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 = 5.3 m. The total mass of the empty balloon and basket is mb = 128.0 kg and the total volume is Vb = 0.074 m3. Assume the average person that gets into the balloon has a mass mp = 70.0 kg and volume Vp = 0.076 m3. 1) What is the volume of helium in the balloon when fully inflated? What is the magnitude of the force of gravity on the entire system (but with no people)? 2) 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 * 4.9 ^ 3 V = 492.8 m^3 Problem Two: Force =============== Mass balloon = 125 kg Mass Helium = d_Helium * Volume = 0.18 kg/m^3 * 492.8 = 88.71 kg Total Mass = 213.71 kg Force = m*g m = 213.71 g = 9.81 F = m*g F = 213.71 * 9.81 F = 2096.5 N Problem Three: Buoyant Force ====================== Mass of displaced volume of air. = d * V d = 1.28 kg/m^3 V = 492.8 m^3 Mass = 1.26 * 492.8 Mass = 621 kg The volume of the basket = 0.059 m^3 The buoyant mass = the volume of air displaced * density The buoyant mass = 0.059m^3 * 1.28 kg/m^3 = 0.07552 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 = 621 - 213.7+ 0.07552 = 407.37 The force = 407.37 * 9.81 = 3996.4 N Problem Four ========== This is really tricky. I think you are intended to use the volume, but you really should not have to. F person (no buoyancy) = 72 * 9.81 = 706.32 N Buoyancy on each person = 0.9543 With buoyancy = 706.32 - 0.076 * 1.28 * 9.81 =705.4 N 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.

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