Physics midterm question: comparing tensions in suspended beams

Physics 205A Midterm 2, fall semester 2015
Cuesta College, San Luis Obispo, CA

Two horizontal uniform beams are each mounted on pivots and suspended by 45° angle cables. The short beam is one-half the length of the long beam, but the short and long beams have the same mass. Discuss why the tension in the short beam cable is equal to the tension in the long beam cable. Explain your reasoning using diagram(s) with locations of forces and perpendicular lever arms, the properties of torques, and Newton's laws.

Solution and grading rubric:
  • p:
    Correct. Complete free-body diagrams with forces and perpendicular lever arms, and discusses/demonstrates:
    1. Newton's first law applies to each case, such that the cw torque of the cable equals the ccw torque of the weight;
    2. for either case, the perpendicular lever arm for the weight force is (L/2) and the perpendicular lever arm for the cable is L·sin(45°), where L is the length of either beam;
    3. from equating torques, for either beam, the tension force T = w/(2·sin(45°)) independent of the length of the beam, and thus is the same for either beam.
    May also instead argue (2)-(3) by recognizing similar triangles, such that tension force T must be the same for either beam given that their weights are the same.
  • r:
    As (p), but argument indirectly, weakly, or only by definition supports the statement to be proven, or has minor inconsistencies or loopholes. May have incorrect or missing cable lever arm.
  • t:
    Nearly correct, but argument has conceptual errors, or is incomplete. Typically equates similiar lever arms (short ℓw = long ℓw, short ℓT = long ℓT), when it is actually the ratios of these lever arms that are equal (short ℓw/short ℓT) = (long ℓw/long ℓT).
  • v:
    Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Some garbled attempt at applying Newton's first law to torques, forces, and perpendicular lever arms.
  • x:
    Implementation of ideas, but credit given for effort rather than merit. Approach other than that of applying Newton's first law to torques, forces, and perpendicular lever arms.
  • y:
    Irrelevant discussion/effectively blank.
  • z:
Grading distribution:
Sections 70854, 70855, 73320
Exam code: midterm02h4W6
p: 13 students
r: 15 students
t: 21 students
v: 20 students
x: 1 student
y: 0 students
z: 0 students

A sample "p" response (from student 2934), comparing relative ratios of lever arms:

A sample "p" response (from student 1793), putting numbers in to find absolute equivalence of cable tension forces:

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