A consumer upset with the latest trend of postal rate increases has decided to t

Question

A consumer upset with the latest trend of postal rate increases has decided to try to send letters by balloon even though they may not reach their intended destination. A 66400 cm3 gas-filled balloon will provide enough lift for a 43.4 g package to be accelerated upward at a rate of 2.95 m/s2. For these circumstances, calculate the density of the gas the consumer fills the balloon with. The acceleration due to gravity is g = 9.81 m/s2 and the density of air is air = 1.16 kg/m3. Neglect the mass of the balloon material and the volume of the package.

The gas station owner balks at your estimate. He neither has the funds nor approval from the city to dig that deeply. However, you tell him that the best a thief could hope to have at his disposal is a 194-mbar vacuum pump. Anything better would be enormously expensive. Given this information, how deep would the gasoline have to sit below the surface to keep it safe from thieves?

Explanation / Answer

According to the given problem,

A)Mass of displaced air M = air*V = 1.16kg/m³ * 0.0664m³ = 0.0770 kg
buoyant force Fb = air*V*g = 1.16kg/m³ * 0.0664m³ * 9.81m/s² = 0.7556 N

mass of gas m = *V = *0.0664m³
weight of gas Fg = *V*g = *0.0664m³*9.81m/s² = *0.6514m/s²

acceleration a = Fnet / totalmass = (Fb - Fg - m*g) / (*V + m)
Dropping units for ease ( is in kg/m³):
2.95 = (0.7556 - *0.6514 - 0.0434*9.81) / (*0.0664 + 0.0434)
2.95(*0.0664 + 0.0434) = 0.329846 - *0.6514
*0.19588 + 0.12803 = 0.329846 - *0.6514
*0.525726 = 0.201816
= 0.384 kg/m³

B)The lowest pressure that a vacuum pump could ever hope to attain is 0 bar...

The height of a column of fluid (assuming the vapour pressure in the tank is 1 bar) then maximally is

Delta(p) = rho g Delta(h)

Delta(h) = Delta(p) / (rho g)
= 10^5 N/m^2 / (766 kg/m^3 * 9.81 m/s^2) = 13.3 meter (43.6 feet)


If Delta(p) is not 1 bar but ( 1 - 0.194 ) bar = 0.806 bar, then the depth needed is reduced by a factor 0.806.

So then

D = 0.806 * 43.6 feet = 35.1416 feet

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