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[Mechanics] Hot Air Balloon

2021-07-30 14:58 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

The density of air is $1.29%20%5C%3B%20kg%2Fm%5E3$ and the density of helium is $0.18%20%5C%3B%20kg%2Fm%5E3$ . The mass of the balloon is $20%20%5C%3B%20kg%20$ . What volume of helium would a hot air balloon needed to lift a $%20180%20%5C%3B%20kg%20$ crate?

【Solution】

The balloon must be able to lift the crate and its own mass. Thus, the buoyancy force $F_B$ is

$F_B%20%3D%20%7Bm%7D_%7Btotal%7Dg$

$F_B%20%3D%20(%7Bm%7D_%7BHe%7D%20%2B%20180%20%2B%2020)g$

Here, the buoyant force is $F_B%20%3D%20%7B%5Crho%7D_%7BHe%7D%20Vg$ and the mass of helium is $%7Bm%7D_%7BHe%7D%20%3D%20%7B%5Crho%7D_%7BHe%7D%20V$

$%7B%5Crho%7D_%7Bair%7D%20Vg%20%3D%20%5Cleft(%7B%5Crho%7D_%7BHe%7D%20V%20%2B%20200%20%5Cright)%20g%20$

$V%20%3D%20%5Cfrac%7B200%7D%7B%7B%5Crho%7D_%7Bair%7D%20-%20%7B%5Crho%7D_%7BHe%7D%7D%20$

$V%20%3D%20%5Cfrac%7B200%7D%7B1.29%20-%200.18%7D%20$

$V%20%3D%20180%20%5C%3B%20m%5E3$

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