I wished to understand a particular case of Euler's equation applied in the following cylindrical body:

enter image description here

where $I_{1,2,3}$ are the moments of inertia. By symmetry, $I_1=I_2=I_T$.

Here I consider that no external forces are applied to the cylinder. Stating the equations by their components, we have that:

$$I_1\dot{\omega}_1−(I_2−I_3)\omega_2\omega_3=0$$ $$I_2\dot{\omega}_2−(I_3−I_1)\omega_3\omega_1=0$$ $$I_3\dot{\omega}_3−(I_1−I_2)\omega_1\omega_2=0$$

where $\omega_i$ is the rotation speed on axis $\hat{b}_i$.

My question is the following: Considering that initially the cylinder spins with speed $\omega_3$, and it also moves around axis $\hat{b}_1$ and/or $\hat{b}_2$, that is, $\omega_1$ and/or $\omega_2$ are different than zero. According to the third equation, $\omega_3$ should remain constant, because $I_1−I_2=0$. But I imagine that to conserve its momentum, as the cylinder is rotated to a "laid" position around axis $\hat{b}_1$ or $\hat{b}_2$, it would keep spinning around $\hat{b}_3$, but with a different speed because $I_3\neq I_T$.

Am I interpreting this equation wrong?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.