Physics midterm question: partially versus fully submerged buoys

Physics 205A Midterm 2, fall semester 2012
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 9.31

Two buoys (same volumes, same densities) used for generating electrical power from wave motion[*] are each anchored to the ocean floor with cables, and are either partially or fully submerged in sea water. Discuss why the tension in the cable attached to the partially submerged buoy is less than the tension in the cable attached to the fully submerged buoy. Explain your reasoning using the properties of densities, volumes, forces, Newton's laws, and Archimedes' principle.

[*] www2.teknat.uu.se/forskning/uu/beskrivning.php?id=31&vetenskapsid=31&lang=en.

Solution and grading rubric:
  • p:
    Correct. Identifies three forces acting on each buoy (tension downwards, weight downwards, and buoyant force upwards). Each buoy is stationary (in translational equilibrium), such that by Newton's first law, the buoyant force is equal in magnitude to the sum of the tension and weight forces. The two buoys have the same weight, due to their same volumes and densities (thus same masses). The partially submerged buoy has a smaller upwards bouyant force acting on it, as it displaces less volume underwater. Thus the downwards tension force on the partially submerged buoy (which is the difference between the buoyant force and weight) must be less than the downwards tension force on the fully submerged buoy.
  • r:
    As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. Identifies weight forces acting on both buoys, but does not explicitly discuss how/why these weights are identical.
  • t:
    Nearly correct, but argument has conceptual errors, or is incomplete. Omits weight forces, effectively discussing a case where the buoys are extremely lightweight compared to the magnitude of the tension and buoyant forces. Still (a) applies Newton's first law to those two forces, and (b) understands how buoyant force depends on submerged volume.
  • v:
    Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. As (t), but does not explicitly explain how bouyant force depends on submerged volume.
  • x:
    Implementation/application of ideas, but credit given for effort rather than merit.
  • y:
    Irrelevant discussion/effectively blank.
  • z:
Grading distribution:
Sections 70854, 70855
Exam code: midterm02gL0u
p: 14 students
r: 5 students
t: 22 students
v: 13 students
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 9494):

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