Reaction wheel are commonly used in spacecraft to change its attitude: an onboard inertia wheel is accelerated or decelerated along an axis to make the spacecraft rotate around the same axis.
While books explain how reaction wheel works with angular momentum, I cannot convince myself that it is entirely true and an important point seems to be missing: the spacecraft and the reaction wheel are necessarily connected through the motor that drive the reaction wheel angular velocity. And for me it is the key point of how a reaction wheel works.
Consider rotations of the reaction wheel (of moment of inertia $I_{RW}$) and the spacecraft (of moment of inertia $I_S$) around the same axis. Without the motor, the reaction wheel and the spacecraft are not connected at all. If the wheel angular velocity $\omega_{RW}$ change because of an external torque $\Gamma$, the spacecraft angular velocity $\omega_S$ should not change as this torque does not apply on it. Consequently, the whole system angular momentum $L$ evolves but the spacecraft angular momentum remains unchanged, which is not how a reaction wheel works :
\begin{eqnarray} L &=& I_{RW}\omega_{RW} + I_{S}\omega_{S}\\ \dfrac{d(I_{RW}\omega_{RW})}{dt} &=& \Gamma\\ \dfrac{d(I_{S}\omega_{S})}{dt} &=& 0\\ \implies \dfrac{dL}{dt}&=& \Gamma \end{eqnarray}
Now, consider a motor linking the reaction wheel and the spacecraft. As the motor apply a torque $\Gamma$ on the reaction wheel, it also apply a torque $-\Gamma$ (of the same intensity and opposite direction) on the spacecraft through action-reaction principle (same as when I screw a screw). So now we have
\begin{eqnarray} \dfrac{d(I_{RW}\omega_{RW})}{dt} &=& \Gamma\\ \dfrac{d(I_{S}\omega_{S})}{dt} &=& -\Gamma\\ \implies \dfrac{dL}{dt}&=& 0 \implies L \text{ is constant} \end{eqnarray}
We get the same explanation as the books: reaction wheel principle is based on the fact that the total angular momentum is constant. We get the same conclusion, but not considering the motor is for me a shortcut.
So I have two questions :
- Is my reasoning correct ?
- If so: How does the action-reaction principle work with a DC motor ? Even though I read about it, I can't find any answer (it does not seem to be the the counter electromotive force).
Thank you by advance for your answers.