WEBVTT

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This is Giancoli Answers
with Mr. Dychko.

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Let's consider just a single
balloon for a moment:

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we have this buoyant force

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equal to the weight of air displaced by
the balloon that's going upwards

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and then we have the weight of
the helium downwards

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and we also have this load that's
pulling the balloon down

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and this load is at a maximum when

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the load plus the weight of the helium
equals the buoyant force upwards.

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So that's considering one balloon

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and this solves for the maximum load force

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that could be exerted downwards
on a single balloon.

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When we consider a person, they are
gonna have <i>n</i> times

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this maximum load force

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so the number of balloons, <i>n</i>,

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times <i>F L</i>—that's gonna be the force
upwards—

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and then that's gonna have to equal
their weight downwards, <i>mg</i>.

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So we have the weight of the helium is
the mass of helium which is

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helium's density times by the volume

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which is four-third's <i>πr cubed</i>

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because this balloon is a sphere

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and that's times by <i>g</i>

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and that equals the density of air

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displaced by the balloon times by
this same volume—

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the volume of air displaced by the balloon
is the volume of the balloon—

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times by <i>g</i>

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and so the maximum load force after you
subtract this term from both sides

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is gonna be four-third's <i>πr cubed g</i>

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which is a common factor between
these two terms

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multiplied by <i>ρ air</i> minus <i>ρ helium</i>—

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density of air minus density of helium—

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and then that we'll substitute down here.

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So we have <i>n</i> times <i>F L</i> equals <i>mg</i>

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so now we are looking at this
picture of a person

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and we'll solve for <i>n</i> by dividing
both sides by <i>F L</i>

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and we have the number of balloons
is <i>mg</i> divided by <i>F L</i>

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which is <i>mg</i> divided by four-third's
<i>πr cubed g</i>

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times the difference in the densities
between air and helium

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and cleaning that up a bit,

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the <i>g</i>'s cancel and this 3 goes on the top

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we have <i>3m</i> over 4<i>πr cubed</i>

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difference in their densities and
then we plug in some numbers.

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And we have 3 times 72 kilograms

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divided by 4<i>π</i> times this diameter

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divided by 2 to get radius

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and then converting the centimeters
into meters by multiplying

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33 by 10 to the minus 2,

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cube that radius times by 1.29 kilograms
per cubic meter—density of air—

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minus 0.179 kilograms per cubic meter—
density of helium—

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and you get about 3400 balloons would be
needed to lift this person.