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Say I have a round object, whose center of mass is NOT in its center. This can be caused due to a hole or non uniform distribution of density. The object rolls on the ground with velocity of $\omega*R$, radius being $R$ & angular velocity $\omega$.
My question is, how do I calculate the object's kinetic energy?
The Kinetic Energy can be calculated as:
$K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$
where $v$ is the velocity of centre of mass, $I$ moment of Inertia about it's centre of mass and $\omega$ being its angular velocity.
Thus Kinetic Energy can be calculated, at that instant by substiuting $v = R\omega$ in the first equation for $K$.
But, wait, the Kinetic energy keeps changing as the body rolls.
Since, the total Energy being conserved, the Potential energy factor $mgh$ has $h$ (height of center of mass above the ground) changing, thus the Kinetic Energy keeps changing.
The instant at when velocity is $R\omega$ only when $h = R$.
