1. The problem statement, all variables and given/known data

Hi,

A question on work done in inflating a square balloon.

2. Relevant equations

Work done = integral of p*dv (see below).

3. The attempt at a solution

I just wanted some pointers on the following. Work done = integral of p*dv. But I don't know if p0 would be taken into account, and if so, how that would be incorporated into the equation? My second question is that, how would the calculation differ between (a) and (b)?

The question is too confusing to admit of an answer.

It asks about inflating the balloon, yet the balloon is being deflated, not inflated.

'energy done in inflating a square balloon.' is meaningless. Is it supposed to say 'work' rather than energy? If so then we need to know work done by what or on what, as a multitude of different interpretations are possible.

What is 'displacement work', and on what, or by what is it being done?

Once the answers to those questions are clarified, answering the question, including determining whether p0 is relevant, should become easy.

For (a) you do not use p0, because the work is the integral of the pressure applied by the internal air to the balloon, over the increments in volume.
For (b) you do use p0 because there is no constraint on what system the balloon is working on. So we take account of the force applied by the balloon both to the external and internal gases. So you net the two to get a net pressure on the balloon and integrate that over the change in volume.