# Work done by an eccentric force on a rigid body

I am hoping someone can explain in simple terms where I might be going wrong in my logic below.

We have a thin rigid rod (see diagram A) where a couple is acting and rotates the rod about its COM by an angle θ .

I am assuming that the work done by the couple is F.r.θ

Now consider diagram C where an eccentric force F is acting on the rod at point r from its COM and continues to do so for a distance d. I understand that an eccentric force will cause a translation of its COM and a couple rotation about its COM.

Let's assume that as the eccentric force F acts over distance d that the rod is rotated an angle θ (see diagram B).

I am assuming that the total work done on the rod by the eccentric force is F.d . But the rod has also been rotated to the same orientation in space as in diagram A which implies angular work has also been done by the couple caused by the eccentric force.

Therefore, doesn't this mean that the total work done on the rod is greater than F.d ?