# Kinetic energy of a rigid body

Gold Member

## Homework Statement

I think the problem was taken out from Landau&Lifgarbagez's book on classical mechanics.
An inhomogeneous cylinder of radius R rolls over a plane. The mass is distributed in such a way that a principal axis is parallel to the rotational axis of the cylinder and the center of mass is at a distance "a" from the rotational axis. The moment of inertia of the cylinder about the rotational axis is I. Calculate the kinetic energy of the cylinder.

## Homework Equations

$T=\frac{m v_{CM} ^2}{2}+ \frac{I \omega _c ^2 }{2}$.

## The Attempt at a Solution

The center of mass suffer from a circular motion of radius a and angular velocity $\omega _c$.
So that $v_{CM}=a \omega _c$ and thus $T=\frac{\omega ^2 _c }{2} (ma^2 + I)$.
This seems wrong to me because when a tends to 0, my kinetic energy equality tells me that there's only a rotational motion and no translational motion from the center of mass, which I believe it totally wrong.
I don't see what I did wrong though... I'd love some help to figure out what's wrong with what I did. Thanks in advance.

ehild
Homework Helper
Rolling means both translational and rotational motion. The centre of symmetry performs translation with velocity Vc. All points of the cylinder translate with Vc and rotate around the axis of symmetry with angular frequency equal Vc/R. The velocity of a point is the resultant of the translation and rotation.

This is not a simple problem...

ehild

Gold Member
You are absolutely right, I forgot about considering the translational motion.
If I consider rolling without slipping, I get that v of translation is worth $-\omega _c R$.
Thus $T=\frac{\omega ^2 _c}{2} (ma^2+R^2+I)$. Is there something I'm missing?

this cant be solved like this...the velocity of cm keeps on changing during motion

the question should give the information of initial position of cm

ehild
Homework Helper
I suggest to read Landau&Lifgarbagez's book on classical mechanics.
The problem is presented and solved in that book. The motion of the cylinder is considered pure rotation about the instantaneously fixed axis (the line where the cylinder touches the ground).

ehild