What Minimum Mass of Lead, Hung From the Wood by a String, Will Cause It to Sink?
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Question:
A 3.65-kg block of wood (SG = 0.50) floats on water. What minimum mass of lead, hung from the woods by a cord, will cause the block to sink?
Giancoli, Douglas C., Physics: Principles with Applications, 7th Ed., ©2014. Reprinted past permission of Pearson Education Inc., New York.
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Giancoli 7th Edition, Chapter x, Problem 40
(iii:52)
Transcript for this Giancoli solution
This is Giancoli Answers with Mr. Dychko. When this forest and pb combination just begins to sink, nosotros have the woods fully submerged and the lead submerged and the forest will accept a buoyant strength upwards, F B w for 'woods', and the lead will besides experience a buoyant forcefulness upwardly, F B and then Pb because that's the symbol for 'lead' and then there's the weight of the lead downwards and the weight of the wood downwards and the total up has to equal the total down for this case— we have just the minimum amount of lead— and then F B w plus F B Atomic number 82 equals F yard w plus F one thousand Lead. Then the buoyant force on the forest is gonna be the weight of water that the wood displaces. And then the wood is fully submerged so information technology displaces a volume of water equal to the volume of the wood and multiply that by the density of water and times by g to become the weight of h2o displaced which will be the buoyant forcefulness upwards on the wood and and then density of water times the volume of the atomic number 82 block times g and that equals mass of the wood times g plus mass of the lead times 1000 gravity downwards in each example there and we want to know the mass of lead needed so we should convert these volumes into masses past using this formula here that says the volume of the woods is the mass of the wood divided by the wood's density and then the volume of the lead is gonna exist the mass of the atomic number 82 divided by the lead's density and nosotros know these density numbers and so this is a skilful strategic thing to do because then we'll have a formula having just masses and densities, which we know and nosotros know the mass of the wood and we can solve for the mass of the lead hither. And then I canceled out the g's besides hither by the style so substituted for the volumes on this line here copied here with the volume substituted and nosotros have mass of the wood divided by specific gravity of the wood because here we have mass of the woods multiplied by density of water over density of the wood and nosotros could instead divide by its reciprocal and then we take mass of the wood divided by the reciprocal of this affair density of wood divided past density of H 2 O and this is specific gravity so we have mass of the wood divided by specific gravity here. Plus you know, we could become mass of pb divided by pb'due south specific gravity too but we don't take that data in whatsoever table here so we'll but leave information technology as it is and then that equals mass of wood plus mass of the lead. And the next line doing a couple of steps here, we accept this term moved to the correct side so that makes information technology a minus and so we can factor out the mass of the lead and it becomes mass of the lead multiplied by one minus this ratio of densities there and then as well switch the sides around then that we take mass of the atomic number 82 on the left side because the unknown should always be on the left just by convention And then nosotros accept mass of the forest moved to the left side which ends upwardly on the correct side after I flip it around but we have mass of the wood so nosotros can factor that out from this term and this term and it becomes mass of wood times 1 over specific gravity of wood minus 1 and then split up both sides by this subclass and you have terminal formula is mass of the lead needed volition be mass of the wood times reciprocal of the wood's specific gravity minus one all divided past 1 minus density of water divided by density of lead. And so I plug in 3.65 kilograms times i over 0.l—specific gravity of wood— minus one divided past one minus i.00 divided by 11.3— I didn't need to put this times 10 to the three business into the calculator considering ten to the 3 divided by 10 to the 3 but cancels anyway so 1 divided by eleven.3 gives the aforementioned ratio— and we take 4.00 kilograms is the mass of lead needed to sink the woods block.Source: https://www.giancolianswers.com/giancoli-physics-7th-edition-solutions/chapter-10/problem-40
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