Question

A helium balloon ride lifts up passengers in a basket. Assume the density of air is 1.28 kg/m³ and the density of helium in the balloon is 0.18 kg/m³. The radius of the balloon (when filled) is R = 4.5 m. The total mass of the empty balloon and basket is m_b = 123 kg and the total volume is V_b = 0.068 m³. Assume the average person that gets into the balloon has a mass m_p = 70 kg and volume V_p = 0.077 m³.

1. What is the volume of helium in the balloon when fully inflated?
2. What is the magnitude of the force of gravity on the entire system (but with no people)? Include the mass of the balloon, basket, and helium.
3. What is the magnitude of the buoyant force on the entire system (but with no people)? Include the volume of the balloon, basket, and helium.
4. What is the magnitude of the force of gravity on each person?
5. What is the magnitude of the buoyant force on each person?
6. How many FULL people can the balloon lift up? (Your answer must be an integer.)

Answer 1

(1) Volume of helium in the balloon,

V = 4/3 pi r^3
V = 1.33 * 3.141 * 4.5 ^ 3
V = 380.7 m^3

(2) Here, mass balloon = 123 kg
Mass Helium = d_Helium * Volume = 0.18 kg/m^3 * 380.7 = 68.53 kg
Total Mass = 123 + 68.53 = 191.53 kg

Force = m*g = 191.53*9.81 = 1879 N

(3) Mass of displaced volume of air. = d * V
d = 1.28 kg/m^3
V = 380.7 m^3
Mass = 1.28 * 380.7 = 487.3 kg

Again, volume of the basket = 0.068 m^3
The buoyant mass = the volume of air displaced * density
The buoyant mass = 0.068m^3 * 1.28 kg/m^3 = 0.087 kg

The buoyant mass is the difference between the balloon full of helium that displaces the air and the helium mass and the basket mass.
m = 487.3 - 191.53 + 0.087 = 295.9 kg

So, buoyant force = 295.9 * 9.81 = 2903 N

(4) Here, F person (no buoyancy) = 70 * 9.81 = 686.7 N
(5) Buoyancy on each person = 0.077*1.28*9.81 N

(6) With buoyancy = 686.7 - 0.077 * 1.28 * 9.81 = 685.7 N

So, the number of people who could be lifted by the system = 2903 / 685.7 = 4.23 people.

Taking integral value, you could lift 4 people.