Physics midterm problem: cable-supported beam

Physics 205A Midterm 2, Spring Semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 8.34

[20 points.] A uniform beam of weight 560 N is held horizontal by a pivot and a cable attached as shown at right. The cable is attached halfway between the center of gravity (CG) of the beam, and the pivot point. Determine the x- and y- components of the force exerted by the pivot point on the end of the beam. Show your work and explain your reasoning.

Solution and grading rubric:

• p = 20/20:
  Correct. Sets up Newton's first law for forces along the x- and y- directions, as well as for torques about an axis that does not include the physical pivot attached to the wall. Solves these three equations for three unknowns T, F_x, and F_y. May directly solve for F_x = 408 N and F_y = 560 N by placing the axis where the cable attaches to the beam. (The tension in the cable is 1,190 N.)
  
• r = 16/20:
  Nearly correct, but includes minor math errors. At least F_x or F_y is correct from applying Newton's first law. May only have correct value for T, with some methodical (but problematic) approach to finding F_x and/or F_y. Or has complete and correct set of Newton's first law equations, but algebra is problematic.
  
• t = 12/20:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. At least some serious attempt at applying Newton's first law, and care in resolving components, measuring lever arms, and calculating torques.
  
• v = 8/20:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. Mainly resolving magnitude of weight into 20-70-90 degree triangle components.
  
• 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:
Sections 30880, 30881
p: 0 students
r: 9 students
t: 19 students
v: 9 students
x: 2 students
y: 0 students
z: 1 student

A sample "r" response (from student 1830), solving successfully only for T and for F_x:

Another sample "r" response (from student 2679), applying Newton's laws to all three degrees of freedom, but bogged down in the ensuing algebra: