Can a force acting on a single point of a resting, freely movable body cause it to spin without causing translational movement?

Question

In his answer to another post user Albertus Magnus describes the situation of a bullet hitting a rod in free space on its tip in a "purely tangential" way causing the rod to spin in a purely rotational movement. He then elaborates that this case is not realistic and should be rather treated as an external force acting on the tip of the rod. While this would remove the question, whats happening with the momentum of the bullet, this does not really make sense to me either. Any force acting on a point should cause a translation of the point and therefore apply some translational momentum to the body. Purely rotational motion seems only possible if another force at any part of the body would compensate the translational momentum, but that is not possible if a finite force is acting on only one point. Since I still don't understand this after some discussion - or rather I still beleave such process is unphysical - I decided to open a new question:

Can a force acting on a single point of a resting, freely movable body cause it to spin without causing translational movement?

If yes, how does it work? What is the meaning of a "purely tangential impulse" in this context?

If no, is there point in bringing up such an unphysical process? Does it have any meaning in any kind of mathematical thought experiment?

Also let me mention, even though this really should be obvoius, that this is no personal attack on anyone in any way.

Answer 1

Consider that there is a system consisting of an incoming projectile (point with mass m) and a rigid rod (one dimensional segment with mass M). For the sake of simplicity let the problem be two dimensional so that the motion is entirely in the $x,y$ plane, otherwise the rod is unconstrained, i.e. it is not fastened at any point and is free to move in any fashion.

We now ask: what happens when the projectile strikes the rod imparting an impulse at some point along the rod's extension, with a particular emphasis on the conservation of momentum.

The situation is that the incoming projectile impinges against the rod imparting to it an impulse while merging with it so that the two bodies are now one. Since the system consists of the projectile and rod, we have two masses each with its own linear and angular momenta (depending upon the coordinates chosen).

Before the collision the projectile has some angular momentum according to the origin of coordinates so chosen. The rod on the other hand is stationary and has no angular momentum. After the collision, the combined rod\projectile system has two kinds of angular momentum that sum to a net total, viz. the angular momentum of the system due to rotation about its center of mass and the angular momentum associated with the translation of the center of mass itself. The principle of conservation of angular momentum demands that the angular momentum of the projectile before collision is equal to the angular momentum of the combined rod\projectile after collision.

This analysis assumes coordinate such that each mass can be characterized entirely in terms of its angular momentum. We could have chosen other coordinates where one would have had to introduce both linear and angular momentum, however, each would still be separately conserved and the resultant motion identical, i.e. the rod would move away from its initial location with both rotational and translational motion.

Do there ever arise any cases where a rigid body can experience an external force and not exhibit translation in addition to rotation. The answer is definitely not. The fact that the body is rigid will not allow for the pure rotation without translation in the absence of some force of constraint. This can be seen clearly from the standpoint that; whereas the configuration manifold of the unconstrained system is two dimensional, requiring two numbers to specify its motion, viz. $x,y$, the purely rotating system requires only one coordinate to specify its motion entirely, i.e. an angle $\theta$ will suffice. Thus, there must be an equation of constraint to eliminate one of the two previously required coordinates. 

I must gratefully acknowledge user Zaph, as their stubborn persistence, in what they knew to be right, was a guiding light for the resolution of the problem.