Question

connecting buoyancy with displacement

metal | density (g/cm³)
--- | ---
aluminum | 2.64
cork | 0.24
iron | 7.50
lead | 11.34
wax | 0.72

which material will displace a volume of water? iron, lead, and aluminum
which material will displace a volume of water less than its own volume? dropdown with options: all of them, cork and wax, iron, lead, and aluminum, none of them
which material will displace a volume of water equal to its own volume?
which material will displace a volume of water greater than its own volume?

Explanation:

Step1: Recall Archimedes' Principle

The density of water is \(1\space g/cm^3\). For a material, if its density (\( ho_{material}\)) is greater than water's density (\( ho_{water} = 1\space g/cm^3\)), it sinks, and the volume of water displaced is equal to the volume of the material. If \( ho_{material}< ho_{water}\), it floats, and the volume of water displaced is less than its own volume (since it displaces water equal to its weight, not volume).

Step2: Analyze each material's density

- Aluminum: \( ho = 2.64\space g/cm^3>1\space g/cm^3\)
- Iron: \( ho = 7.50\space g/cm^3>1\space g/cm^3\)
- Lead: \( ho = 11.34\space g/cm^3>1\space g/cm^3\)
- Cork: \( ho = 0.24\space g/cm^3<1\space g/cm^3\)
- Wax: \( ho = 0.72\space g/cm^3<1\space g/cm^3\)

For the question "Which material will displace a volume of water less than its own volume?", we need materials with \( ho_{material}< ho_{water}\). But the options given are about the ones with \( ho_{material}> ho_{water}\) (iron, lead, aluminum) or others. Wait, no—wait, the question is "Which material will displace a volume of water less than its own volume?" Wait, no, the dropdown is for "Which material will displace a volume of water less than its own volume?" Wait, no, let's re - check. Wait, the first question was "Which material will displace a volume of water equal to its own volume?" (sinking materials, density >1). Then the second question (the dropdown) is "Which material will displace a volume of water less than its own volume?" Wait, no, the user's question is about the dropdown for "Which material will displace a volume of water less than its own volume?" Wait, no, the options are: all of them, cork and wax, iron, lead, and aluminum, none of them.

Wait, no—wait, when a material floats (\( ho < 1\)), the volume of water displaced is less than its own volume (because \(F_b= ho_{water}gV_{displaced}=mg= ho_{material}gV_{material}\), so \(V_{displaced}=\frac{ ho_{material}}{ ho_{water}}V_{material}\). If \( ho_{material}< ho_{water}\), then \(V_{displaced}<V_{material}\)). When it sinks (\( ho > 1\)), \(V_{displaced} = V_{material}\) (since it's fully submerged).

So for "displace less than its own volume", we need \( ho < 1\) (cork and wax). But the dropdown options: the option "iron, lead, and aluminum" have \( ho>1\), so they displace equal to their volume. The option "cork and wax" have \( ho < 1\), so they displace less. But the question in the dropdown—wait, maybe the user's question is about the second question: "Which material will displace a volume of water less than its own volume?". Wait, no, the first question was "Which material will displace a volume of water equal to its own volume?" (answer: iron, lead, aluminum). Then the next question: "Which material will displace a volume of water less than its own volume?" The materials with \( ho < 1\) are cork and wax. But the options given in the dropdown are: all of them, cork and wax, iron, lead, and aluminum, none of them.

But wait, the user's problem is about the dropdown for "Which material will displace a volume of water less than its own volume?". Wait, no, maybe I misread. Wait, the first question was "Which material will displace a volume of water equal to its own volume?" (iron, lead, aluminum—correct, since their density >1, they sink, displace equal volume). Then the next question: "Which material will displace a volume of water less than its own volume?" The answer should be cork and wax (since their density <1, they float, displace less volume). But the dropdown options include "cork and wax" and "iron, lead, and aluminum" etc.

But maybe the user's question is about the second dropdown (the one with the options: all of them, cork and wax, iron, lead, and aluminum, none of them) for the question "Which material will displace a volume of water less than its own volume?".

Wait, let's re - express:

- For a material, if \( ho_{material}> ho_{water}\) (water is \(1\space g/cm^3\)): it sinks, \(V_{displaced}=V_{material}\) (displace equal volume).
- If \( ho_{material}< ho_{water}\): it floats, \(V_{displaced}<V_{material}\) (displace less volume).

So materials with \( ho < 1\): cork (0.24), wax (0.72). Materials with \( ho>1\): aluminum (2.64), iron (7.50), lead (11.34).

So the question "Which material will displace a volume of water less than its own volume?" should be answered with "cork and wax". But the options include "cork and wax" as an option.

But wait, the user's problem is about the dropdown. Let's assume the question is "Which material will displace a volume of water less than its own volume?" and the options are as given. Then the correct option is "cork and wax". But wait, no—wait, the first question was "Which material will displace a volume of water equal to its own volume?" (answer: iron, lead, aluminum). Then the next question (the dropdown) is "Which material will displace a volume of water less than its own volume?"—so the answer is "cork and wax".

But maybe the user's question is about the dropdown for "Which material will displace a volume of water less than its own volume?".

Answer:

The correct option is "cork and wax" (from the dropdown options: "cork and wax").