91Ó°ÊÓ

The given information can be listed below,

• The volume of hot air balloon is $V=2200m^3$
• The weight of balloon fabric is,mg=900N
• The additional weight on balloon is,w=2400
• The density of outside air is,${\mathrm{p}}_{\mathrm{air}}=1.23\mathrm{kg}/{\mathrm{m}}^3$

The gravitational force causes a pressure gradient. This has to be equal to the force of gravity on the amount of gas or fluid that would fit in the volume of the immersed object.

Hence, an upward force always acts on the object by the gas/fluid.

The buoyant force on the balloon is equal to the total weight of the balloon fabric, the basket, other contents and the gas inside the balloon, which can be given by,

$B=p_{air}Vg$

Here,$p_{air}$is the density of air and V is the volume of a hot air balloon, and g is the acceleration due to gravity whose value is$\left(9.8m/s^2\right)$.

The mass of the gas can be given by,

$m_{gas}=p_{gas}V$

The total weight of the balloon can be calculated as,

$$\begin{gathered} \mathrm{W}=900\mathrm{N}+1700\mathrm{N}+3200\mathrm{N} \\ =5800\mathrm{N} \end{gathered}$$

$W=p_{gas}Vg=p_{air}Vg$

By using the above equation,

$$\begin{gathered} W=Vg\left(p_{air}-p_{gas}\right) \\ \frac{W}{Vg}=p_{air}-p_{gas} \\ {p}_{gas}=p_{air}-\frac{W}{Vg} \end{gathered}$$

Substitute all the values above equation.

$$\begin{gathered} {\mathrm{ÒÏ}}_{gas}=1.23\mathrm{kg}/{\mathrm{m}}^3-\frac{5800\mathrm{N}}{\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)\left(2200{\mathrm{m}}^3\right)}\left(\frac{1\mathrm{kg}.\mathrm{m}/{\mathrm{s}}^2}{1\mathrm{N}}\right) \\ {\mathrm{ÒÏ}}_{\mathrm{gas}}=1.23\mathrm{kg}/{\mathrm{m}}^3-0.26\mathrm{kg}/{\mathrm{m}}^3 \\ {\mathrm{ÒÏ}}_{\mathrm{gas}}=0.960\mathrm{kg}/{\mathrm{m}}^3 \end{gathered}$$

Thus, the average density of gas in the heated envelope is $0.96\mathrm{kg}/{\mathrm{m}}^3$.