Ten sense-making tasks on density, pressure, and buoyancy. Explain your reasoning for every answer. The explanation counts as much as the answer.
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Six containers, all open to the air, are filled with water. The dot in each container marks a point in the water. The depth $h$ of each point below the water's surface is given. (Container D is a bottle with a narrow neck. Point F is on the side wall of its tank.)
Rank the points by the absolute pressure at each point, from greatest to least. If any are equal, show the tie. If the ranking cannot be determined, say so.
Greatest 1 ____ 2 ____ 3 ____ 4 ____ 5 ____ 6 ____ Least
Explain your reasoning. Would your ranking change if you ranked gauge pressure instead?
Back to topA rock hangs from a string, fully under water. It is lowered from Position 1, just below the surface, to Position 2, near the bottom of the tank. Water is (nearly) incompressible. Three students discuss the buoyant force on the rock.
Ava: "The buoyant force is bigger at Position 2. Pressure increases with depth, so the water pushes harder on the rock down there."
Ben: "The buoyant force is the same at both positions. It equals the weight of the water the rock displaces, and the rock displaces the same volume at both depths."
Cam: "The buoyant force is the same, but only because the rock is light. A heavy rock of the same size would feel a bigger buoyant force."
Which student(s), if any, do you agree with? Explain what is right or wrong about each statement. Ava's statement about pressure is true, so explain why it doesn't lead to her conclusion.
Back to topA wooden block floats at rest in a pool of water, half submerged. A student draws this free-body diagram for the block:
Student's explanation: "The buoyant force has to be bigger than the weight, or the block would sink. That's what floating means."
What, if anything, is wrong with this free-body diagram and explanation? If something is wrong, explain the error and describe the correct diagram. If nothing is wrong, explain why it is correct.
Back to topA wooden block floats at rest in fresh water. For each change, choose what happens to the quantity listed, compared with the original situation:
A. increases B. decreases C. stays the same D. cannot be determined
| Change | Choice |
|---|---|
| (a) The block is moved to salt water (denser). Buoyant force on the block: | |
| (b) The block is moved to salt water. Volume of the block below the surface: | |
| (c) A coin is placed on top of the block (it still floats). Buoyant force: | |
| (d) A hand pushes the block completely under water and holds it there. Buoyant force: | |
| (e) The block is cut in half and one half floats by itself. Fraction of that half that is submerged: | |
| (f) The pool of water and the block are taken to the Moon. Fraction of the block submerged: | |
| (g) The block is replaced by a different block of the same size. Buoyant force: |
Explain the rule you used. Which choices did you make by using "floating means $F_B = F_g$"?
Back to topTwo cubes have the same volume, $1.0 \times 10^{-3}\ \text{m}^3$. One is wood (density $600\ \text{kg/m}^3$) and floats. The other is aluminum (density $2700\ \text{kg/m}^3$) and rests on the bottom of the tank.
For each quantity, state whether it is greater for the wood, greater for the aluminum, or the same for both. If it's different, give the ratio.
| Quantity | Answer |
|---|---|
| (a) Weight of the cube | |
| (b) Volume of water displaced | |
| (c) Buoyant force on the cube | |
| (d) Net force on the cube |
Explain each answer, and find the actual values. Many students say "the wood floats, so it has the bigger buoyant force." Respond to that idea.
Back to topAn aluminum block (volume $1.0 \times 10^{-3}\ \text{m}^3$) hangs from a spring scale, which reads 27 N. Below it, a beaker of water sits on a scale that reads 50 N. The block is then lowered until it is completely under water, without touching the beaker.
The equation below is Newton's 2nd law for an object at rest in some physical situation:
$$(1000\ \text{kg/m}^3)(10\ \text{m/s}^2)(2.0 \times 10^{-3}\ \text{m}^3) - (1.2\ \text{kg})(10\ \text{m/s}^2) - T = 0$$
The graph shows the absolute pressure vs. depth in an open container of an unknown Liquid X.
A hydraulic lift has a small piston of area $0.010\ \text{m}^2$ and a large piston of area $0.50\ \text{m}^2$. A car weighing 15 000 N sits on the large piston. Ignore the masses of the pistons, and assume both pistons start at the same height. Find (a) the force on the small piston needed to hold the car up, and (b) how far the car rises when the small piston is pushed down 0.50 m.
Student solution:
(a) Pascal's principle: $\dfrac{F_1}{A_1} = \dfrac{F_2}{A_2}$, so $F_1 = (15\,000\ \text{N})\dfrac{0.010\ \text{m}^2}{0.50\ \text{m}^2} = 300\ \text{N}$
(b) "The pistons are connected by the same fluid, so they move the same distance." The car rises $0.50\ \text{m}$.
Identify the error in the solution. Explain the physics of why it's an error, then find the correct answer. Use energy to check your answer.
Back to topA log with a volume of $0.50\ \text{m}^3$ and a density of $600\ \text{kg/m}^3$ floats at rest in a lake.
Morgan writes: $F_B = \rho_{water} V g = (1000\ \text{kg/m}^3)(0.50\ \text{m}^3)(10\ \text{m/s}^2) = 5000\ \text{N}$, so the buoyant force on the log is 5000 N.