Buoyant Force Is Greatest On A Submerged

• Buoyant Force Is Greatest On A Submerged Area
• How To Find The Buoyant Force Of An Object Submerged In Water
• Buoyant Force Is Greatest On A Submerged Wave
• Buoyant Force Is Greatest On A Submerged Plane
• Buoyant Force Is Greatest On A Submerged Water
• How Does The Buoyant Force On A Submerged Object

When you rise from lounging in a warm bath, your arms feel strangely heavy. This is because you no longer have the buoyant support of the water. Where does this buoyant force come from? Why is it that some things float and others do not? Do objects that sink get any support at all from the fluid? Is your body buoyed by the atmosphere, or are only helium balloons affected (Figure (PageIndex{1}))?

Answers to all these questions, and many others, are based on the fact that pressure increases with depth in a fluid. This means that the upward force on the bottom of an object in a fluid is greater than the downward force on the top of the object. There is a net upward, or buoyant force on any object in any fluid (Figure (PageIndex{2})). If the buoyant force is greater than the object’s weight, the object will rise to the surface and float. If the buoyant force is less than the object’s weight, the object will sink. If the buoyant force equals the object’s weight, the object will remain suspended at that depth. The buoyant force is always present whether the object floats, sinks, or is suspended in a fluid.

Defintion: Buoyant Force

Buoyant force is equal to the weight of the water displaced. The weight of the water displaced has nothing to do with the weight of the object (10N), it has to do with the volume of the object. (5 votes) See 1 more reply.

1. A sphere of radius 10.0 cm floats in equilibrium partially submerged in water with its lowest point 5.00 cm below the water's surface. (a) What is the buoyant force acting on the sphere? Homework Equations F = pvg The Attempt at a Solution F= 1000. v. 9.8 V= 4/3.pi.(r)^3 and then I multiplied this by 1/2 because half of it is submerged.
2. Buoyant force is greatest on a submerged: A. 1-kg block of lead. 1-kg block of aluminum. Is the same on each. Buoyant force is greatest on a submerged: A. 1-cubic centimeter block of lead. 1-cubic centimeter block of aluminum. Is the same on each 11. A consequence of the surface tension of water is: A. Capillary action.

The buoyant force is the net upward force on any object in any fluid.

Just how great is this buoyant force? To answer this question, think about what happens when a submerged object is removed from a fluid, as in Figure (PageIndex{3}).

The space it occupied is filled by fluid having a weight (w_{fl}). This weight is supported by the surrounding fluid, and so the buoyant force must equal (w_{fl}), the weight of the fluid displaced by the object. It is a tribute to the genius of the Greek mathematician and inventor Archimedes (ca. 287–212 B.C.) that he stated this principle long before concepts of force were well established. Stated in words, Archimedes’ principle is as follows: The buoyant force on an object equals the weight of the fluid it displaces. In equation form, Archimedes’ principle is

[F_B = w_{fl},]

where (F_B) is the buoyant force and (w_{fl}) is the weight of the fluid displaced by the object. Archimedes’ principle is valid in general, for any object in any fluid, whether partially or totally submerged.

Archimedes’ Principle

According to this principle the buoyant force on an object equals the weight of the fluid it displaces. In equation form, Archimedes’ principle is [F_B = w_{fl},] where (F_B) is the buoyant force and (w_{fl}) is the weight of the fluid displaced by the object.

Humm … High-tech body swimsuits were introduced in 2008 in preparation for the Beijing Olympics. One concern (and international rule) was that these suits should not provide any buoyancy advantage. How do you think that this rule could be verified?

Making Connections: Take-Home Investigation

The density of aluminum foil is 2.7 times the density of water. Take a piece of foil, roll it up into a ball and drop it into water. Does it sink? Why or why not? Can you make it sink?

Floating and Sinking

Drop a lump of clay in water. It will sink. Then mold the lump of clay into the shape of a boat, and it will float. Because of its shape, the boat displaces more water than the lump and experiences a greater buoyant force. The same is true of steel ships.

Example (PageIndex{1}): Calculating buoyant force: dependency on shape

1. Calculate the buoyant force on 10,000 metric tons ((1.00 times 10^7 , kg)) of solid steel completely submerged in water, and compare this with the steel’s weight.
2. What is the maximum buoyant force that water could exert on this same steel if it were shaped into a boat that could displace (1.00 times 10^5 , m^3) of water?

Strategy for (a)

To find the buoyant force, we must find the weight of water displaced. We can do this by using the densities of water and steel given in [link]. We note that, since the steel is completely submerged, its volume and the water’s volume are the same. Once we know the volume of water, we can find its mass and weight.

Solution for (a)

First, we use the definition of density (rho = frac{m}{V}) to find the steel’s volume, and then we substitute values for mass and density. This gives

[V_{st} = dfrac{m_{st}}{rho_{st}} = dfrac{1.00 times 10^7 , kg}{7.8 times 10^3 , kg/m^3} = 1.28 times 10^3 , m^3.]

Because the steel is completely submerged, this is also the volume of water displaced, (V_W). We can now find the mass of water displaced from the relationship between its volume and density, both of which are known. This gives

[m_W = rho_WV_W = (1.000 times 10^3 , kg/m^3)(1.28 times 10^3 , m^3)]

[= 1.3 times 10^6 , kg.]

By Archimedes’ principle, the weight of water displaced is (m_Wg), so the buoyance force is

[F_B = w_W = m_Wg = (1.28 times 10^6space kg)(9.80 , m/s^2)]

[= 1.3 times 10^7 , N.]

The steel’s weight is (m_W g = 9.80 times 10^7 , N),

which is much greater than the buoyant force, so the steel will remain submerged. Note that the buoyant force is rounded to two digits because the density of steel is given to only two digits.

Strategy for (b)

Here we are given the maximum volume of water the steel boat can displace. The buoyant force is the weight of this volume of water.

Buoyant Force Is Greatest On A Submerged Area

Solution for (b)

The mass of water displaced is found from its relationship to density and volume, both of which are known. That is,

[m_W = rho_WV_W = (1.000 times 10^3 , kg/m^3)(1.00 times 10^5 , m^3)]

[= 9.80 times 10^8 , kg.]

The maximum buoyant force is the weight of this much water, or

[F_B = w_W = m_W g = (1.00 times 10^8 , kg)(9.80 , m/s^2)]

[= times 10^8 , N.]

Discussion

The maximum buoyant force is ten times the weight of the steel, meaning the ship can carry a load nine times its own weight without sinking.

Making Connections: Take-Home Investigation

• A piece of household aluminum foil is 0.016 mm thick. Use a piece of foil that measures 10 cm by 15 cm. (a) What is the mass of this amount of foil? (b) If the foil is folded to give it four sides, and paper clips or washers are added to this “boat,” what shape of the boat would allow it to hold the most “cargo” when placed in water? Test your prediction.

Density and Archimedes’ Principle

Density plays a crucial role in Archimedes’ principle. The average density of an object is what ultimately determines whether it floats. If its average density is less than that of the surrounding fluid, it will float. This is because the fluid, having a higher density, contains more mass and hence more weight in the same volume. The buoyant force, which equals the weight of the fluid displaced, is thus greater than the weight of the object. Likewise, an object denser than the fluid will sink.

The extent to which a floating object is submerged depends on how the object’s density is related to that of the fluid. In Figure (PageIndex{4}), for example, the unloaded ship has a lower density and less of it is submerged compared with the same ship loaded. We can derive a quantitative expression for the fraction submerged by considering density. The fraction submerged is the ratio of the volume submerged to the volume of the object, or

[fraction , submerged = dfrac{V_{sub}}{V_{obj}} = dfrac{V_{fl}}{V_{obj}}.]

The volume submerged equals the volume of fluid displaced, which we call (V_{fl}). Now we can obtain the relationship between the densities by substituting (rho = frac{m}{V}) into the expression. This gives

[dfrac{V_{fl}}{V_{obj}} = dfrac{m_{fl}/rho_{fl}}{m_{obj}/overline{rho}_{obj}},]

where (overline{rho}_{obj}) is the average density of the object and (rho_{fl}) is the density of the fluid. Since the object floats, its mass and that of the displaced fluid are equal, and so they cancel from the equation, leaving

[fraction , submerged = dfrac{overline{rho}_{obj}}{rho_{fl}}.]

We use this last relationship to measure densities. This is done by measuring the fraction of a floating object that is submerged—for example, with a hydrometer. It is useful to define the ratio of the density of an object to a fluid (usually water) as specific gravity:

[specific , gravity = dfrac{overline{rho}}{rho_W},] where (overline{rho}) is the average density of the object or substance and (rho_W) is the density of water at 4.00°C. Specific gravity is dimensionless, independent of whatever units are used for (rho). If an object floats, its specific gravity is less than one. If it sinks, its specific gravity is greater than one. Moreover, the fraction of a floating object that is submerged equals its specific gravity. If an object’s specific gravity is exactly 1, then it will remain suspended in the fluid, neither sinking nor floating. Scuba divers try to obtain this state so that they can hover in the water. We measure the specific gravity of fluids, such as battery acid, radiator fluid, and urine, as an indicator of their condition. One device for measuring specific gravity is shown in Figure (PageIndex{5}).

Definition: Specific Gravity

Specific gravity is the ratio of the density of an object to a fluid (usually water).

Example (PageIndex{2}): Calculating Average Density: Floating Woman

Suppose a 60.0-kg woman floats in freshwater with (97.0%) of her volume submerged when her lungs are full of air. What is her average density?

Strategy

We can find the woman’s density by solving the equation

[fraction , submerged = dfrac{overline{rho}_{obj}}{rho_{fl}}]

for the density of the object. This yields

[overline{rho}_{obj} = overline{rho}_{person} = (fraction , submerged) cdot rho_{fl}.]

We know both the fraction submerged and the density of water, and so we can calculate the woman’s density.

Solution

Entering the known values into the expression for her density, we obtain

[overline{rho}_{person} = 0.970 cdot left(10^3 , dfrac{kg}{m^3}right) = 970 , dfrac{kg}{m^3}.]

Discussion

Her density is less than the fluid density. We expect this because she floats. Body density is one indicator of a person’s percent body fat, of interest in medical diagnostics and athletic training. (See Figure (PageIndex{7}))

There are many obvious examples of lower-density objects or substances floating in higher-density fluids—oil on water, a hot-air balloon, a bit of cork in wine, an iceberg, and hot wax in a “lava lamp,” to name a few. Less obvious examples include lava rising in a volcano and mountain ranges floating on the higher-density crust and mantle beneath them. Even seemingly solid Earth has fluid characteristics.

More Density Measurements

One of the most common techniques for determining density is shown in Figure (PageIndex{7}). An object, here a coin, is weighed in air and then weighed again while submerged in a liquid. The density of the coin, an indication of its authenticity, can be calculated if the fluid density is known. This same technique can also be used to determine the density of the fluid if the density of the coin is known. All of these calculations are based on Archimedes’ principle.

Archimedes’ principle states that the buoyant force on the object equals the weight of the fluid displaced. This, in turn, means that the object appears to weigh less when submerged; we call this measurement the object’s apparent weight. The object suffers an apparent weight loss equal to the weight of the fluid displaced. Alternatively, on balances that measure mass, the object suffers an apparent mass loss equal to the mass of fluid displaced. That is

[apparent , weight , loss = weight , of , fluid , displaced] or

[apparent , mass , loss = mass , of , fluid , displaced.]

The next example illustrates the use of this technique.

Example (PageIndex{3}): Calculating Density: Is the Coin Authentic?

The mass of an ancient Greek coin is determined in air to be 8.630 g. When the coin is submerged in water as shown in Figure (PageIndex{7}), its apparent mass is 7.800 g. Calculate its density, given that water has a density of (1.000 , g/m^3)

and that effects caused by the wire suspending the coin are negligible.

Strategy

To calculate the coin’s density, we need its mass (which is given) and its volume. The volume of the coin equals the volume of water displaced. The volume of water displaced (rho = frac{m}{V}) for (V).

Solution

The volume of water is (V_W = frac{m_W}{rho_W}) where (m_W) is the mass of water displaced. As noted, the mass of the water displaced equals the apparent mass loss, which is (m_W = 8.630 , g - 7.800 , g = 0.830 , g). Thus the volume of water is (V_W = frac{0.830 , g}{1.000 , g/cm^3} = 0.830 , cm^3). This is also the volume of the coin, since it is completely submerged. We can now find the density of the coin using the definition of density:

[rho_c = dfrac{m_c}{V_c} = dfrac{8.630 , g}{0.830 , cm^3} = 10.4 , g/cm^3.]

Discussion

You can see from [link] that this density is very close to that of pure silver, appropriate for this type of ancient coin. Most modern counterfeits are not pure silver.

This brings us back to Archimedes’ principle and how it came into being. As the story goes, the king of Syracuse gave Archimedes the task of determining whether the royal crown maker was supplying a crown of pure gold. The purity of gold is difficult to determine by color (it can be diluted with other metals and still look as yellow as pure gold), and other analytical techniques had not yet been conceived. Even ancient peoples, however, realized that the density of gold was greater than that of any other then-known substance. Archimedes purportedly agonized over his task and had his inspiration one day while at the public baths, pondering the support the water gave his body. He came up with his now-famous principle, saw how to apply it to determine density, and ran naked down the streets of Syracuse crying “Eureka!” (Greek for “I have found it”). Similar behavior can be observed in contemporary physicists from time to time!

PhET Explorations: Buoyancy

When will objects float and when will they sink? Learn how buoyancy works with blocks. Arrows show the applied forces, and you can modify the properties of the blocks and the fluid.

Summary

• Buoyant force is the net upward force on any object in any fluid. If the buoyant force is greater than the object’s weight, the object will rise to the surface and float. If the buoyant force is less than the object’s weight, the object will sink. If the buoyant force equals the object’s weight, the object will remain suspended at that depth. The buoyant force is always present whether the object floats, sinks, or is suspended in a fluid.
• Archimedes’ principle states that the buoyant force on an object equals the weight of the fluid it displaces.
• Specific gravity is the ratio of the density of an object to a fluid (usually water).

Glossary

Archimedes’ principle

the buoyant force on an object equals the weight of the fluid it displaces

buoyant force

the net upward force on any object in any fluid

specific gravity

the ratio of the density of an object to a fluid (usually water)

Contributors and Attributions

• Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).

Multiple Choice Questions

1) Water pressure is greatest against the

a. the top of a submerged object.
b. bottom of a submerged object.
c. sides of a submerged object.
d. is the same against all surfaces.

Reasoning: The bottom of the object is at a higher depth. Since pressure depends on depth, the bottom is going to be acted by a higher pressure. See page 247.

2) A dam is thicker at the bottom than at the top partly because

a. water is denser at deeper levels.
b. water pressure is greater with increasing depth.
c. surface tension exists only on the surface of liquids.
d. it looks better.

3) The pressure in a liquid depends on liquid

a. density.
b. depth.
c. both of these.
d. neither of these.

4) The pressure at the bottom of a jug filled with water does NOT depend on

a. the acceleration due to gravity.
b. water density.
c. the height of the liquid.
d. surface area of the water.

5) One liter of water has a mass of

a. 1 kilogram.
b. 1 newton.
c. both of these.
d. neither of these.

Reasoning: One liter means 1000 cm3 of water. Density of water is 1 g/cm3. Thus 1000 cm3 of waer has a mass of: (1000 cm3)x(1 g/cm3) = 1000 g = 1 kg.

6) The mass of a cubic meter of water is

a. 1 kg.
b. 10 kg.
c. 100 kg.
d. 1000 kg.

7) Pumice is a volcanic rock that floats. Its density is

a. less than the density of water.
b. equal to the density of water.
c. more than the density of water.

Reasoning: To float, the density of the object must be less than the density of the fluid.

8) The reason that buoyant force acts upward on a submerged object is that

a. it acts in a direction to oppose gravity.
b. if it acted downward, nothing would float.
c. the weight of fluid displaced reacts with an upward force.
d. upward pressure against the bottom is greater than downward pressure against the top of the submerged object.

Reasoning: This issue is discussed in the computer lesson on buoyancy.

9) A completely submerged object always displaces its own

a. volume of fluid.
b. weight of fluid.
c. density of fluid.
d. all of these.

10) A fish normally displaces its own

a. volume of water.
b. weight of water.
c. both of these.
d. neither of these.

Reasoning: See page 254 for more details. Fish have a density close to the density of water. Thus the displaced volume of water has the same weight as the fish.

11) When a boat sails from fresh water to salt water, the boat will float

a. lower in the water.
b. higher in the water.
c. at the same water level.

Reasoning: In salt water, the density of water is higher. Thus less volume of water is needed to produce the same buoyant force.See page 255.

12) What is the buoyant force acting on a 10-ton ship floating in the ocean?

a. less than 10 tons
b. 10 tons
c. more than 10 tons
d. depends on density of sea water

How To Find The Buoyant Force Of An Object Submerged In Water

13) What is the weight of water displaced by a 100-ton floating ship?

a. less than 100 tons
b. 100 tons
c. more than 100 tons
d. 100 cubic meters

14) When an object is partly or wholly immersed in a liquid, it is buoyed up

Buoyant Force Is Greatest On A Submerged Wave

a. by a force equal to its own weight.
b. by a force equal to the weight of liquid displaced.
c. and floats because of Archimedes principle.
d. but nevertheless sinks.

15) The buoyant force on an object is least when the object is

a. partly submerged.
b. submerged near the surface.
c. submerged near the bottom.
d. none of these.

16) The buoyant force on a rock is least when the rock is submerged

a. near the surface.
b. halfway to the bottom.
c. near the bottom.
d. all of the above.
e. none of the above

Buoyant Force Is Greatest On A Submerged Plane

Reasoning: The buoyant force will remain the same as long as the object is totally submerged. This is because the buoyant force is the weight of the displaced water. As the object sinks deeper, the displaced water remains the same and so is the buoyant force.

17) The reason a life jacket helps you float is

a. the jacket makes you weigh less.
b. the jacket has the same density as an average human.
c. the jacket repels water.
d. if you sink, the jacket sinks.
e. the density of both you and the jacket together is less than your density alone.

18) Blood pressure is normally greater in your

Buoyant Force Is Greatest On A Submerged Water

a. ears.
b. feet.
c. same in each.

19) Three holes are drilled in a water tower, which is filled with water, at the positions shown. Water will spurt the greatest horizontal distance from hole

a. A.
b. B.
c. C.
d. The horizontal distance will be the same for all three holes.

20) Lobsters live on the bottom of the ocean. The density of a lobster is

How Does The Buoyant Force On A Submerged Object

a. greater than the density of sea water.
b. equal to the density of sea water.
c. less than the density of sea water.

21) A lobster crawls onto a bathroom scale submerged at the bottom of the ocean. Compared to its weight above the surface, the lobster will have an apparent weight under water that is

a. greater.
b. less.
c. the same.

22) Two life preservers have identical volumes, but one is filled with styrofoam while the other is filled with sand. When the two life preservers are fully submerged, the buoyant force is greater on the one filled with

a. styrofoam.
b. sand.
c. same on each as long as their volumes are the same.

23) When an ice cube in a glass of water melts, the water level

a. rises.
b. falls.
c. remains the same.

24) A rock suspended by a weighing scale weighs 5 N out of water and 3 N when submerged in water. What is the buoyant force on the rock?

a. 8 N
b. 5 N
c. 3 N
d. 2 N

25) An egg is placed at the bottom of a bowl filled with water. Salt is slowly added to the water until the egg rises and floats. From this experiment, one concludes

a. calcium in the egg shell is repelled by sodium chloride.
b. the density of salt water exceeds the density of egg.
c. buoyant force does not always act upward.
d. salt sinks to the bottom.