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 = 5.4 m. The total mass of the empty balloon and basket is m_b = 122 kg and the total volume is V_b = 0.064 m³. Assume the average person that gets into the balloon has a mass m_p = 73 kg and volume V_p = 0.079 m³.

1) What is the volume of helium in the balloon when fully inflated?

m³

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.

N

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.

N

4) What is the magnitude of the force of gravity on each person?

N

5) What is the magnitude of the buoyant force on each person?

N

6) How many FULL people can the balloon lift up? (Your answer must be an integer.)

people

Answer 1

Problem One: volume
================
V = 4/3 pi r^3
V = 1.3333 * 3.141592 * 5.4 ^ 3
V = 657.6 m^3

Problem Two: Force
===============
Mass balloon = 122 kg
Mass Helium = d_Helium * Volume = 0.18 kg/m^3 * 657.6 = 118.3 kg
Total Mass = 240.3 kg

Force = m*g
m = 240.3
g = 9.81
F = m*g
F = 240.3 * 9.81
F =2358.01 N

Problem Three: Buoyant Force
======================
Mass of displaced volume of air. = d * V
d = 1.28 kg/m^3
V = 657.6 m^3
Mass = 1.28 * 492.8
Mass = 841.7 kg

The volume of the basket = 0.079 m^3
The buoyant mass = the volume of air displaced * density
The buoyant mass = 0.079m^3 * 1.28 kg/m^3 = 0.10112 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 = 841.7 - 240.3+ 0.10112 = 601.50

The force = 601.5 * 9.81 = 5900.72 N

Problem Four
==========

F person (no buoyancy) = 73 * 9.81 = 716.13 N
Buoyancy on each person = 0.967

With buoyancy = 716.13 - 0.079 * 1.28 * 9.81 =715.13N
the actual force is close enough without the buoyancy, but I gave you both.

The number of people who could be lifted by the system = 3996.4/705.4 = 5.66 people. In this case you have to round downward, so you could lift 5 people.