Chapter 5.P, Problem 6P

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 ${\rho }_0$ floating partly submerged in a fluid of density ${\rho }_f$, the buoyant force is given by $F={\rho }_{f}g{\displaystyle {\int }_{-h}^{0}A\left(y\right)dy}$, where $g$ is the acceleration due to gravity and $A\left(y\right)$ is the area of a typical cross-section of the object (see the figure). The weight of the object is given by

$W={\rho }_{0}g{\displaystyle {\int }_{-h}^{L-h}A\left(y\right)dy}$

(a) Show that the percentage of the volume of the object above the surface of the liquid is

$100\frac{{\rho }_f-{\rho }_0}{{\rho }_f}$

(b) The density of ice is 917 kg/m3 and the density of seawater is 1030 kg/m3. 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 0.4 m 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 1000 kg/m3.