11.7 Archimedes’ principle  (Page 6/9)

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Conceptual questions

More force is required to pull the plug in a full bathtub than when it is empty. Does this contradict Archimedes’ principle? Explain your answer.

Do fluids exert buoyant forces in a “weightless” environment, such as in the space shuttle? Explain your answer.

Will the same ship float higher in salt water than in freshwater? Explain your answer.

Marbles dropped into a partially filled bathtub sink to the bottom. Part of their weight is supported by buoyant force, yet the downward force on the bottom of the tub increases by exactly the weight of the marbles. Explain why.

Problem exercises

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

$\text{91}\text{.}7%\text{}$

Logs sometimes float vertically in a lake because one end has become water-logged and denser than the other. What is the average density of a uniform-diameter log that floats with $20.0%$ of its length above water?

Find the density of a fluid in which a hydrometer having a density of $0\text{.}\text{750 g}\text{/mL}$ floats with $\text{92.0%}$ of its volume submerged.

$\text{815 kg}{\text{/m}}^{3}$

If your body has a density of $\text{995 kg}{\text{/m}}^{3}$ , what fraction of you will be submerged when floating gently in: (a) Freshwater? (b) Salt water, which has a density of $\text{1027 kg}{\text{/m}}^{3}$ ?

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 $\text{45.0 g}$ 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 its average density?

(a) 41.4 g

(b) $\text{41}\text{.}4\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$

(c) $1\text{.}\text{09 g}{\text{/cm}}^{3}$

A rock with a mass of 540 g in air is found to have an apparent mass of 342 g when submerged in water. (a) What mass of water is displaced? (b) What is the volume of the rock? (c) What is its average density? Is this consistent with the value for granite?

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 [link] (c) Calculate the fluid’s density and identify it.

(a) 39.5 g

(b) $\text{50}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$

(c) $0\text{.}\text{79}\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}$

It is ethyl alcohol.

In an immersion measurement of a woman’s density, she is found to have a mass of 62.0 kg in air and an apparent mass of 0.0850 kg when completely submerged with lungs empty. (a) What mass of water does she displace? (b) What is her volume? (c) Calculate her density. (d) If her lung capacity is 1.75 L, is she able to float without treading water with her lungs filled with air?

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 $\text{1015}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ ?

8.21 N

(a) Calculate the buoyant force on a 2.00-L helium balloon. (b) Given the mass of the rubber in the balloon is 1.50 g, what is the net vertical force on the balloon if it is let go? You can neglect the volume of the rubber.

(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) $\text{960}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$

(b) $6.34%$

She indeed floats more in seawater.

A certain man has a mass of 80 kg and a density of $\text{955}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ (excluding the air in his lungs). (a) Calculate his volume. (b) Find the buoyant force air exerts on him. (c) What is the ratio of the buoyant force to his weight?

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?

(a) $0\text{.}\text{24}$

(b) $0\text{.}\text{68}$

(c) Yes, the cork will float because ${\rho }_{\text{obj}}<{\rho }_{\text{ethyl alcohol}}\left(0\text{.}\text{678}\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}<0\text{.}\text{79}\phantom{\rule{0.25em}{0ex}}{\text{g/cm}}^{3}\right)$

What fraction of an iron anchor’s weight will be supported by buoyant force when submerged in saltwater?

Scurrilous con artists have been known to represent gold-plated tungsten ingots as pure gold and sell them to the greedy 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?

The difference is $0.006%.$

A twin-sized air mattress used for camping has dimensions of 100 cm by 200 cm by 15 cm when blown up. The weight of the mattress is 2 kg. How heavy a person could the air mattress hold if it is placed in freshwater?

Referring to [link] , 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 $\left({h}_{2}-{h}_{1}\right)A$ .

${F}_{\text{net}}={F}_{2}-{F}_{1}={P}_{2}A-{P}_{1}A=\left({P}_{2}-{P}_{1}\right)A$

$=\left({h}_{2}{\rho }_{\text{fl}}g-{h}_{1}{\rho }_{\text{fl}}g\right)A$

$=\left({h}_{2}-{h}_{1}\right){\rho }_{\text{fl}}\text{gA}$

where ${\rho }_{\text{fl}}$ = density of fluid. Therefore,

${F}_{\text{net}}=\left({h}_{2}-{h}_{1}\right){\mathrm{A\rho }}_{\text{fl}}g={V}_{\text{fl}}{\rho }_{\text{fl}}g={m}_{\text{fl}}g={w}_{\text{fl}}$

where is ${w}_{\text{fl}}$ the weight of the fluid displaced.

(a) A 75.0-kg man floats in freshwater with $3.00%$ of his volume above water when his lungs are empty, and $5.00%$ of his volume above water when his lungs are full. Calculate the volume of air he inhales—called his lung capacity—in liters. (b) Does this lung volume seem reasonable?

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