Rolling on frictionless surface

I guess it will then both translate and rotate:

$$F = m ddot{x}$$

in the horizontal direction and

$$Fr = I ddot{phi}$$

about the CM.

Or, by statics, the equivalent system of forces about the CM is $F$ pointing toward it and a torque with value $Fr$.

It is always true for a system of particles (rigid body or not) that $F_{net enspace external} = Ma_{CM}$ where $F_{net enspace external}$ is the net external force, $M$ is the total mass, and $a_{CM}$ is the acceleration of the center of mass. For your case, the force moves the center of mass.

So it must translate not just rotate. But why?

It must translate because the only external force acting on the ball is $F$ in the horizontal direction, or $F_x$. Thus the COM of the ball will have translational motion with acceleration of $a_{x}=F_{x}/M$.

Hope this helpsl

There is a conservation of energy in play here. The more spin the force generates, the less it can transform to forward motion and vice versa.

In essence, as the r increases, the more of the force is transformed into rotational momentum. With shorter r values, the more it will translate to forward acceleration.

I guess this is the same what Petrini was referring to.

If the force were directly in line with the COM it would cause translation without rotation. When the force is not directly in line with the center of mass it will cause rotation and translation. If you had the force applied on top of the ball, as in your diagram, and you also apply an equal and opposite force on the bottom diametrically opposed to the top force, you will have rotation without translation. This is because the top force to the right and the bottom force to the left are both acting equally to rotate the ball, but the opposing forces cancel out translation of the COM.

A way of answering your question is to add two extra forces which are equal in magnitude and opposite in direction (ie net force zero) at the centre of mass as shown in the diagram below.

You now have a force (black $F$) acting at the centre of mass which produces a translational acceleration of the centre of mass of the body and a couple (red $FR$) which produces an angular acceleration of the body.