radius R an moment of inertia I, that rolls without slipping straight down an inclined plane which is at an angle a from the horizontal. Use as your generalized coordinate the cylinder's distance x measure down the plane from its starting point. Write down the Lagrange equation and solve it for the cylinder's acceleration.

The Attempt at a Solution

I tried to find the Lagrangian, this is what i have so far, is it right?
i said that T=1/2mv²+1/2Iw²
and U= -mgxsina so my Lagrangian would be
L=1/2mv²+1/2Iw²+mgxsina.
I think this is correct, but I'm not positive. If it is right then should i write I and w (omega) differently? and then continue with the partial differentiation?

 

[Irid]

Everything is correct, except that you must express all the dynamical variables (v, w) in terms of x and its derivatives, as the problem tells you. Also you can explicitly specify the value for I, since you know the shape of the body.

 

[Jumblebee]

so if i remember right v=x' but i am not sure what I and w are in terms of x and derivatives. doesn't I have something to do with R, the radius?

 

[Irid]

w is the angular velocity, which for non-slipping case is given by

$$\omega = \frac{v}{R} = \frac{\dot{x}}{R}$$

and I hope you can figure out the moment of inertia I yourself.

 

[Jumblebee]

I= 1/2 m R²?
Just checking before I go on with the partial derivatives.

 

[Irid]

Yes, for a uniformly filled cylinder.