Forces Producing Rotational & Translatory Motion: Exploring Mechanics

[NANDHU001]

Why does a force applied exactly perpendicular to one end of a rigid rod produce translatory motion along with rotational motion about center of mass?
I know the mathematics F=m(center of mass)*a but I want a mechanical answer.
Please help me analyze the mechanics of it.
I think analyzing the following thought experiment will be of some help though i don't
know how to proceed with it.
Consider two spherical bobs attached to two ends of an elastic string.(rigid rod as a limiting case of elasticity)
When an exact perpendicular force is applied at one bob the bob will accelerate and move along a straight line while the other bob remains at rest. Now due to an increase in length
of the string a tension will be developed and will pull both the bobs toward its center of mass.
I don't know how to proceed with it and can't see why the rotation is about center of mass.

 

[mfb]

Why do you think it should not produce translatory motion? You push the rod in one direction. After your push, one end moves, the other does not, so the center of mass moves, too.

 

[A.T.]

can't see why the rotation is about center of mass

Because you need a centripetal force for circular motion. At the center of mass the centripetal radial forces of both sides balance.

 

[NANDHU001]

Thanks but please consider analyzing the thought experiment I have mentioned.

 

[mfb]

What is so special about that experiment? You will get translation, rotation and oscillation of the string length at the same time (assuming it has compression forces as well).

If the string does not give any force below a certain length, your objects will come closer, but miss each other until they extend the string again, and the whole thing repeats.