11.7 Archimedes’ Principle

What fraction of ice is submerged when it floats in freshwater, given the density of water at 0°C is very close to $\text{1000 kg} /\text{m}^{3}$?

Find the density of a fluid in which a hydrometer having a density of $0 . \text{750 g} /\text{mL}$ floats with $\text{92}.\text{0}\%$ of its volume submerged.

Bird bones have air pockets in them to reduce their weight—this also gives them an average density significantly less than that of the bones of other animals. Suppose an ornithologist weighs a bird bone in air and in water and finds its mass is $\text{45}.\text{0 g}$ and its apparent mass when submerged is $3.60 g$ (the bone is watertight). (a) What mass of water is displaced? (b) What is the volume of the bone? (c) What is…

Archimedes’ principle can be used to calculate the density of a fluid as well as that of a solid. Suppose a chunk of iron with a mass of 390.0 g in air is found to have an apparent mass of 350.5 g when completely submerged in an unknown liquid. (a) What mass of fluid does the iron displace? (b) What is the volume of iron, using its density as given in the reference (c) Calculate the fluid’s densit…

Some fish have a density slightly less than that of water and must exert a force (swim) to stay submerged. What force must an 85.0-kg grouper exert to stay submerged in salt water if its body density is $\text{1015} \text{kg}/\text{m}^{3}$?

(a) What is the density of a woman who floats in freshwater with $4.00$ of her volume above the surface? This could be measured by placing her in a tank with marks on the side to measure how much water she displaces when floating and when held under water (briefly). (b) What percent of her volume is above the surface when she floats in seawater?

A simple compass can be made by placing a small bar magnet on a cork floating in water. (a) What fraction of a plain cork will be submerged when floating in water? (b) If the cork has a mass of 10.0 g and a 20.0-g magnet is placed on it, what fraction of the cork will be submerged? (c) Will the bar magnet and cork float in ethyl alcohol?

Scurrilous con artists have been known to represent gold-plated tungsten ingots as pure gold and sell them at prices much below gold value but deservedly far above the cost of tungsten. With what accuracy must you be able to measure the mass of such an ingot in and out of water to tell that it is almost pure tungsten rather than pure gold?

Referring to the figure, prove that the buoyant force on the cylinder is equal to the weight of the fluid displaced (Archimedes’ principle). You may assume that the buoyant force is $F_{2} - F_{1}$ and that the ends of the cylinder have equal areas $A$. Note that the volume of the cylinder (and that of the fluid it displaces) equals $\left(h_{2} - h_{1}\right) A$.