# Kinetic Energy and Moment of Inertia

In this video, at around 12:00, it is said that spinning about the axis with the smallest moment of inertia gives the most kinetic energy. But, isn’t rotational kinetic energy equal to $$(1/2)(I)(ω)^2$$ . Thus, shouldn’t kinetic energy increase with increase in moment of inertia?

• Your formula would indicate minimal kinetic energy when spinning about the axis with smallest moment of inertia if $\omega$ were constant as the axis adjusts over time. However, as Bhavay's answer indicates, the equations of motion preserve $I \omega$ in this process rather than just $\omega$. Commented Apr 28, 2021 at 21:36

It is because the angular momentum is conserved while the kinetic energy is not.

So: $$I_1\omega_1=I_2\omega_2$$

When $$I_2$$ decreases, $$\omega_2$$ increases. $$\omega$$ is squared in the expression of kinetic energy and hence the net kinetic energy increases.

• If the kinetic energy is not conserved, what is it converted to/from (in this case)? Commented Apr 28, 2021 at 21:07
• @PeterMortensen it depends on the setup. In the video it's converted to kinetic energy within the contained fluid and, ultimately, heat via viscous dissipation.
– Kyle
Commented Apr 28, 2021 at 21:11