Hi,
If you have a rod with no mass it doesnt matter where you hit it with a point force, it's going to move in that direction and it wont rotate at all because mass is what gives it inertia and since there is no inertia then there is no point in the rod that has inertia of rest and that is what makes it rotate. But unlike a rod with mass, it will not continue to rotate and/or translate once the force is removed because again there is no inertia. For a rod WITH mass it would rotate or translate or both rotate and translate depending on exactly where it was struck, and it would continue this action even once the force is removed because some parts of the rod will have inertia of motion.
You can figure this out for a rod with mass by dividing the rod up into small sections of length dL and enumerate the sections, and each section becomes a point mass at the signed distance d from the center and each section represents the mass of the rod at the center of each dL. In a uniform rod the masses will all be equal, but their distance from the center will be different. Start with 3 sections. When you hit it on the end (section 1) with a perpendicular force that end is going to move a fraction of an inch ds in time dt according to the laws of motion. This is going to cause a rotation dr in section 2 which is going to make it move and rotate because of the fact that section 3 acts as a fulcrum, and that is going to cause a rotation only in section 3 because section 3 in the time dt is still considered fixed and there is no translational force. Since we hit it on the end there is no secondary lever action so the rod rotates and does not translate. Note this would be very hard to actually do in real life. If we hit it in the middle section (section 2) this would be different because we have inertia of rest for sections 1 and 3, and so we have secondary lever action. This causes the rod to translate and not rotate.
If we use 5 sections and hit it in section 2, there will be partial secondary lever action so both ends of the rod will move, but the end nearest the point force will move farther in time dt than the end farther away from the force, so the rod will translate and rotate.
Dividing the rod up into 1000 equally sized pieces leads to a refinement of the process, and in the limit as the length of each piece dL goes to zero you'll get the formula.
Just to recap though, a rod with no mass will not behave this way because there is no real inertia for a rod with no mass and the whole calculation is based on some real finite inertia, so it is like applying the force to a vacuum which doesnt do much
Things like this are often what theory itself is made up of so there are usually predetermined rules to follow, but we dont make up the rules for fictitious objects without having some observed measurements to guide us and a more important reason for doing so.
For the rod with mass, you use classical mechanics and the laws of motion. Hitting it perfectly perpendicular and at the *very* end causes a rotation only because there is no inertia to make the other end rotate, and hitting it in the middle causes it to translate only because there is no inertia to make either end rotate, and hitting it between the center and end causes it to translate and rotate because the shorter end (from the point force) acts to partially rotate the other end which makes it move.
It is interesting that hitting it even a tiny tiny fraction in from the end causes at least some small translation as well as rotation.
A simpler way to look at it is that when hit at the very end we have only one lever action to consider, but when hit somewhere in from the end we have two lever actions to consider.