The question titles arose when solving the following problem:
Figure 5 is a schematic of a centrifuge that rotates with angular velocity $\Omega$. Solid material is first fed into the centrifuge at a radius $r_0$ where it is accelerated up to the speed $r_0 \Omega$ by friction with the rough inner wall.
Determine the work done by the centrifuge to accelerate a point mass $m$ to the speed $r_0 \Omega$. The mass is initially stationary. The cylindrical section of the centrifuge has a rough inner surface with a coefficient of friction $\mu$.
And this is the beginning of the solution provided to us (where $mr_0\dot\theta^2$ and $mr_0\ddot\theta$ are d'Alembert forces):
The work in the solution is given by: $$Work\ done = \int_0^T F\Omega r_0 dt$$ However, the force $F$ acts on the mass $m$ which moves with velocity $\dot\theta r_0$. Hence, the work equation I would expect would be: $$Work\ Done = \int_0^T F\dot\theta r_0 dt$$
Where $\dot\theta r_0$ is the distance that the force moves.
Why does this approach yield a different answer than when considering the force as acting on the centrifuge (and ignoring that the point of application moves relative to the centrifuge)?