Concept #1: Intro to Buoyancy & Buoyant Force

Example #1: Comparing Buoyant Forces

Practice: When an object of unknown mass and volume is fully immersed in large oil (800 kg/m^{3} ) container and released from rest, it stays at rest. Calculate the density of this object.

Concept #2: Buoyancy / Three Common Cases

Practice: A block floats with 40% of its volume above water. When you place it on an unknown liquid, it floats with 30% of its volume above. What is the density of the unknown liquid?

Example #2: Is Crown Made of Gold? (Buoyancy)

Practice: An 8,000 cm^{3} block of wood is fully immersed in a deep water tank, then tied to the bottom. When the block is released and reaches equilibrium, you measure the tension on the string to be 12 N. What is the density of the wood?

Example #3: Maximum Load on Floating Board

Practice: You want to build a large storage container, with outer walls and an open top, as shown, so that you can load things into it, while it floats on fresh water, without any water getting inside. If the bottom face of the container measures 3.0 m by 8.0 m, how high should the side walls be, such that the combined mass of container and inside load is 100,000 kg?

A rectangular block of wood, 12 cm x 18 cm x 42 cm, has a specific gravity of 0.60. Determine the buoyant force that acts on the block when it is placed in a pool of fresh water.

Two wooden boxes of equal mass but different density are held beneath the surface of a large container of water. Box A has smaller average density than box B. When the boxes are released, they accelerate upward to the surface. Which box has the greater acceleration?
a. Box A
b. Box B
c. They are the same.
d. We need to know the actual densities of the boxes in order to answer the question.
e. It depends on the contents of the boxes.

A block of density ρ1 and volume V1 is submerged in a liquid of density ρL. A second block of density ρ2 and volume V2 is placed on top of the first block. The two blocks are floating in the manner shown in the figure below.
Find V2 such that the two blocks are just submerged, as shown in figure above.

Suppose that a volleyball A and a bowling ball B are completely submerged in water and have the same volume, as in the figure. (Of course, you would have to hold the volleyball beneath the water to keep it from popping up to the surface.)
Which feels a greater buoyant force?
A. bowling ball B
B. Unable to determine
C. They feel the same buoyant force.
D. volleyball A

Consider a rectangular block of ice floating in water in an open vessel. Let a denote the height of the ice block showing above the water surface, and b the height of the submerged part of the ice block.
Given a, b, the density ρwater of liquid water and the density ρice of the ice, which relationship is correct?

A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a 45 kg woman to be able to stand on it without getting her feet wet?

A diver wishes to recover a treasure chest she found at the bottom of the sea, 60 m below the surface. To do this, she inflates a plastic balloon to a radius of 40 cm with the air from her compressed air tanks. The mass of the treasure chest is 200 kg and its dimensions are 20 cm x 40 cm x 10 cm. Take the density of sea water as 1025 kg/m3. How does the acceleration change (qualitatively) as the balloon and chest float to the top? Does it get larger, smaller, or stay the same? Justify your answer with equations.

A boat's hull displaces 1600 L of water when floating. What is the weight of the boat?

An object floats on the surface of water. If the object's specific gravity is 0.7, what percentage of the object's volume is above the water-line?

A boat is floating on the surface of the water when the captain throws the anchor overboard. When the anchor is thrown out of the boat, which of the following is true?
(a) The weight of the boat decreases, so the buoyant force increases
(b) The weight of the boat decreases, so the buoyant force decreases
(c) The weight of the boat increases, so the buoyant force increases
(d) The weight of the boat increases, so the buoyant force decreases

A barge is 10.0 m wide and 60.0 m long and has vertical sides. The bottom of the hull is 1.20 m below the water surface. What is the weight of the barge and its cargo, if it is floating in fresh water?
(A) 3.53 MN
(B) 6.39 MN
(C) 7.06 MN
(D) 6.82 MN
(E) 9.54 MN

A rectangular block of wood, 12 cm x 18 cm x 42 cm, has a specific gravity of 0.60. What fraction of the block is submerged?(Express your answer to two significant figures.)

A cube of side s is completely submerged in a pool of fresh water. After drawing a free-body diagram, derive an algebraic expression for the net force on the cube.a. ΣFy = Pfluidgs - Mcubegb. ΣFy = Pfluidgs3 - Mcubegc.. ΣFy = Pcubegs3 - Mfluidgd. ΣFy = Pfluidgs3

A cube of side s is completely submerged in a pool of fresh water. What is the weight of the displaced water when the cube is submerged?a. wwater = ρfluidsgb. wwater = ρcubes3gc. wwater = ρfluids3gd. wwater = mcubeg

A woman floats in a region of the Great Salt Lake where the water is about four times saltier than the ocean and has a density of about 1130 kg/m3. The woman has a mass of 56 kg, and her density is 971 kg/m3 after exhaling as much air as possible from her lungs. Determine the percentage of her volume that will be above the waterline of the Great Salt Lake. (Express your answer to three significant figures.)

A beaker of mass 1 kg containing 2.5 kg of water rests on a scale. A 3.1 kg block of a metallic alloy of density 4900 kg/m3 is suspended from a spring scale and is submerged in the water of density 1000 kg/m3 as shown in the figure.
What does the hanging scale read? The acceleration due to gravity is 9.8 m/s 2.
A. 21.3715
B. 16.043
C. 23.0577
D. 28.3774
E. 14.3733
F. 31.6615
G. 27.8826
H. 24.18
I. 23.8994
J. 27.6033

A spar buoy consists of a circular cylinder, which floats with its axis oriented vertically. One such buoy has a radius of 1.00 m, a height of 2.00 m and weighs 40.0 kN. What portion of it is submerged when it is floating in fresh water?
[A] 1.35 m
[B] 1.30 m
[C] 1.25 m
[D] 1.20 m
[E] 1.50 m

Suppose the vessel containing the water and the ice is full: The water level is at the vessel’s rim. What happens once the ice melts?
A. The water overflows.
B. The level of the water remains at the rim.
C. There is not enough information given, the outcome is not definite.
D. The level of the water drops below the rim.

A solid aluminum cylinder of length L, radius r and density pal is suspended vertically from a thin wire and slowly lowered into a still lake. Let z denote the depth of the bottom of the cylinder. (When z < 0 no portion of the cylinder is submerged in water. Note that z is assumed here to increase downwards.) Assume that both the mass of the wire and the density of air are negligibly small. Let pω denote the density of water. What is the tension T in the wire
a) for z < 0?
b) for 0 < z < L?
c) for z > L?
d) Plot T(z).

A table-tennis ball has a diameter of 3.80 cm and average density of 0.0840 g/cm 3. What force is required to hold it completely submerged under water?

A 1.00-kg beaker containing 2.00 kg of oil (density = 916 kg/m 3) rests on a scale. A 2.00-kg block of iron is suspended from a spring scale and is completely submerged in the oil (Fig. P9.43). Find the equilibrium readings of both scales.

A geologist finds that a Moon rock whose mass is 9.30 kg has an apparent mass of 6.30 kg when submerged in water. What is the density of the rock?

Archimedes principle can be used not only to determine the specific gravity of a solid using a known liquid; the reverse can be done as well. As an example, a 3.80-kg aluminum ball has an apparent mass of 2.30 kg when submerged in a particular liquid: calculate the density of the liquid. Derive a formula for determining the density of a liquid using this procedure.

A cube of side length 13.0 cm and made of unknown material floats at the surface between water and oil. The oil has a density of 810 kg/m3.a) If the cube floats so that it is 73 % in the water and 27 % in the oil, what is the mass of the cube?b) What is the buoyant force on the cube?

A 3.60-kg piece of wood (exttip{SG}{SG}SG = 0.51) floats on water. What minimum mass of lead, hung from the wood by a string, will cause it to sink?

Oil having a density of 930 kg/m3 floats on water. A rectangular block of wood 4.00 cm high and with a density of 960 kg/m3 floats partly in the oil and partly in the water. The oil completely covers the block. How far below the interface between the two liquids is the bottom of the block?

A 7.4 kg solid sphere made of metal whose density is 2300 kg/m3, is suspended by a cord. When the sphere is immersed in a liquid of unknown density, the tension in the cord is 18 N. The density of the liquid is closest to:
A. 1700 kg/m3
B. 1600 kg/m3
C. 1500 kg/m3
D. 1400 kg/m3
E. 1300 kg/m3

A diver wishes to recover a treasure chest she found at the bottom of the sea, 60 m below the surface. To do this, she inflates a plastic balloon to a radius of 40 cm with the air from her compressed air tanks. The mass of the treasure chest is 200 kg and its dimensions are 20 cm x 40 cm x 10 cm. Take the density of sea water as 1025 kg/m3. The diver attaches the inflated balloon to the treasure chest using a rope. Calculate the initial acceleration of the balloon and the chest. Ignore the mass of the plastic of the balloon and of the rope connecting the balloon and the chest.

A rock with density 2300 kg/m3 is suspended from the lower end of a light string. When the rock is in air, the tension in the string is 43.0 N. What is the tension in the string when the
rock is totally immersed in a liquid with density 750 kg/m3?

A hydrometer consists of a spherical bulb and a cylindrical stem with a cross-section area of 0.400 cm2. The total volume of bulb and stem is 13.2 cm3. When immersed in water, the hydrometer floats with 8.00 cm of the stem above the water surface. When immersed in an organic fluid, 3.20 cm of the stem is above the surface. (Note: This illustrates the precision of such a hydrometer. Relatively small density differences give rise to relatively large differences in hydrometer readings.) Find the density of the organic fluid.

A toy floats in a swimming pool. The buoyant force exerted on the toy depends on the volume ofa. water in the pool.b. the pool.c. the toy under water.d. the toy above water.e. none of the above choices.

An object floats in water with 5/8 of its volume submerged. The ratio of the density of the object to that of water isa. 8/5.b. 5/8.c. 1/2.d. 2/1.e. 3/8.

An ice cube floats in a glass of water. As the ice melts, what happens to the water level?a. It rises.b. It remains the samec. It falls by an amount that cannot be determined from the information given.d. It falls by an amount proportional to the volume of the ice cube.e. It falls by an amount proportional to the volume of the ice cube that was initially above the water line.

An iron anchor with mass 35.5 kg and density 7860 kg/m3 lies on the deck of a small barge that has vertical sides and floats in a freshwater river. The area of the bottom of the barge is 8.05 m2 . The anchor is thrown overboard but is suspended above the bottom of the river by a rope; the mass and volume of the rope are small enough to ignore. After the anchor is overboard and the barge has finally stopped bobbing up and down, has the barge risen or sunk down in the water? By what vertical distance?

A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a 72.0 kg woman to be able to stand on it without getting her feet wet?

A hollow, plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.660 m3 and the tension in the cord is 2120 N.a) Calculate the buoyant force exerted by the water on the sphere.b) What is the mass of the sphere?c) The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged?