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phy131studiof15:lectures:chapter15 [2015/10/21 09:31] mdawber [Helium Balloon] |
phy131studiof15:lectures:chapter15 [2015/10/21 09:40] (current) mdawber [Floating ball] |
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I want to inflate a ball so that it will float in water with exactly half of the ball above the surface and the other half below when the temperature is 20$\mathrm{^{o}C}$. The ball will be 20 cm in diameter and made from rubber which is 1 cm thick and has density 1800 $\mathrm{kg\, m^{-3}}$. The density of air at 20$\mathrm{^{o}C}$ under normal conditions, such as those that exist above the surface of the water, is 1.2 $\mathrm{kg\, m^{-3}}$. The density of water is 1000 $\mathrm{kg\, m^{-3}}$. The inside of the ball is to be filled with pressurized air, which will have a higher mass density than that outside the ball. What should the mass density of air inside the ball be so that ball floats as desired? | I want to inflate a ball so that it will float in water with exactly half of the ball above the surface and the other half below when the temperature is 20$\mathrm{^{o}C}$. The ball will be 20 cm in diameter and made from rubber which is 1 cm thick and has density 1800 $\mathrm{kg\, m^{-3}}$. The density of air at 20$\mathrm{^{o}C}$ under normal conditions, such as those that exist above the surface of the water, is 1.2 $\mathrm{kg\, m^{-3}}$. The density of water is 1000 $\mathrm{kg\, m^{-3}}$. The inside of the ball is to be filled with pressurized air, which will have a higher mass density than that outside the ball. What should the mass density of air inside the ball be so that ball floats as desired? | ||

+ | Weight of displaced fluid | ||

+ | $(\frac{1}{2}1000+\frac{1}{2}1.2)\frac{4}{3}\pi0.1^{3}$ | ||

+ | |||

+ | Weight of ball including dense air | ||

+ | |||

+ | $1800\frac{4}{3}\pi(0.1^{3}-0.09^{3})+\rho_{air}\frac{4}{3}\pi0.09^{3}$ | ||

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+ | At equilibrium these two are equal, so | ||

+ | |||

+ | $\rho_{air}=\frac{(\frac{1}{2}1000+\frac{1}{2}1.2)\frac{4}{3}\pi0.1^{3}-1800\frac{4}{3}\pi(0.1^{3}-0.09^{3})}{\frac{4}{3}\pi0.09^{3}}$ | ||

+ | |||

+ | $\rho_{air}=17.55\mathrm{kg\,m^{-3}}$ | ||

===== Helium Balloon ===== | ===== Helium Balloon ===== | ||