Archimedes’ Principle states that the buoyant force on an object partially or fully submerged in a fluid is equal to the weight of the fluid that the object displaces. Thus, for an object of density ρ0ρ0 floating partly submerged in a fluid of density ρfρf, the buoyant force is given by
![](data:image/gif;base64,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)
where g is the acceleration due to gravity and A(y) is the area of a typical cross-section of the object (see the figure). The weight of the object is given by
![](data:image/gif;base64,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)
(a) Show that the percentage of the volume of the object above the surface of the liquid is
.
(b) The density of ice is
and the density of seawater is
. What percentage of the volume of an iceberg is above water?
(c) An ice cube floats in a glass filled to the brim with water. Does the water overflow when the ice melts?
(d) A sphere of radius
and having negligible weight is floating in a large freshwater lake. How much work is required to completely submerge the sphere? The density of the water is
.
![](https://s3.amazonaws.com/prealliance-textbook-qa.oneclass.com/qa_images/homework_help/question/qa_images/816812979302762/question/7.png?img=38cfc509a5adae1e5635a3244f1d72ce)