I don't know is the simple answer. I'm pretty certain the COM will move since the applied force has a non-zero impulse and that must change the linear momentum of the object i.e. the COM must change momentum.
But I'm not sure how to write down the equation of motion and solve it.
Consider applying the force for a small but non-zero time Δt. In this time the rod will rotate a bit, but if Δt is small it won't rotate much. So the direction o the applied force F doesn't change very much. OK so far?
The point is that if the force doesn't change much we can approximate it as constant for a non-zero time Δt, and that means there is a non-zero linear impulse Δp = FΔt
So the linear momentum of the COM must change i.e. the COM must move. It cannot remain stationary.
Thanks @satan29 I was able to understand till why the centre of cone was instantaneous axis of rotation afterwards it was difficult to follow the video I will watch this tomorrow again some it might be able to understand more part of it
@PrateekMourya no, the h Bar room was created at the same time the Physics SE was created, a little over ten years ago. I didn't join the Physics SE until 2012.
consider a length dl , (and consider the angle maide by the radius with the x axis)along the arc, find the B due to the dipole, V due to rotation as a function of theta. dl=rd(theta)
evaluate the vector expression properly, vary theta from zero to 90