Physics final exam problem: pushed, stacked boxes

Physics 5A Final Exam, fall semester 2007
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 1/e, Problem 4.52

[20 points.] A horizontal force of 18.0 N is required to keep a 2.00 kg box moving across the floor at constant speed. A box of unknown mass is then stacked on top of the 2.00 kg box. A horizontal force of 56.0 N, applied to the 2.00 kg box, is required to keep both stacked boxes moving across the floor at constant speed. What is the mass of the unknown box? Show your work and explain your reasoning.

Solution and grading rubric:

• p = 20/20:
  Correct. Sets up Newton's first law for each of two situations (2.00 kg box sliding at constant speed; 2.00 kg + m₂ boxes sliding at constant speed). Reduces these two equations to solve for the two unknowns μ_k and m₂ (or eliminates μ_k to find only m₂). May instead appeal directly to a ratio of 2.00 kg to (2.00 kg + m₂), in relation to the ratio of kinetic friction forces in order to find the unknown mass.
  
• r = 16/20:
  Nearly correct, but includes minor math errors. As (p), but solves for the combined mass of (2.00 kg + m₂), and not for the unknown mass m₂ itself.
  
• t = 12/20:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. As (p), but uses Newton's second law, and eliminates acceleration common to both second law equations to find unknown mass.
  
• v = 8/20:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. May use work-energy conservation concepts.
  
• x = 4/20:
  Implementation of ideas, but credit given for effort rather than merit.
  
• y = 2/20:
  Irrelevant discussion/effectively blank.
  
• z = 0/20:
  Blank.
  

Grading distribution:
p: 18 students
r: 8 students
t: 2 students
v: 6 students
x: 0 students
y: 0 students
z: 1 student

A sample of a "p" response (from student 3153):
Another sample of a "p" response (from student 8181), who sets up a free-body diagram and applies Newton's laws, and then applies proportional reasoning:
One last "p" response (from student 1587), appealing directly to the ratio of forces and masses: