Chapter 6, Problem 6PP

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 ρ₀, floating partly submerged in a fluid of density ρ_f, the buoyant force is given by $F={\rho }_{f}g{\displaystyle {\int }_{-h}^{0}A(y)dy}$, 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

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

(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.

FIGURE FOR PROBLEM 6