- #1

fluidistic

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## Homework Statement

Using the corresponding constraints conditions, calculate the kinetic energy of

1)A homogeneous cylinder of radius a that rolls inside a cylindrical surface of radius R>a.

## Homework Equations

My toughts: I hope they meants "roll without slipping". Let's consider this case!

[itex]T=\frac{mv ^2_{CM}}{2}+\frac{I_1\omega ^2 _x+I_2\omega ^2 _y+I_3 \omega ^2 _z}{2}[/itex].

## The Attempt at a Solution

Ok I already found out that [itex]I_1=I_2=\frac{m}{4} \left ( \frac{h^2}{3}+a ^2 \right )[/itex] and [itex]I_3=\frac{ma^2}{2}[/itex].

The constraint is that the line of the rolling cylinder in contact with the cylindrical surface does not slide. Mathematically I'm not really sure how to write this down. First, I'd set my coordinate system in the center of the rolling cylinder. (I remember in introductory physics that we dealt with this by setting the relative speed of the cylinders to be 0 m/s, but here I'm not sure)

Considering the axis along the height of the rolling cylinder as the z-axis, I think that [itex]\omega _x =\omega _y =\vec 0[/itex]. While [itex]\omega _z \neq 0 \neq \text{constant}[/itex]. It would be worth [itex]\omega _z (t)= \frac{d \theta (t)}{dt}[/itex] but I'm not sure I should use "theta" as angle. I've read in Goldstein that I should use Euler angles for this, but I have absolutely no idea how.

If the cylinder starts rolling at [itex]t=0[/itex] from its maximum height, [itex]\theta (0)=0[/itex], which is an initial condition for the motion equation of the cylinder.

Any help is appreciated, as usual.

Edit: Just checked out the picture of Euler angles in wikipedia and I think that instead of theta I should use beta.

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