Physics midterm question: floating ebony-balsa wood cubes

Physics 205A Midterm 2, fall semester 2018
Cuesta College, San Luis Obispo, CA

A wooden cube is made by gluing ebony (denser) and balsa (less dense) pieces together. Both pieces have the same volume. The total density of the cube is less than that of water. The cube is carefully placed into water such that it floats "top-heavy" (ebony on top of balsa). The cube is then turned over such that it floats "bottom-heavy" (balsa on top of ebony). Discuss which orientation will float higher (or if there is tie), and why. (Ignore any water that may soak into the wood pieces, and the thin layer of glue between the two wood pieces.) Explain your reasoning using the properties of densities, volumes, forces, Newton's laws, Archimedes' principle (buoyant forces), and free-body diagrams.

Solution and grading rubric:

• p:
  Correct. Recognizes that:
  1. each block ("bottom-heavy" or "top-heavy") has two vertical forces acting on it:

     Weight force of Earth on block (downwards, magnitude w = m⋅g),
     Buoyant force of water on block (upwards, magnitude F_B = ρ_water⋅g⋅V_sub);

     and
  2. because each block ("bottom-heavy" or "top-heavy") is stationary in the vertical direction, then its downwards weight force must have the same magnitude as its upwards buoyant force, due to Newton's first law; and
  3. since the mass of each block ("bottom-heavy" or "top-heavy") does not matter which type of wood is stacked above the other, the magnitude of the weight is the same, making the magnitudes of the buoyant forces the same; such that
  4. the amount submerged volume underwater for both blocks must be the same.
  
  Thus the buoyant forces on each block ("bottom-heavy" or "top-heavy") are equal, and thus the amount of volume submerged for either block must be the same.
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. May somehow claim that the cube will float differently when "bottom-heavy" or "top-heavy," or does not explicitly conclude that the cube will float at the same water level whether "bottom-heavy" or "top-heavy."
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. At least recognizes that the weight force on the block is unchanged whether "bottom-heavy" or "top-heavy," but somehow has different buoyant forces acting (thus Newton's first law would not apply to at least one of the blocks); or has different weights and different buoyant forces acting on the blocks, but for each block these forces are balanced via Newton's first law.
• v:
  imited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some constructive attempt at relating the buoyant force to the density of the fluid and volume displaced (Archimedes' principle) and/or Newton's first law.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Appeals to some other properties of fluids and densities other than Archimedes' principle and Newton's laws.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.
  

Grading distribution:
Sections 70854, 70855
p: 21 students
r: 17 students
t: 13 students
v: 6 students
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 0921):