A helium balloon ride lifts up passengers in a basket. Assume the density of air

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 = 5.2 m. The total mass of the empty balloon and basket is mb = 128 kg and the total volume is Vb = 0.072 m3. Assume the average person that gets into the balloon has a mass mp = 74 kg and volume Vp = 0.079 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.)

Explanation / Answer

Problem One: volume

V = 4/3 pi r^3
V = 1.3333 * 3.141592 * 5.2 ^ 3
V = 588.97 m^3

Problem Two: Force
===============
Mass balloon = 128 kg
Mass Helium = d_Helium * Volume = 0.18 kg/m^3 * 588.97 = 106 kg
Total Mass = 234 kg

W0 = m*g
m = 234
g = 9.81
W0 = m*g
W0 = 234 * 9.81
W0 = 2295.5 N

Problem Three: Buoyant Force
======================

The volume of the basket Vb = 0.072 m^3

d = 1.28 kg/m^3
V = 588.97 m^3
The unoccupied balloon (filled with helium) has a volume equal to:
Vb + V
= 0.072m3 + 588.97m3 = 589.04 m3
The corresponding buoyancy force (weight of air displaced), that acts upon it:
B1 = d*g*(Vb + V)

B1 = 1.28*9.81*589.04

B1 = 7396.48N

Problem Four
==========
It simply is the weight of each person. Weight doesn't care what other set of forces acts on a body. It just cares about the mass and gravitational field.
W1 = mp*g

W1 = 74kg * 9.81 = 725.94N

Problem Five
==========

Magnitude of buoyant force on each person, based upon air displaced by each person. Remember, this assumes that the balloon is tethered to the ground so it is NOT accelerating.
B1 = d*Vp

B1 =  1.28 kg/m3*0.079 m3

B1 = 0.101 N

Problem six
==========

Total weight of system:
W = W0 + n*W1

Total static buoyant force on system:
B = B0 + n*B1

In order for the balloon to lift, B > W.
B0 + n*B1 > W0 + n*W1

We will solve for n, pretending it is an equals sign, and give an answer that is floored to the nearest integer.
B0 + n*B1 = W0 + n*W1
B0 - W0 = n*W1 - n*B1
n*(W1 - B1) = (B0 - W0)
n = (B0 - W0)/(W1 - B1)

n = 7396.48N - 2295.5N / (725.94N - 0.101N)

n = 7.02

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