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 = 4.9 m. The total mass of the empty balloon and basket is mb = 121 kg and the total volume is Vb = 0.071 m3. Assume the average person that gets into the balloon has a mass mp = 72 kg and volume Vp = 0.076 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

Explanation / Answer

1)

The volume of the balloon is given by

Vb =(4/3)pir3 =(4/3)*3.14*(4.9)3 =492.55m3

2)

Mass of the helium is given by

mHe =(0.18 kg/m3)(492.55m3)=88.659kg

The force of gravity on the system is

F =(mHe+mballon)g

=(88.659+121)*9.81 =2056.75N

3)

Mass of the displaced volume of air is

m= pV = (1.28kg/m3)(492.55m3)=630.464kg

Buoyant mass is given by

m =Vb *p =( 0.071 m3)(1.28kg/m3) =0.09088kg

Buoyant mass is

mbuoyant =630.464kg -(88.659kg+0.09088kg)

= 541.714kg

Now the buoyant force is F =mbuoyant*g =(541.714)(9.81) =5314.215N

4)

The magnitude of the force of gravity on each person is F =mg =(72kg)(9.81m/s2) =706.32N

5)

The Boyant force on each person is

F =706.32N-(0.09088kg)(1.28kg/m3)(9.81m/s2)=705.178N

5)

The number of people lifted by the system is =5314.215N/705.178N=7.535 nearly 8 persons

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