Question

A hollow sphere of diameter 3 in. is filled with air at 200 atm, 60°F. The shell of the sphere has a mass of 0.2 lbm and is of negligible thickness. If the sphere is submerged in water at 60° F, determine the magnitude of the net vertical force, in lbf. Is the force acting upward or downward? Assume the air can be modeled as in ideal gas.

Answer 1

`f_(net) = 0.1853 lbf` {upward}

Step-by-step explanation:

Byoncy force is given as

`fb =  \gamma_(water) v_(ball)`

`fb = \gamma_(water) (4)/(3) \pi R^3`

At 60 degree F
`\gamma_(water) = 62.37 lb/ft^3`

R is 1.5 inch = 1.5/12 ft

so solving for Fs we get

fb = 0.5102 lb

By using ideal gas equation calculate mass of air in sphere

`m = (pV)/(R_(air)T)`

p = 200 atm = 2939.19 psi

`R_(air) = 0.3704 psi ft^3 /lbm -R`

`m = (2939.19 psi a* [(4)/(3) \pi [(1.5)/(12)]^2])/(0.3704 * (90+459.67)`

m = 0.1249 lbm

total weight of air and ball os

wa +wb = g(ma+mb)

`= 32.174 (0.1249+0.2)* (1 lbf)/(32.174\ lbm ft/s^2)`

wa+wb = 0.3249 lbf

net foce on the ball is

`f_(net) = fb - (wa+wb)`

`f_(net) = 0.5102 - 0.249`

`f_(net) = 0.1853 lbf`