Archimedes' principle and upthrust

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

When you get into a swimming pool you feel lighter, as if the water were supporting you. What force appears when you submerge yourself? What exerts it and in which direction does it push?

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 upthrust 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 upthrust than the other? Justify your answer by looking at the quantities on which upthrust depends.

A log of $0.01\ \text{m}^3$ is fully submerged, first in a lake and then in the sea. Take $g = 10\ \text{m/s}^2$.

1. a) Calculate the upthrust it receives in the lake ($d = 1\,000\ \text{kg/m}^3$).
2. b) Calculate the upthrust it receives in the sea ($d = 1\,025\ \text{kg/m}^3$).
3. c) Using these results, explain why swimmers float "higher" in the sea than in a swimming pool.

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 a body 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.

An egg of $60\ \text{g}$ has a volume of $55\ \text{cm}^3$. Take $g = 10\ \text{m/s}^2$.

1. a) Calculate its weight in newtons (take care with the units).
2. b) Calculate the upthrust it receives when fully submerged in tap water ($d = 1\,000\ \text{kg/m}^3$) and deduce whether it floats or sinks.
3. c) On dissolving a large amount of salt, the water reaches a density of $1\,200\ \text{kg/m}^3$: calculate the new upthrust and explain why the egg now floats.

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 upthrust 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 litres.

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.

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 upthrust 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 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 newton meter 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 newton meter reading, calculate the actual upthrust 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.

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 upthrust: $d_{hielo}\times V\times g = d_{agua}\times V_{sum}\times g$. 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 shipping 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 upthrust 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 litres.
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 tonnes 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 upthrust 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 upthrust 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 upthrust, and why this causes it to rise.

A hot-air balloon has a volume of $2\,000\ \text{m}^3$, as shown in the figure. The outside air has a density of $1.2\ \text{kg/m}^3$ and the hot air inside has a density of $1.0\ \text{kg/m}^3$. Take $g = 10\ \text{m/s}^2$.

1. a) Calculate the upthrust exerted by the outside air on the balloon.
2. b) Calculate the weight of the hot air filling the balloon.
3. c) The difference between the two is the force available to lift the envelope, basket, and passengers: what is the maximum combined mass they can have for the balloon to ascend?

A party balloon contains $5\ \text{L}$ of helium ($0.005\ \text{m}^3$). Given data: $d_{aire} = 1.2\ \text{kg/m}^3$, $d_{helio} = 0.18\ \text{kg/m}^3$; take $g = 10\ \text{m/s}^2$.

1. a) Calculate the upthrust exerted by the air on the balloon.
2. b) Calculate the weight of the helium filling it.
3. c) The balloon's envelope has a mass of $3\ \text{g}$: what maximum extra mass (ribbon, decorations) can it carry to keep ascending?
4. d) Explain why the balloon stops rising once it reaches a certain height (think about how the density of air changes with altitude).

A yacht of $4\,000\ \text{kg}$ floats in the sea ($d = 1\,025\ \text{kg/m}^3$). Take $g = 10\ \text{m/s}^2$.

1. a) Calculate its weight and reason what upthrust it receives when floating in equilibrium.
2. b) Calculate the submerged volume of the hull.
3. c) The yacht enters a freshwater river ($d = 1\,000\ \text{kg/m}^3$): calculate the new submerged volume and explain whether the boat floats higher or lower in the water.