Multiple choice buoyancy problem

In summary: I am guessing you did not include the summary because it was quite long and detailed. It is always a good idea to provide a brief summary for readers who may not have the time or interest to read the entire conversation. In summary, the conversation discusses a problem involving a wooden cube floating in water with a block placed on top of it. After the block is removed, there is a difference in height between the cube's upper face and the water surface. The task is to determine the length of the cube's edge. Through the use of equations and considering the forces at play, it is determined that the correct answer is (B) 10 cm and the density of the wood is 800 kg/m^3.
  • #1
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Homework Statement


A wooden cube is floating in water. The upper face of the cube meets the surface of a pool of water when a block with mass 0.200 kg is placed on top of the cube. When you remove the block, there's a 2 cm difference in height between the upper face and the water surface.

564180098c263a8d204f9a427bebd39e.png


What's the length of the cube's edge?

Possible answers are:
(A) 9 cm
(B) 10 cm (this is the correct answer)
(C) 18 cm
(D) this problem is unsolvable without knowing the density of the wood used

Homework Equations


$$F=mg \\
B=\rho g V$$ The length of a cube's edge is the cube root of its volume

The Attempt at a Solution


Notice that the volume of the cube is equal to the volume of the water displaced in the first picture.
Since forces are at rest, they will sum to zero Newton. (Newton's second law)
$$\rho_{water}V_{cube}g=g(0.200\mathrm{\ kg}+\rho_{wood}V_{cube})\\
\Leftrightarrow V_{cube}=\frac{0.200 \mathrm{\ kg}}{\rho_{water}-\rho_{wood}}$$ Since the density of wood is unknown, I would suppose (D) is the correct answer.
 
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  • #2
You haven't stated for what question the choices A-D are the answers.
 
  • #3
Sorry, I added it.
 
  • #4
I checked and (B) is right as far as I can see.

You are only using one of the two situations in the figure to derive an equation. You may obtain a second equation by using the other situation that is displayed. From this system of equations you will see that the density of wood is not needed. (In these equations, it may also help to express the volume of the cube directly in terms of the length of its edges.)
 
  • #5
Further to Krylov's post:
Because you obtained an equation in which the cube's dimensions depend on the wood's density, you assumed answer D. But it turns out that there is enough information here to find the wood's density.
 
  • #6
Thank you all for your insightful replies. This has tremendously helped me develop my intuition in buoyancy problems.

In the right figure, notice that the forces are at rest and thus, by Newton's second law: $$\sum F_y = 0$$ The only considerable forces are gravity and the buoyant force. $$\Leftrightarrow m_{cube}=\rho_{wood}s^3=\rho_{water}(s^3-0.02\mathrm{\ m} \cdot s^2)$$ By solving the system of equations: $$s^3=\frac{0.200 \mathrm{\ kg}}{\rho_{water}-\rho_{wood}}\\\rho_{wood}s^3=\rho_{water}(s^3-0.02\mathrm{\ m} \cdot s^2)$$ the desired result follows: $$s=0.10 \mathrm{\ m} \land \rho_{wood}=800 \mathrm{\ \frac{kg}{m^3}}$$
Therefore, (B) is the correct answer.
 
  • #7
Well done, and thank you for letting us know that you solved it, that is always nice.
 

Related to Multiple choice buoyancy problem

1. What is buoyancy and how does it relate to multiple choice problems?

Buoyancy is the upward force exerted by a fluid on a submerged object. In multiple choice buoyancy problems, the concept of buoyancy is often used to determine the correct answer choice.

2. How do I approach solving a multiple choice buoyancy problem?

First, identify the given information and what is being asked in the problem. Then, apply the principles of buoyancy, such as Archimedes' principle, to determine the correct answer choice.

3. What are some common pitfalls to avoid when solving a multiple choice buoyancy problem?

One common mistake is confusing weight with mass. Remember that buoyancy is determined by the weight of the fluid displaced, not the mass of the object. Also, pay attention to the units given in the problem and make sure to convert them if necessary.

4. Are there any helpful tips for solving multiple choice buoyancy problems?

Yes! It can be helpful to draw a diagram of the problem and label all given information. Also, make sure to read each answer choice carefully and eliminate any that do not make sense or do not follow the principles of buoyancy.

5. Can I use a calculator to solve multiple choice buoyancy problems?

In most cases, yes. However, make sure to check the problem for any specific instructions or limitations on the use of a calculator. Also, be mindful of rounding errors and make sure to use the appropriate number of significant figures in your final answer.

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