Physics midterm question: tension forces on static beams

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

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 8.32

[10 points.] A uniform beam attached to a pivot is supported by a horizontal cable attached to its end, or a 45° cable attached to its midpoint. Which cable (if any) has a greater tension? Explain your answer using Newton's laws and properties of torque.

Solution and grading rubric:

• p = 10/10:
  Correct. Shows quantitatively from Newton's first law that the magnitude of the cw torques from either cable are identical (as they support the ccw weight torques of identical beams), such that F_1*r_perp1 and F_2*r_perp2, but since r_perp1 is larger (sqrt(2)*L) vs. r_perp2 (L/2), then F_1 < F_2. (Alternately may set F_perp1*L and F_perp2*(L/2) equal to each other, with F_perp1 = F_1*sin(45 degrees) and F_perp2 = F_2, such that (0.707)*F_1*L = (0.5)*F_2*L, proving that F_1 must be smaller than F_2.
  
• r = 8/10:
  As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. States that cable 2 has more tension as it has a smaller lever arm, but does not explicitly show that r_perp2 is smaller than r_perp1 (i.e., no clear distinction between r and r_perp).
  
• t = 6/10:
  Nearly correct, but argument has conceptual errors, or is incomplete. Garbled attempt at implementing Newton's first law for rotations and the F_perp*r or F*r_perp torque definition. May indicate that distance from CG as important.
  
• v = 4/10:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. May indicate that the length of the cable is important.
  
• x = 2/10:
  Implementation/application of ideas, but credit given for effort rather than merit.
  
• y = 1/10:
  Irrelevant discussion/effectively blank.
  
• z = 0/10:
  Blank.
  

Grading distribution:
Sections 70854, 70855, 72177
p: 5 students
r: 20 students
t: 32 students
v: 3 students
x: 1 student
y: 0 students
z: 0 students

A sample of a "p" response (from student 3026):