Physics midterm problem: package transported uphill

Physics 5A Midterm 1, Spring Semester 2008
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 1/e, Chapters 1-5 Review and Synthesis Review Exercise 20

[20 points.] A Physics 5A student is going to drive a 15,000 kg flat-bed truck at constant speed up a hill that makes a certain angle with respect to the horizontal direction. A 40.0 kg package sits in the back of the truck. The coefficient of static friction between the package and the truck is 0.310. What is the maximum angle of the hill that the truck can travel up at constant speed without the package falling off of the back? Neglect air resistance. Show your work and explain your reasoning.

Solution and grading rubric:

• p = 20/20:
  Correct. Draws free-body diagram for _only_ the package with tilted axes, with the weight force broken up into w_x = m*g*cos(theta) and w_y = m*g*sin(theta) components. Newton's first law is applied in _both_ x- and y- directions. For the y-direction, Newton's first law gives N = w_y = m*g*cos(theta), and for the y-direction, Newton's first law gives f_s = w_x, or mu_s*N = m*g*sin(theta). Solving for and substituting in for N allows theta to be solved for, resulting in theta = Arctan(mu_s).
  
• r = 16/20:
  Nearly correct, but includes minor math errors. Typically uses N = m*g, but otherwise has f_s = m*g*sin(theta), but otherwise shows complete systematic FBD -> Newton's laws -> algebra approach.
  
• t = 12/20:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. Free-body diagrams, application of Newton's laws, and algebra are problematic, but still demonstrates some application of this systematic approach.
  
• v = 8/20:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. Some attempt at a free-body diagram, applying Newton's laws, or algebra.
  
• 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: 7 students
r: 6 students
t: 5 students
v: 19 students
x: 6 students
y: 0 students
z: 0 students

A sample of a "p" response (from student 2012) is shown below:

A sample of a "r" response (from student 7137) is shown below, where N = m*g, but otherwise all else is correct: