Suppose a small particle of mass $m$ moving at a velocity $v$ collides with a rod at the end in a perpendicular direction to the rod. I am sure that the rod translated in the direction same as the direction of velocity of particle (by linear momentum conservation), but for the rotation of the rod, I'm not quite sure why it does always rotate about center of mass, can it rotate about any other point? Or can it rotate about every point on the rod depending on the frame we choose on rod?
Rotational mechanics in collision of a mass and a rod
collisionreference framesrotational-dynamicstorque
Best Answer
In general, if you pick any point $p$ of a rigid body, the instantaneous motion of the body can be decomposed into the sum of the translational motion of $p$ and rotation about an axis passing through $p$. So, in this sense, the object can be considered to be instantaneously "rotating" about any of its points. But only instantaneously, because the velocity of $p$ at subsequent times need not remain the same as the inertial frame comoving with $p$ at that instant.
However, in this case, after the collision, there is no net force acting on the rod. Its center of mass will move inertially with constant velocity. No other point does. So its motion as a whole is translation of the center of mass and rotation about the center of mass. Using the center of mass and an inertial frame simplifies calculations significantly.
See this post for more information.