Question

A hot-air balloon plus cargo has a mass of 1890 and a volume of 11430m3. the balloon is floating at a constant height of 6.25m above the ground. What is the density of the hot air in the balloon?
Please dont just solve. Explain it. If you derive an equation show me how you derived it.

A couple of things I don't get: is the hot air accounted for in the 1890kgs?

because it floats I understand the Fbouy has to be equal to (Density of air)(11430)(9.81)
but that doesnt really get me anywhere.

Answer 1

I've seen this problem solved two ways both which I don't understand.
finding the density of the cargo and the hot air balloon 1890/11430 and subtract it from the density of air. Error in this solution if the air is accounted for in the 1890kg that makes no sense. Looking at the answer it doesnt seem to be included.

Second way: someone states that the mass of the air should be (mass of the air displaced)-(mass of the cargo and hot air balloon) then divide by volume. I'm not too certain how that makes sense.

Looking at the answer the hot air is not taken into account in the 1890kg, where in the problem does it state that?

Answer 2

This problem makes total sense to me if I knew in the first place that the hot air was not accounted for.

Answer 3

So you finished it?

Is it like this?Then the buoyant force is opposite the weight of the basket: \(-1\times (1890\ [kg])\ (-9.8\ [m/s^2])=F_B\).

And the buoyant force is \(F_B=V\Large\frac{\rho_{\text{hot air}}}{\rho_{\text{normal air}}}\)? That's what I gather... And so\[\rho_{\text{hot air}}=\frac{F_B}{V}\rho_{\text{normal air}}\]Wikipedia says that the density of air at sea level is about \(1.225\ [kg/m^3]\), so then\[\rho_{\text{hot air}}=\frac{(-1\times (1890\ [kg])\ (-9.8\ [m/s^2])}{11,430\ [m^3]}1.225\ [kg/m^3]\]

Is that how you did that? I'm curious, because I've never had such a problem.

In that, I see the conflict. You ignore hot air mass and the volume is of only the hot air. Those assumptions make everything easier. Those are the quantities you would want to solve this problem, though. There would be error from assuming there's no gravitational force on the hot air, unless I am mistaken.

Answer 4

My formula for \(F_B\) is way off..

Answer 5

Which means that all of my solution that follows it is also wrong.