Irodov Problem 1.59

First to give some background. An aerostat is a dirigible sort of like a balloon. There's the balloon's buoyant force pushing it up and the force of gravity of the mass attached to the balloon pulling it down. The way a balloon can be made to go up is either by throwing off some mass (ballast mass) attached to the balloon or by heating the air in the balloon.

Let the buoyant force of the aerostat pushing it up be F. Let the mass attached to the balloon be m. Since the balloon is accelerating downwards with at a rate w. We can write the force equations as,




Suppose that we have to throw a mass to make sure that the balloon accelerates upwards at a rate w then once again from newton's laws,