Archimedes' principle and buoyant force

State Archimedes' principle in your own words: what buoyant force is, in what direction it acts, and what its magnitude is equal to.

A cube fully submerged in water displaces a volume of $0.002\ \text{m}^3$ (about $2\ \text{L}$), as shown in the figure.

Calculate the buoyant force it receives using $E = d\times V\times g$. Data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 10\ \text{m/s}^2$.

Two identical bottles filled with sand are fully submerged in water, one at a depth of $1\ \text{m}$ and the other at $5\ \text{m}$, as shown in the figure.

Does one receive more buoyant force than the other? Justify your answer by looking at the quantities on which buoyant force depends.

On the table you have a cork stopper ($d = 240\ \text{kg/m}^3$) and an iron nut ($d = 7\,870\ \text{kg/m}^3$). Without carrying out calculations, predict which one will float and which will sink in water ($d = 1\,000\ \text{kg/m}^3$) and explain the criterion you used.

Ice cubes float in a glass of water. Knowing that the density of ice is about $900\ \text{kg/m}^3$, explain why they float and state the condition that the density of an object must meet to float in a liquid.

A block of $3\ \text{kg}$ has a volume of $0.004\ \text{m}^3$ and is fully submerged in water ($d = 1\,000\ \text{kg/m}^3$), as shown in the figure. Take $g = 10\ \text{m/s}^2$.

1. a) Calculate its weight.
2. b) Calculate the upthrust it receives.
3. c) Does it float or sink? State what net force acts on it and in which direction.

A metal part of $2\ \text{kg}$ has a volume of $0.0005\ \text{m}^3$ ($0.5\ \text{L}$). Data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 10\ \text{m/s}^2$.

1. a) Calculate its weight.
2. b) Calculate the buoyant force it receives if fully submerged in water.
3. c) Does it float or sink? Also calculate its density and compare it with that of water to confirm your answer.

An object weighs $12\ \text{N}$ in air and, when fully submerged in water, the force meter reads $8\ \text{N}$. Data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 10\ \text{m/s}^2$.

1. a) What is the buoyant force it receives?
2. b) Calculate the volume of the object.
3. c) Using its true weight and volume, calculate its density in $\text{kg/m}^3$.

A stone of $5\ \text{kg}$ and a volume of $0.001\ \text{m}^3$ hangs from a force meter and is fully submerged in water, as shown in the figure. Data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 9.8\ \text{m/s}^2$.

1. a) Calculate its true weight.
2. b) Calculate the upthrust from the water.
3. c) What reading does the force meter show with the stone submerged? Explain why this "apparent weight" is smaller.

A crown of $1\ \text{kg}$ should be made of pure gold ($d_{oro} = 19\,300\ \text{kg/m}^3$). You weigh it submerged in water and the spring scale reads $9.2\ \text{N}$. Given data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 10\ \text{m/s}^2$.

1. a) What volume would the crown occupy if it were pure gold? Express it in $\text{cm}^3$.
2. b) From the spring scale reading, calculate the actual buoyant force and the actual volume of the crown.
3. c) Calculate the actual density of the crown and determine whether it is pure gold or alloyed with a less dense metal, as Archimedes suspected.

A wooden block of $6\ \text{kg}$ floats in equilibrium on the water of a swimming pool ($d = 1\,000\ \text{kg/m}^3$), as shown in the figure. Take $g = 10\ \text{m/s}^2$.

1. a) If it floats in equilibrium, what is the buoyant force it receives? Justify your answer.
2. b) Calculate the volume of water it displaces, that is, its submerged volume. Express it in $\text{m}^3$ and in liters.

An ice cube ($d_{hielo} = 900\ \text{kg/m}^3$) floats in a glass of water ($d_{agua} = 1\,000\ \text{kg/m}^3$), as shown in the figure.

1. a) In equilibrium, the weight of the ice equals the buoyant force: $d_{hielo}\times V\times g = d_{agua}\times V_{sum}\times g$. Simplify the expression and solve for the submerged fraction $\frac{V_{sum}}{V}$.
2. b) Calculate what percentage of the ice cube remains under water and what percentage sticks out above the water.
3. c) Relate the result to the phrase "this is just the tip of the iceberg".

An iceberg of $10\,000\ \text{m}^3$ floats in the sea. Data: $d_{hielo} = 900\ \text{kg/m}^3$, $d_{mar} = 1\,025\ \text{kg/m}^3$.

1. a) Calculate what fraction of its volume remains submerged (remember: in equilibrium, $\frac{V_{sum}}{V} = \frac{d_{hielo}}{d_{mar}}$).
2. b) What volume of the iceberg sticks out above the water?
3. c) Using these results, explain why icebergs are so dangerous to ships even though the visible part seems small.

A cork raft ($d = 240\ \text{kg/m}^3$) has a volume of $0.5\ \text{m}^3$ and floats in water ($d = 1\,000\ \text{kg/m}^3$). Take $g = 10\ \text{m/s}^2$.

1. a) What fraction of the raft is submerged when it floats on its own?
2. b) Calculate the maximum buoyant force that the raft can provide, i.e. when fully submerged.
3. c) Calculate the weight of the raft itself and deduce the maximum mass it can carry on top without sinking completely.

A person of $70\ \text{kg}$ floats in equilibrium, first in a swimming pool ($d = 1\,000\ \text{kg/m}^3$) and then in the Dead Sea ($d = 1\,240\ \text{kg/m}^3$).

1. a) Write down the equilibrium condition $P = E$ using the person's mass and the submerged volume, and verify that simplifying gives $V_{sum} = \frac{m}{d}$: you do not even need to know $g$.
2. b) Calculate the submerged volume in the swimming pool and express it in liters.
3. c) Calculate the submerged volume in the Dead Sea and explain, by comparing both results, why people float "like a cork" there and it is almost impossible to sink.

A steel ship weighing thousands of metric tons floats, but a small steel ball sinks to the bottom. If both are made of the same material, explain what enables the ship to float (think about its hollow shape and average density).

A hot-air balloon rises when the burner heats the air inside it. Explain what happens to the density of the air when heated and why the buoyant force from the cool outside air ends up lifting the balloon.

A submarine of $800\ \text{t}$ ($800\,000\ \text{kg}$) travels fully submerged, in equilibrium, in the sea ($d = 1\,025\ \text{kg/m}^3$), as shown in the figure. Take $g = 10\ \text{m/s}^2$.

1. a) What is the buoyant force it receives? Justify your answer using the equilibrium condition.
2. b) Calculate the volume of water it displaces, i.e. the volume of the submarine.
3. c) To surface, the submarine expels sea water from its ballast tanks and replaces it with air: explain which quantity changes, whether weight or buoyant force, and why this causes it to rise.