Exercises: Archimedes' principle: upthrust

State Archimedes' principle in your own words: what buoyant force is, in what direction 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 buoyant force it receives using $E = d\times V\times g$. Data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 10\ \text{m/s}^2$.

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.

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).

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.

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 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 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 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 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.

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

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

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

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

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

1. a) Calculate its weight and reason what buoyant force 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.

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 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.

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 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 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 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 buoyant force 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 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.

Unit challenge. How does a bathyscaphe control its buoyancy to rescue objects from the seabed? Design a technical report solving this problem using a homemade prototype.

1. Starting data: Check the current atmospheric pressure ($p_0$) of your area on the National Weather Service website. Build a Cartesian diver by placing a pen cap ballasted with modeling clay inside a $1.5\text{ L}$ plastic bottle filled with water.

2. Data collection and graph: Squeeze the bottle to keep the diver in equilibrium at $5$ different depths. Measure each depth $h$ with a ruler ($\pm 0.1\text{ cm}$). Create a data table recording $h$ and calculating the pressure due to the atmosphere and the water column, excluding the extra pressure added by squeezing ($p = p_0 + \rho \times g \times h$, with $\rho = 1000\text{ kg/m}^3$ and $g = 9.8\text{ m/s}^2$). Plot $p$ versus $h$ displaying the uncertainties.

3. Reasoned proposal: Write a brief report explaining how the applied pressure is transmitted through the water (Pascal's principle), how the air is compressed, altering the buoyant force (Archimedes' principle), and the influence of hydrostatic pressure on the descent.

Evaluation criteria:

• Rigor in taking home data, estimating uncertainties, and graphing.

• Correct application of Pascal's and Archimedes' principles, and the calculation of hydrostatic pressure.

• Quality of the report's conclusions and appropriate use of scientific language.