A hot-air balloon must support its own weight plus aperson for a total weight of 1300 N. The balloon material has amass of 60 g/m2. Ambient air is at 25°C and 1 atm. The hot airinside the balloon is at 70°C and 1 atm. What diameter sphericalballoon will just support the weight? Neglect the size of the hot-airinlet vent.4950
Question
A hot-air balloon must support its own weight plus aperson for a total weight of 1300 N. The balloon material has amass of 60 g/m2. Ambient air is at 25°C and 1 atm. The hot airinside the balloon is at 70°C and 1 atm. What diameter sphericalballoon will just support the weight? Neglect the size of the hot-airinlet vent.4950
Solution
To solve this problem, we need to use the principle of buoyancy, which states that the buoyant force (the force that supports the balloon) is equal to the weight of the air displaced by the balloon.
Step 1: Calculate the volume of air displaced by the balloon
The buoyant force is equal to the weight of the balloon, which is 1300 N. We can use the equation for the buoyant force:
F_b = ρ_air * g * V_displaced
where ρ_air is the density of the air, g is the acceleration due to gravity, and V_displaced is the volume of the air displaced. We need to solve for V_displaced:
V_displaced = F_b / (ρ_air * g)
We know that F_b = 1300 N and g = 9.81 m/s^2. The density of air at 25°C and 1 atm is approximately 1.184 kg/m^3. So:
V_displaced = 1300 N / (1.184 kg/m^3 * 9.81 m/s^2) = 110.3 m^3
Step 2: Calculate the diameter of the balloon
The balloon is spherical, so its volume is given by the equation:
V_balloon = 4/3 * π * (d/2)^3
where d is the diameter of the balloon. We can solve for d:
d = (6 * V_balloon / π)^(1/3)
Substituting V_balloon = 110.3 m^3, we get:
d = (6 * 110.3 m^3 / π)^(1/3) = 6.2 m
So, the diameter of the balloon that will just support the weight is approximately 6.2 meters.
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