As shown in the following, a rigid body experiences an external force $\overrightarrow{F}$ away from its center of mass C:
The rigid body is assumed to be free in space, experiencing no other force.
According to the theorem for the center of mass, the center of mass should have an moment acceleration $\frac{\overrightarrow{F}}{M}$, in which $M$ is the mass of the rigid body.
But since the center of mass C is the rotation center of the rigid body, $\overrightarrow{F}$ can be decomposed according to line $\overrightarrow{CP}$ into $\overrightarrow{Fr}$ to cause the rotation, and $\overrightarrow{Fm}$ to cause the momentum acceleration.
To reconciliate these two pictures, I have to assume that in additional to causing rotation, $\overrightarrow{Fr}$ has to be as efficient as $\overrightarrow{Fm}$ to cause the motion of the center of mass. It is hard for me to understand this. Anyone can explain?