To solve the problem it is necessary to apply the definition of Newton's second Law and the definition of density.
Density means the relationship between volume and mass:
[tex]\rho = \frac{m}{V}[/tex]
While Newton's second law expresses that force is given by
F = ma
Where,
m = mass
a= acceleration (gravity at this case)
In the case of the given data we have to,
[tex]m_b = 3Kg[/tex]
[tex]r = 1.5m\\V = \frac{4}{3}\pi r^3 \\V = \frac{4}{3} \pi 1.5^3\\V = 14.13m^3[/tex]
In equilibrium, the entire system is equal to zero, therefore
[tex]\sum F = 0[/tex]
[tex]F_g +F_h-F_b = 0[/tex]
Where,
[tex]F_g =[/tex] Weight of balloon
[tex]F_h =[/tex] Weight of helium gas
[tex]F_b =[/tex] Bouyant force
Then we have,
[tex]mg+V\rho g -V\rho_a g = 0[/tex]
[tex]\rho = \rho_0-\frac{m}{V}[/tex]
Replacing the values we have that
[tex]\rho = 1.19kg/m^3 -\frac{3Kg}{14.13m^3}[/tex]
[tex]\rho = 0.978kg/m^3[/tex]
Now by ideal gas law we have that
[tex]PV=nRT[/tex]
[tex]P\frac{\rho}{m} = nRT[/tex]
[tex]P = \rho \frac{n}{m}RT[/tex]
But the relation \frac{n}{m} is equal to the inverse of molar mass, that is
[tex]P = \frac{\rho}{M_0} RT[/tex]
[tex]P = \frac{0.978kg/m^3}{0.04kg/mol}*8.314J/K.Mol * 305K[/tex]
[tex]P = 619995.7Pa[/tex]
Therefore the pressure of the helium gas assuming it is ideal is 0.61Mpa
