Physics midterm question: anchored foam block with rising water level

Physics 205A Midterm 2, fall semester 2017
Cuesta College, San Luis Obispo, CA

A foam block is anchored by a string to the bottom of a container of water. As more water is poured into this container, discuss why the tension in the string increases. (Ignore any stretching of the length of the string, such that the block remains at the same level while the water level rises.) 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. the block has three vertical forces acting on it:

     Weight force of Earth on block (downwards, magnitude w = m⋅g),
     Tension force of container on block (downwards, magnitude T),
     Buoyant force of water on block (upwards, magnitude F_B = ρ_water⋅g⋅V_sub = ρ_water⋅g⋅(0.50⋅V_block); and

  2. because the block is stationary in the vertical direction, the sum of the two downwards forces is equal and opposite in direction to the upwards force due to Newton's first law; and
  3. as more water is poured into the container, the volume of the block that is submerged increases, increasing the amount of upwards buoyant force of water on the block from F_B = ρ_water⋅g⋅(0.50⋅V_block) to ρ_water⋅g⋅(0.75⋅V_block); such that
  4. the downwards tension force of the container on the block must also increase (as the weight force remains constant), in order for the two downwards forces to still balance out the increase in the upwards force on the block.
  May either draw a free-body diagram, and/or discuss these forces and Newton's laws in words.
• r:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. Excludes weight force, or adds extraneous normal force, but still recognizes why buoyant force increases, thus increasing the tension force.
• t:
  Nearly correct, but argument has conceptual errors, or is incomplete. Excludes weight force and adds extraneous normal force.
• 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
Exam code: midterm02bu2Z
p: 30 students
r: 16 students
t: 4 students
v: 0 students
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 1902):
Another sample "p" response (from student 7164):