A balloon with a mass of 90 kg is filled with helium (density = 0.179 kg/m3). The balloon is spherical and has a diameter of 11.0 m. What is the maximum mass that can be attached to the bottom of this this helium balloon, for it to be able to fly? The density of air is 1.29 kg/m^3.
m=90 kg
ρ(He) =0.179 kg/m³
D=11 m => R=5.5 m.
ρ(air) =1.29 kg/m³
( 4πR³/3)•g• ρ(air)= (4 πR³/3 )•g• ρ(He) + (M+m)g,
M = (4πR³/3)•{ρ(air) - ρ(He)} – m =….
To determine the maximum mass that can be attached to the bottom of the helium balloon for it to be able to fly, we need to calculate the buoyant force acting on the balloon.
The buoyant force is given by the formula:
Buoyant force = (density of fluid) x (volume of fluid displaced) x (acceleration due to gravity)
In this case, the fluid is air, and the balloon is displacing air as it rises. Therefore, the buoyant force acting on the balloon is:
Buoyant force = (density of air) x (volume of air displaced) x (acceleration due to gravity)
To calculate the volume of air displaced by the balloon, we need to find the volume of the balloon itself.
The volume of a sphere is given by the formula:
Volume = (4/3)πr³
Given that the diameter of the balloon is 11.0 m, the radius (r) is half the diameter, so r = 11.0 m / 2 = 5.5 m.
Plugging in the values, we have:
Volume = (4/3) x π x (5.5 m)³
Now, let's calculate the buoyant force:
Buoyant force = (density of air) x (volume of air displaced) x (acceleration due to gravity)
Buoyant force = (1.29 kg/m³) x [(4/3) x π x (5.5 m)³] x (9.8 m/s²)
The buoyant force is acting in the opposite direction to the force of gravity to make the balloon float. Therefore, the maximum mass that can be attached to the balloon is equal to the mass of the balloon itself.
The mass of an object is given by the formula:
Mass = (density) x (volume)
In this case, the mass of the balloon is:
Mass = (density of helium) x (volume of balloon)
Mass = (0.179 kg/m³) x [(4/3) x π x (5.5 m)³]
So, the maximum mass that can be attached to the bottom of the helium balloon for it to be able to fly is equal to the mass of the balloon itself, which is the value obtained from the above calculation.