# Rigid body kinematics problem: finding the velocity of the center of mass

1. Jul 15, 2012

### atat1tata

I have to deal with the problem of finding an expression for the kinetic energy of a rigid body. One of its point is pivoted to a point that moves arbitrarily. So in order to find an expression for the kinetic energy I use König's theorem, but I need the velocity of the center of mass. I use Euler angles to describe the motion of the fixed frame of reference attached to the principal axes of the body, with the origin of the system in the center of mass of the body.

I know how to find an expression for $\vec\omega$ in the fixed frame of reference, but what I lack is an expression for the velocity of the center of mass (let's say it's a distance $l$ from the pivot) in terms of the velocity of the pivot, the Euler angles and their derivatives

This should be a common problem but I have seen no reference to it in the Internet. So I would be thankful if someone suggests me a book/website/reference to these kinds of problems

2. Jul 15, 2012

### Staff: Mentor

Instead of calculating everything in the center of mass system, what about calculating this in the system of the fixed point? To calculate the kinetic energy, it is sufficient to have two values: The rotation around the fixed point, and one thing which I would call "internal rotation" (around an axis through the fixed point).
I would expect that both are independent of each other, and without external forces, both values stay the same. The third degree of freedom is irrelevant here, as it just defines the plane where the rotation happens.

3. Jul 15, 2012

### atat1tata

Maybe I didn't understand. For fixed point you intend the pivot? The problem is that it is not fixed, really, so I thought that all the expression for kinetic energy around a fixed point do not apply here

4. Jul 15, 2012

### atat1tata

Possibly I should get an expression of $\vec v_{CM}$ with its components in the inertial frame of reference basis that should be a sum of the relative velocity of the center of mass from the pivot and the velocity of the pivot

5. Jul 17, 2012

### Staff: Mentor

Oh sorry, I misread your post, I thought the other part of the body could move freely. I would still use this pivot, I think.

6. Jul 17, 2012

### Filip Larsen

Perhaps you can describe your problem a bit more. Is the pivot axis fixed with respect to the rotating body, i.e. body frame? If so, do you have a kinematic model for how the pivot axis moves in your reference frame? What are your dependent and independent variables?

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