# Kinetic energy for rotating round body having its COM not at the centre

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 $$K$$.

# Extra :

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 $$h = R$$.