A recreational (open) hot air balloon (i.e., Pinside is approximately the same as Poutside) has a volume of 2227 m3 when fully inflated. The total weight of the balloon, basket, ballast and pilot is 1775.6 N (399 lb). How much less dense must the air in the balloon be than the surrounding atmosphere in order to keep the balloon floating level near the ground?
To solve this problem, we need to use the concept of buoyancy. The principle of buoyancy states that an object will float when the buoyant force exerted on it by the surrounding fluid (in this case, the air) is equal to or greater than the gravitational force acting on it.
In this case, the balloon needs to be less dense than the surrounding air in order to float. Let's assume that the density of the surrounding air is ρ_air, and the density of the air inside the balloon is ρ_balloon.
The buoyant force on the balloon can be calculated using the formula:
Buoyant force = (ρ_air - ρ_balloon) * g * V
where:
ρ_air is the density of the surrounding air,
ρ_balloon is the density of the air inside the balloon,
g is the acceleration due to gravity (approximately 9.8 m/s^2), and
V is the volume of the balloon.
The gravitational force acting on the balloon can be calculated by multiplying the total weight of the balloon, basket, ballast, and pilot by the acceleration due to gravity:
Gravitational force = Total weight * g
To keep the balloon floating level near the ground, the buoyant force must be equal to the gravitational force. Therefore, we can set up an equation:
(ρ_air - ρ_balloon) * g * V = Total weight * g
We can then solve for ρ_air - ρ_balloon:
ρ_air - ρ_balloon = (Total weight * g) / (V * g)
Simplifying the equation:
ρ_air - ρ_balloon = Total weight / V
Now, let's plug in the given values:
Total weight = 1775.6 N
V = 2227 m^3
ρ_air - ρ_balloon = 1775.6 N / 2227 m^3
Simplifying further, we get:
ρ_air - ρ_balloon = 0.796 N/m^3
Therefore, the air inside the balloon must be approximately 0.796 N/m^3 less dense than the surrounding air in order to keep the balloon floating level near the ground.