![]() This means that $V_1$ is smaller than $V_2$.Īs the percent $p$ of gold, by volume, in the crown varies, from $0$ to $100$ Note that this agrees with the answer in part (a) which deals with the casesįirst note that gold is almost twice as dense as silver and so an equal mass of gold Therefore the amount of water displaced would be Gold crown and $1-p$ of a solid silver crown. Reasoning as in part (a), this crown would contain a percent $p$ of a solid So the total amount of waterĭisplaced by a crown which is half silver and half gold would be Is half silver and half gold by volume, the silver half would displace $\frac$. While a solid silver crown would displace a volume $V_2$ of water. We are given that a solid gold crown would displace a volume $V_1$ of water Of the crown which is the same as the volume of the water displaced when it Measuring the two water levels indicated and subtracting gives the volume If not, then the lighter material in the crown would displace more water and the submerged balance would tilt toward the side with the gold.Ī picture showing how to find the volume of the crown is given below: If the two sides remained balanced, then the crown was made of solid gold. Though it is not possible to know with certainty how Archimedes made his discovery, one likely possibility is that he took a balance with the crown on one side and an equal mass of gold on the other and then submerged the entire balance in water. Teachers may also wish to discuss, in this same regard, how it is that an aircraft carrier, whose weight is phenomenally large, is able to float on water. ![]() Submerging the tube means displacing a large volume of water and of course water is relatively dense. The air occupies a lot of space with very little density. To submerge an inner tube filled with air under water comes from the fact that For example, the tremendous force required ![]() More generally, this method of relating volumes and density relates to otherĮxperiences in students' lives. Not change but the graph would no longer be linear in part (c). The question could also be formulated in terms of the relative percentages of ![]() Relative percentages of silver and gold by volume in the crown. The problem asks students to find the amount of water displaced given the The teacher may wish to share the picture at the beginning of the solution with students to help them visualize Archimedes' idea. Work as there would be a third volume $V_3$ involved and a more complicatedĪlgebraic expression relating the composition of the crown to its density. Lead, a much cheaper but also very dense material, the argument would no longer That the crown is made of gold and silver alone. In practice, Archimedes' method requires knowing The problem combines the ideas of ratio and proportion within the context of density of matter. Shouting ''Eureka!!!'' (I have found it) reportedly occurred after he solved this problem. The famous story of Archimedes running through the streets of Syracuse (in Sicily during the third century bc) ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |