Physics midterm question: weights of comparable floaters

Physics 205A Midterm 2, fall semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 9.27

Two solid objects of different sizes both float with 75% of their volumes below water. It is unknown whether these objects are made of the same material. Which object (if any) has the greater weight? 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. Both objects are stationary, so Newton's first law applies to each object, where the downwards weight force is balanced by the upwards buoyant force. Since the larger object has more submerged volume, then it will have a greater buoyant force, and thus a greater weight force.
  
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. At least recognizes that the bouyant force on object 1 is greater, but does not carry through Newton's first law to determine that the weight of object 1 is greater; or similar argument as (p) with a minor contradiction or omission in reasoning.
  
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. At least recognizes three of the following as relevant factors in discussion: F_B = ρ_water·g·V, density, specific gravity, and Newton's first law; but typically claims that both objects have the same weight.
  
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. As (t), but only two of relevant factors are recognized and discussed in a systematic manner.
  
• x:
  Implementation/application of ideas, but credit given for effort rather than merit.
  
• y:
  Irrelevant discussion/effectively blank.
  
• z:
  Blank.
  

Grading distribution:
Sections 70854, 70855
p: 29 students
r: 5 students
t: 13 students
v: 2 students
x: 0 students
y: 0 students
z: 0 students

A sample of a "p" response (from student 3276):
A sample of a "t" response (from student 8128), arguing that the objects must have the same weight because they have the same density: