# Between a solid and a hollow cylinders of the same mass, which one has the greater rotational kinetic energy?

I know that rotational kinetic energy is $\frac{1}{2}I\omega^2$. Therefore, the rotational kinetic energy will depend on the moment of inertia.

I came to the conclusion that since both have the same mass, both should have the same moment of inertia as
$$I = \sum_\text{over all mass} m r^2$$

But the answer says that since the hollow cylinder has greater moment of inertia, it has greater rotational kinetic energy.

How can the hollow cylinder have greater moment of inertia even though the masses of both the bodies is same?

• I think you mean to add the qualifier "at the same angular speed". – Selene Routley Dec 13 '13 at 11:44
• Google "radius of gyration" and you will find that $I = m r_{gyr}^2$ – John Alexiou Dec 13 '13 at 13:32

As you write the formula for the moment of inertia, it depends on the distribution of the mass. The further away the mass is from the rotation axis, the more contributes to the moment of inertia (as in distance squared $r^2$).
You've failed to take into consideration that $r$ is the radius of a piece of mass $\delta m$ rotating about an axis. So that the product of $\delta m$ and $r$ must be done first before the sum, or more probably the integral.