# Determine the dilation temperature so as to double the speed

There's a metallic rod of length $l_1$ which is spinning around a vertical ax which passes through its center. The ends of the rod are spinning with $\omega_1$ angular speed. Determine the temperature at which to heat the metal rod so that the angular speed of the end $\omega_2$ doubles.

The only idea I have is the following: $\omega_1$ and $\omega_2$ are constant, so the net force is 0, and so the angular momentum $\vec{r} \times m\vec{v}$ is conserved. How should I approach this? And in general, for problems relating to torque or angular momentum, how should I approach those?

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The momentum of intertia, $I$, of a rod pivoted about its centre is $I = mL^2/12$. The angular momentum is constant and equal to $I\omega$, so $\omega$ is inversely proportional to $I$. However heating the rod will increase $I$ and therefore decrease $\omega$, so I don't see how the question can be answered. – John Rennie Oct 3 '12 at 6:00