# Force needed to push a syringe plunger: does one add force associated with downstream back-pressure to frictional plunger force?

I am trying to figure out how much force $F$ is needed to push a syringe plunger. The plunger needs to overcome the friction force $F_1$ and (a much smaller) inertia force $F_2=ma$, giving the total $F$ needed: $F=F_1+F_2$. Now there is a constriction downstream creating a back-pressure $dP$ (or force $F_3=dP \times A$). This is where I am getting confused.

Does the back pressure mean that the total force needed to push the plunger is $F=F_1+F_2+F_3$ or is it just $F=F_1+F_2$ provided $F_1+F_2 > F_3$?

The force you have to exert on the plunger is $F_1 - F_2 + F_3$. If you didn't include the back pressure term it would be just as easy to squirt treacle as it would be to squirt water, which obviously isn't the case.