Is My Buoyant Force Calculation Correct?

[hanag]

Homework Statement

A recreational (open) hot air balloon (i.e., Pinside is approximately Poutside) has a volume of 2107 m3 when fully inflated. The total weight of the balloon, basket, ballast and pilot is 1832.6 N (412 lbs). By how much must the density of the air in the balloon be smaller than that of the surrounding atmosphere in order to keep the balloon floating level near the ground?

Homework Equations

Buoyant force= volume x g x rho

The Attempt at a Solution

I got the right solution, but it was explained much differently in class (much more complicated), and I didn't even use all the values given in the problem. I'm not sure if my logic is correct.

I figured that the buoyant force must be equal to gravity because the balloon is not accelerating.
v x rho x g = m x g

2107 m^3 x rho x g= 1832.6 N
2107 m^3 x rho = 187 kg
difference in rho= 0.0888 kg/m^3

Does what I did make any sense at all? Thanks in advance!

 

[nasu]

The logic is a little loose even though the answer is probably right.
You label the density by rho in the buoyant force. And then you change the meaning to a density difference. This is not good practice and is missing some steps.

 

[hanag]

So how would I go about doing this the right way? I don't even know where to start.

 

[haruspex]

So how would I go about doing this the right way? I don't even know where to start.

Put in an unknown density for the cold outside air, $\rho_c$, and write the thing you want to find, the difference, as $\Delta \rho$. In terms of those write the density of the hot air. Then write an expression for the buoyant force in terms of those, and observe the cancellation of the unknown $\rho_c$.

 

[HalfLostAndSearching]

Your logic is correct. The buoyant force must be equal to the weight of the balloon and its contents in order for it to float at a constant level. In order to find the density difference between the air in the balloon and the surrounding atmosphere, you correctly used the equation for buoyant force and set it equal to the weight of the balloon and its contents. You then solved for the density difference, which is the correct approach. Your solution is also consistent with the given values in the problem, so it seems like you have a solid understanding of the concept of buoyant force. Good job!