A 3.65-kg block of wood (SG = 0.50) floats on water. What minimum mass of lead, hung from the wood by a string, will cause the block to sink?

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. When this wood and lead combination just begins to sink, we have the wood fully submerged and the lead submerged and the wood will have a buoyant force upwards, F B w for 'wood', and the lead will also experience a buoyant force upwards, 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 so F B w plus F B Pb equals F g w plus F g Pb. So the buoyant force on the wood is gonna be the weight of water that the wood displaces. So the wood is fully submerged so it 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 get the weight of water displaced which will be the buoyant force upwards on the wood and then density of water times the volume of the lead block times g and that equals mass of the wood times g plus mass of the lead times g gravity downwards in each case there and we want to know the mass of lead needed so we should convert these volumes into masses by using this formula here that says the volume of the wood is the mass of the wood divided by the wood's density and then the volume of the lead is gonna be the mass of the lead divided by the lead's density and we know these density numbers and so this is a good strategic thing to do because then we'll have a formula having just masses and densities, which we know and we know the mass of the wood and we can solve for the mass of the lead here. So I canceled out the g's as well here by the way and then substituted for the volumes on this line here copied here with the volume substituted and we have mass of the wood divided by specific gravity of the wood because here we have mass of the wood multiplied by density of water over density of the wood and we could instead divide by its reciprocal so we have mass of the wood divided by the reciprocal of this thing density of wood divided by 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 go mass of lead divided by lead's specific gravity too but we don't have that data in any table here so we'll just leave it 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 have this term moved to the right side so that makes it a minus and then we can factor out the mass of the lead and it becomes mass of the lead multiplied by 1 minus this ratio of densities there and then also switch the sides around so that we have mass of the lead on the left side because the unknown should always be on the left just by convention And then we have mass of the wood moved to the left side which ends up on the right side after I flip it around but we have mass of the wood and then we 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 divide both sides by this bracket and you have final formula is mass of the lead needed will be mass of the wood times reciprocal of the wood's specific gravity minus 1 all divided by 1 minus density of water divided by density of lead. And so I plug in 3.65 kilograms times 1 over 0.50—specific gravity of wood— minus 1 divided by 1 minus 1.00 divided by 11.3— I didn't need to put this times 10 to the 3 business into the calculator because 10 to the 3 divided by 10 to the 3 just cancels anyway so 1 divided by 11.3 gives the same ratio— and we have 4.00 kilograms is the mass of lead needed to sink the wood block.