I found this question in a book:
Is the acceleration of the block of mass m the same in part (a) and (b)?
At first I thought that the force applied on the right side is the same in both cases so the answer should be the same. But the answer was actually that they are different.
After a bit of thinking I realised that in part (a) the tension in the wire is not $2mg$ but less than that. However in part (b) we would take tension in the wire equal to $2mg$.
That got me thinking:
What exactly does it mean (physically/experimentally) to apply a force $2mg$ on a wire and how is it different from attaching a block of mass $2m$ onto it?
Clarification: What if I am appying the 2mg force by hand .. why doesnt the mass of my hand need to be taken into account?