WEBVTT

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This is Giancoli Answers
with Mr. Dychko.

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When this metal is submerged in water,
there are two forces upwards:

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one is the buoyant force equal to
the weight of the water displaced;

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and the other is the apparent weight

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or the tension exerted by some spring scale
that's holding it up

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and then there's this gravity downwards
on the metal sample as well.

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So the total up forces have to equal
the total down forces

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when this thing is stationary

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and so the apparent weight up plus
the buoyant force up equals

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the weight down which is <i>mg</i>

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and the buoyant force upwards is
the mass of

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water displaced times <i>g</i> to give
the weight of water displaced.

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So that's the density of water—

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I have written <i>ρ F</i> for 'fluid' here—

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times the volume of the piece of metal

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which is also the volume of water displaced
because the metal is totally submerged

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and then we have the apparent mass times
<i>g</i> to give the apparent weight.

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And we'll solve this for <i>V</i> because
then we can find the density of

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the material by going mass
divided by the volume.

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So the <i>g</i>'s cancel everywhere and then

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we take this <i>m prime</i> to the right side
by subtracting it from both sides

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so this gives <i>m</i> minus <i>m prime</i>

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and then afterwards divide by

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the density of the fluid—density of water
in this case—

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so that gives the volume of this thing.

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And then its density, which is how we'll identify it,

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because we'll look at this table [10-1]
on page 261

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and look at the densities there

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and we take the mass multiplied by
the reciprocal of the volume—

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instead of dividing by a fraction,
we'll multiply by its reciprocal—

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so we are taking mass times density of
the fluid divided by

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the true mass minus the apparent mass

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and the true mass is 63.5 grams, sort of.

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I say sort of because this is the mass
measured in air

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and technically that's also affected by
the fact that there's a buoyant force

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upwards due to the air

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but the density of air is so small compared
to the density of water

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and compared to the density of
the material,

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it's gonna have no significant effect.

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So we can consider this to be the mass in
a vacuum, the true mass.

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So we have 63.5 grams

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times density of water—1.00 times 10 to
the 3 kilograms per cubic meter—

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divided by 63.5 grams minus 55.4 grams

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and that gives 7840 which appears
to be iron or steel

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when you look at this table of densities
on page 261.